Chapter 8
Atmospheric Transport Schemes: Desirable
Properties and a Semi-Lagrangian View on
Finite-Volume Discretizations
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8.2 The continuous equation
8.2.1 Representation of mass in atmospheric models
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.+&- ,+(0*+ -($$ l 4 ,& -#?#12 +(,#& 45
MOK
X(7+#,@*1 *, (35
!& $&38* (19 &. ,"* 0&1,#17#,9 *A7(,#&1$ 2#8*1 (%&8* ,"* "&; !*3/ v -7$, %*
2#8*15 !"* 0&1,#17#,9 *A7(,#&1 .&+ (#+ =[5O> #$ 0&7)3*/ ;#," ,"* -&-*1,7- *A7(4
,#&1$ (1/ ,"*+-&/91(-#0 *A7(,#&1$5 T*10* ,"* ,"*+-&/91(-#0 8(+#(%3*$ (1/ &,"*+
)+&21&$,#0 8(+#(%3*$ .**/ %(0' &1 ,"* 8*3&0#,9 !*3/ ;"#0"6 #1 ,7+16 .**/$ %(0' &1
,"* $&37,#&1 ,& ,"* 0&1,#17#,9 *A7(,#&15 G, .&33&;$ ,"(, ,"* 0&1,#17#,9 *A7(,#&1 .&+ (#+
0(11&, %* $&38*/ #1 #$&3(,#&1 (1/ &1* -7$, &%*9 ,"* -(?#-7- (33&;(%3* ,#-*4$,*)
+*$,+#0,#&1$ #-)&$*/ %9 ,"* .($,*$, ;(8*$ #1 ,"* $9$,*- Z =$** 0"(),*+ M (1/ a>5 !"*
)($$#8* ,+(0*+ ,+(1$)&+, *A7(,#&1 =[5ML> &+ =[5MM> 0(1 %* $&38*/ #1 #$&3(,#&1 2#8*1
)+*$0+#%*/ ;#1/$ (1/ (#+ /*1$#,#*$6 (1/ #$ ,"*+*.&+* 1&, $7$0*),#%3* ,& ,"* $,+#0,*+
,#-*4$,*) +*$,+#0,#&1$ #-)&$*/ %9 ,"* .($,*$, ;(8*$ #1 ,"* $9$,*- %7, `&139Q ,& ,"*
3*$$ +*$,+#0,#8* (/8*0,#8* 8*3&0#,#*$ P 5 T*10*6 #. .&+ $,(%#3#,9 +*3(,#8*39 $"&+, ,#-*4$,*)$
-7$, %* 7$*/ .&+ ,"* 0&1,#17#,9 *A7(,#&1 .&+ (#+6 &1* /&*$ 1&, 1*0*$$(+#39 1**/ ,& 7$*
$"&+, ,#-*4$,*)$ .&+ ,"* ,+(0*+$ =(, 3*($, 1&, .&+ $,(%#3#,9 +*($&1$>5 !"(, #$6 &1* 0(1
$&38* ,"* ,+(0*+ ,+(1$)&+, *A7(,#&1$ ;#," ,#-*4$,*)$ 3&12*+ ,"(1 ;"(, (+* (33&;*/
.&+ $,(%#3#,9 #1 =[5O>5 !"#$ ,*0"1#A7* #$ +*.*++*/ ,& ($ ;5>K2J2*$)E6 ,"(, #$6 -73,#)3*
0903*$ &. /91(-#0$ =(#+ 0&1,#17#,9 *A7(,#&1> (+* )*+.&+-*/ ;#,"#1 &1* ,#-*4$,*) &.
,"* ,+(0*+$5 G1 /&#12 $& 0(+* -7$, %* ,('*1 ,& +*,(#1 ,"* 0&1$#$,*109 %*,;**1 ,+(04
*+$ (1/ (#+5 I&+ *?(-)3*6 #. 4 = M ,"*1 =[5ML> +*/70*$ ,& =[5O> (1/ (//#,#&1(3 0(+*
-7$, %* ,('*1 ,& *1$7+* 0&1$#$,*109 %*,;**1 ,"*$* *A7(,#&1$ #1 ,"* /#$0+*,#@(,#&15
S)*0#!0 *?(-)3*$ (1/ /*,(#3$ &1 $7%40903#12 (+* 2#8*1 3(,*+ =$*0,#&1 [5h5K>5 I#+$,6
3*, 7$ 0&1$#/*+ #-)&+,(1, /*$#21 &%]*0,#8*$ .&+ ,+(0*+ ,+(1$)&+, $0"*-*$ #1,*1/*/ .&+
(,-&$)"*+#0 ())3#0(,#&1$5
8.3 Desirable properties
C"*1 /*8*3&)#12 ( 1*; ,+(1$)&+, =&+ (19 &,"*+> (32&+#,"- &1* #$ 7$7(339 $,+#8#12
.&+ ( $0"*-* ,"(, *1$7+*$ $#-73(,#&1 #-:"2$?J5 G1 &,"*+ ;&+/$6 ( 17-*+#0(3 -*,"&/
$"&73/ %* /*$#21*/ $& ,"(, $#-73(,#&1$ 7$#12 #, (+* ($ ,+7,".73 ($ )&$$#%3*5 G1 -(,"4
*-(,#0(3 3#,*+(,7+* $#-73(,#&1 #-:"2$?J #$ &.,*1 $91&19-&7$ ;#," (007+(09 ;"#0" #$
($$&0#(,*/ ;#," ,"* (%$&37,* ,+7,"5 D&18*+2*10*6 ,+710(,#&1 *++&+ (1/ *++&+ 1&+-$
(+* (33 ($$&0#(,*/ ;#," A7(1,#,(,#8* -*($7+*$ &. 0&1.&+-#,9 ,& ,"* ,+7,"5 G1 -&$,
+*(3#$,#0 (,-&$)"*+#0 -&/*3 $*,,#12$6 "&;*8*+6 ,"* ,+7," #$ 71'1&;1 #1 (1 (%$&37,*
$*1$* =,"* *?(0, $&37,#&1 #$ 1&, '1&;1>5 I&+ #1$,(10*6 #1 -&$, (,-&$)"*+#0 ())3#0(4
,#&1$ (1 #10+*($* #1 +*$&37,#&1 ;#33 &.,*1 +*$&38* !1*+ $0(3*$ (1/ 1*; )"*1&-*1(
())*(+ -('#12 #, )+&%3*-(,#0 ,& /*!1* 0&18*+2*10* #1 ( $,+#0, -(,"*-(,#0(3 $*1$*5
://#12 ,& ,"* 0&-)3*?#,9 #$ ,"* .(0, ,"(, ,"* $9$,*- #$ 0"(&,#0 (1/ ,"*+*.&+* 1&, /*4
,*+-#1#$,#0 %*9&1/ ML /(9$ &+ $& =X&+*1@6 MO[K>6 $& (19 (,,*-), ,& ($$*$$ (%$&37,*
(007+(09 #1 $#-73(,#&1$ %*9&1/ ,"* )+*/#0,(%#3#,9 3#-#, -7$, %* %($*/ &1 $,(,#$,#0(3
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[ I#1#,*4E&37-* !+(1$)&+, S0"*-*$
MOZ
G1 (336 $#-73(,#&1 8*+(0#,9 #1 (1 (,-&$)"*+#0 -&/*3#12 0&1,*?, #$ -&+* ,"(1 (04
07+(09 #1 ( $,+#0, -(,"*-(,#0(3 $*1$*5 <*+"()$ %*0(7$* ,"*+* #$ 3#,,3* A7(1,#,(,#8*
'1&;3*/2* &. ,"* ,+7* $&37,#&1 ( 3&, &. *-)"($#$ #$ )3(0*/ &1 )"9$#0(3 )+&)*+,#*$ &.
,"* $&37,#&1 -*,"&/5 I&+ *?(-)3*6 ;* /& '1&; ,"(, ,"* 17-*+#0(3 $&37,#&1 $"&73/
#/*(339 &%*9 /#$0+*,#@*/ *A7#8(3*1,$ &. )+&)*+,#*$ ;* 0(1 /*+#8* .+&- ,"* 0&1,#14
7&7$ $*, &. *A7(,#&1$ $70" ($ 0&1$*+8(,#&1 R &. -($$ (1/ "#2"*+ -&-*1,$6 $"()*4
)+*$*+8(,#&1 =#1037/#12 -&1&,&1#0#,96 )&$#,#8#,9 (1/ 1&14&$0#33(,&+9 )+&)*+,9>6 0&+4
+*3(,#&1 )+*$*+8(,#&16 (1/ $& &15 :3$&6 $7%42+#/4$0(3* )(+(-*,*+#@(,#&1$ 7$7(339 +*4
A7#+* )"9$#0(3 +*(3#@(%3* (,-&$)"*+#0 $,(,*$ .+&- ,"* +*$&38*/ $0(3* /91(-#0$5 I+&( 0&-)7,(,#&1(3 )&#1, &. 8#*; )+&)*+,#*$ $70" ($ )(+(33*3 *.!0#*1096 2*&-*,+#0 "*?4
#%#3#,96 *,05 (+* (3$& 8*+9 #-)&+,(1, )+&)*+,#*$ &. ,"* !1(3 17-*+#0(3 (32&+#,"-5
C"(, .&33&;$ #$ ( 3#$, &. /*$#+(%3* )+&)*+,#*$ .&+ ,+(0*+ ,+(1$)&+, $0"*-*$ ,"(, (+*
(33 =()(+, .+&- ,"* )+&)*+,#*$ +*3(,*/ ,& *.!0#*109> *$$*1,#(3 #12+*/#*1,$ &. $#-73(,#&1
8*+(0#,9^
8.3.1 Accuracy (error norms)
:007+(09 /*$0+#%*$ ,"* /*2+** &. 03&$*1*$$ &. ,"* $#-73(,*/ =17-*+#0(339 0&-)7,*/>
$&37,#&1 ,& #,$ ,+7* =*?(0,> $,(,* $)*0#!*/ #1 ,*+-$ &. *++&+ 1&+-$ =17-*+#0 8(37*$>5
!"* *++&+ -*($7+*$ 0(1 *#,"*+ %* ($$*$$*/ (, ( !?*/ +*$&37,#&1 =(%$&37,* *++&+> &+ ($
( .710,#&1 &. +*$&37,#&1 =0&18*+2*10*>5 I&+ 3#1*(+#@*/ *A7(,#&1$ (1/ ())+&?#-(,#&1$
( )+&?9 .&+ 0&18*+2*10* 0(1 %* $&72", %9 0&-)7,#12 ,"* .&+-(3 &+/*+ &. (007+(09
&. ,"* 17-*+#0(3 -*,"&/ ,"+&72" !(93&+ $*+#*$ *?)(1$#&1$5 N&,* ,"(, .&+-(3 &+/*+
&. (007+(09 /&*$ 1&, 1*0*$$(+#39 27(+(1,** (007+(,* $&37,#&1$ .&+ /#$,+#%7,#&1$V"&;$
;#," 1*(+ /#$0&1,#17#,#*$ =$"&0'$ (1/ .+&1,$> 1&+ /&*$ #, 27(+(1,** (007+(09 (, ( )(+4
,#073(+ +*$&37,#&15 I&+ -(19 23&%(3 ;*(,"*+ (1/ 03#-(,* ())3#0(,#&1$ (%$&37,* (0074
+(09 (, ( )(+,#073(+ +(12* &. +*$&37,#&1$ #$ )*+"()$ -&+* #-)&+,(1, ,"(1 "#2"4&+/*+
0&18*+2*10* +(,*$5 \*3&; #$ ( 3#$, &. $&-* #/*(3#@*/ ,*$, 0($*$ 7$*/ ,& A7(1,#,(,#8*39
($$*$$ $#-73(,#&1 8*+(0#,9^
8.3.1.1 Linear test cases
F++&+ 1&+-$ (+* ;*33 /*!1*/ ;"*1 ,"* *?(0, $&37,#&1 #$ '1&;1 ;"#0" #$ 7$7(339 &139
,"* 0($* .&+ 3#1*(+ )+&%3*-$5 D&--&139 7$*/ 3#1*(+ ,*$, 0($*$6 ;"*+* ,"* (1(39,#0(3
$&37,#&1 #$ '1&;1 (, (33 ,#-*$ ?6 0(1 %* /#8#/*/ #1,& ,;& 0(,*2&+#*$^ !+(1$3(,#&1(3
(1/ /*.&+-(,#&1(35 T*+* ;* .&07$ &139 &1 23&%(3 ,*$, 0($*$ #1 $)"*+#0(3 2*&-*,+95
U&$, ,*$, 0($*$ (+* .&+-73(,*/ ;#," 1&14/#8*+2*1, "&; !*3/$ .&+ ;"#0" ,"* (/4
8*0,#8* .&+- &. ,"* 0&1,#17#,9 *A7(,#&1 .&+ ( ,+(0*+ =[5MM>6 ,"(, 7$*$ -#?#12 +(,#& 4 ($
,"* )+&21&$,#0 *A7(,#&16 #$ *A7#8(3*1, ,& ,"* "7?4.&+- 8*+$#&1 =[5ML> %($*/ &1 ,+(0*+
-($$ l 45 !"(, #$6 4 &+ l 4 #$ $*, *A7(3 ,& ,"* $(-* $)(,#(3 /#$,+#%7,#&1 (1/ ,"* -&/*3*+
R
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0"(),*+ MM
MOP
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(a)
(b)
Fig. 8.1 F?(0, $&37,#&1$ .&+ ,"* =(> $&3#/ %&/9 (/8*0,#&1 &. ( 0&$#1* %*33 ,*$, 0($* (, ? = L =0*1,*+
&. )3&,> (1/ ? = PP" =.&+ ( `"&; +&,(,#&1 (123*Q &. PR◦ >6 (1/ =%> ,"* $,(,#0 8&+,*? ,*$, 0($* (, /(9 a5
#$ #-)3#0#,39 ($$7-#12 ,"(, l #$ &1* *8*+9;"*+* (1/ $#10* ,"* "&; #$ 1&14/#8*+2*1,
l ;#33 +*-(#1 &1* ,"+&72"4&7, ,"* $#-73(,#&1 (, 3*($, #1 ,"* (1(39,#0(3 0($*5 T*10*
,"* -&/*3*+ #$ 1&, .&+0*/ ,& /#$,#127#$" %*,;**1 ,+(0*+ -($$ l 4 (1/ -#?#12 +(,#& 45
T&;*8*+6 .&+ ( /#8*+2*1,V0&18*+2*1, "&; &139 4 #$ 0&1$,(1, (3&12 )(+0*3 ,+(]*0,&+#*$
;"*+*($ ,+(0*+ -($$ l 4 ;#33 #10+*($*V/*0+*($* #1 (+*($ &. 0&18*+2*10*V/#8*+2*10*5
I&+ ( .733*+ /#$07$$#&1 $** N(#+ (1/ X(7+#,@*1 =KLML>5
Translational. <+&%(%39 ,"* -&$, 0&--&139 7$*/ #/*(3#@*/ ,*$, 0($* #1 ,"* -*4
,*&+&3&2#0(3 3#,*+(,7+* #$ ,"* $&3#/ %&/9 +&,(,#&1 &. ( 0&$#1* %*33 =I#27+* [5M(> =*5256
,*$, 0($* &1* &. ,"* ;#/*39 7$*/ ,;&4/#-*1$#&1(3 ,*$, $7#,* &. C#33#(-$&1 *, (3 MOOK
.&+ ,"* $"(33&;4;(,*+ *A7(,#&1$>5 !"* *?(0, $&37,#&1 #$ $#-)39 ,"* ,+(1$3(,#&1 &. ,"*
#1#,#(3 0&1/#,#&1 (1/ $,(1/(+/ *++&+ 1&+-$ 0(1 %* 0&-)7,*/ (, *8*+9 ,#-*4$,*)5 !"#$
,;&4/#-*1$#&1(3 ,*$, 0($* "($ %**1 *?,*1/*/ ,& ,"+** /#-*1$#&1$ #1 J(%3&1&;$'#
*, (3 =KLML>5 :1&,"*+ ,"+** /#-*1$#&1(3 ,*$, 0($* &1 ,"* $)"*+* ;"*+* ,"* (1(39,#0
$&37,#&1 #$ '1&;1 ;($ )+&)&$*/ %9 f7%&8 *, (3 =MOOO>5
I&+ 0&18*+2*10* $,7/#*$ 7$*/ ,& ($$*$$ ,"* .&+-(3 &+/*+ &. (007+(09 &. ( $0"*-*
,"* ,+(1$3(,*/ /#$,+#%7,#&1 $"&73/ %* $7.!0#*1,39 $-&&,"5 I&+ *?(-)3*6 ,"* 0&$#1*
%*33 /#$,+#%7,#&1 -(9 ())*(+ $-&&," %7, #, #$ &139 / M (, ,"* %($* &. ,"* %*335 D&14
$*A7*1,396 $0"*-*$ ,"(, (+* "#2"4&+/*+ (007+(,* #1 ,*+-$ &. ( !(93&+ S*+#*$ (1(39$#$
-(9 1&, $"&; ,"#$ "#2"4&+/*+ .&+-(3 0&18*+2*10* +(,* ;"*1 7$#12 ,"* 0&$#1* %*33
#1#,#(3 0&1/#,#&15 !& ($$*$$ `#/*(3Q 0&18*+2*10* +(,*$ #, #$ (/8#$*/ ,& 7$* / ' .7104
,#&1$ $70" ($ W(7$$#(1 $7+.(0*$ =X*89 *, (36 KLLh>5
Deformational. !"* ,+(1$3(,#&1(3 ,*$, 0($* /*$0+#%*/ (%&8* "($ ( 3(+2* /*2+**
&. $9--*,+9 (1/ )*+"()$ #$ 1&, 0"(33*12#12 *1&72" ,& ,"&+&72"39 ,*$, ( 17-*+#4
0(3 (32&+#,"-5 Y*(3 ;&+3/ "&;$ (3$& "(8* /*.&+-(,#&1(36 0&18*+2*1,V/#8*+2*1, (1/
+&,(,#&1(3 0&-)&1*1,$ ,"(, /*.&+-6 *?)(1/ (1/ +&,(,* ,"* #1#,#(3 /#$,+#%7,#&15 : )&)4
73(+ )7+*39 /*.&+-(,#&1(3 ,*$, 0($* =1&14/#8*+2*1,> #$ ,"* 0903&2*1*$#$ ,*$, 0($* #14
,+&/70*/ #1 -*,*&+&3&29 %9 B&$;*33 =MO[P> (1/ 7$*/ ($ ( ,*$, 0($* .&+ ,+(1$)&+,
$0"*-*$ %9 17-*+#&7$ (7,"&+$ =*5256 Y(1#0#s0 MOOK6 N(#+ (1/ U(0"*1"(7*+ KLLK>5
!"* *?(0, $&37,#&1 (, /(9 a #$ $"&;1 &1 I#25 [5M% a5 :$ 0(1 %**1 $**1 #1 ,"* I#24
a
,"* /#-*1$#&1(3#@(,#&1 &. ,"* 8&+,*? )+&%3*- 7$*/ "*+* .&33&;$ N(#+ (1/ J(%3&1&;$'# =KLL[>
[ I#1#,*4E&37-* !+(1$)&+, S0"*-*$
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=3*.,o /(9 L>6 "(3. =-#//3*o /(9 a> (1/ &1* =+#2",o /(9 MK> +*8&37,#&15 !"* 0&1,#1*1,$ (+* $"&;1 .&+
+*.*+*10* )7+)&$*$ &1395
7+* ,"* 8&+,*? `07+3$ 7)Q (1/ 2*1*+(,*$ 3&12 ,"#1 !3(-*1,$ #1 ,"* )+&0*$$5 !"*$*6 #1
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:1&,"*+ 7$*.73 ())3#0(,#&1 &. ,"#$ ,*$, 0($* #$ ,& 7$* #,$ 8*3&0#,9 !*3/ %7, #1$,*(/
&. ,+(1$)&+,#12 (1 #1#,#(3 0&1/#,#&1 $70" ($ $"&;1 &1 I#25 [5K #1 ,"* 0903&2*1*$#$
,*$, 0($*6 #1$,*(/ ,+(1$)&+, ( 0&1$,(1, -($$ !*3/ l = l L 5 S#10* ,"* "&; #$ 1&14
/#8*+2*1, (19 17-*+#0(3 $0"*-* $"&73/ #/*(339 )+*$*+8* ( 0&1$,(1, -($$ !*3/5 :3$&
,"* $&3#/4%&/9 +&,(,#&1 "&; !*3/ #$ 1&14/#8*+2*1, (1/ $& .&+ ,"#$ ;#1/ !*3/ ( 0&14
$,(1, -($$4!*3/ $"&73/ +*-(#1 0&1$,(1, ,"+&72"&7, ,"* $#-73(,#&15 T&;*8*+6 ,"*
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,*$, $#10* #, #$ /*.&+-(,#&1(3 =713*$$ ,"* $,+*(- .710,#&1 .&+ ,"* 8*3&0#,9 !*3/ #$ 7$*/
,& -('*$ $7+* ,"(, ,"* /#8*+2*10* ,"(, ,"* $0"*-* `$**$Q #$ @*+&>5 S&-* $0"*-*$
-#2", )+*$*+8* ( 0&1$,(1, -($$ !*3/ .&+ $&3#/ %&/9 (/8*0,#&1 %7, .(#3 ,& )+*$*+8*
( 0&1$,(1, -($$ !*3/ .&+ ,"* /*.&+-(,#&1(3 ;#1/ !*3/5 _1.&+,71(,*39 +*$73,$ .+&$70" ,*$,$ (+* +(+*39 )+*$*1,*/ #1 ,"* 3#,*+(,7+*5
Translational and deformational. :3,"&72" ,"* #/*(3#@*/ 0903&2*1*$#$ ,*$,
0($* /*$0+#%*/ (%&8* #$ 0"(33*12#12 #, 3(0'$ ( ,+(1$3(,#&1(3 0&-)&1*1,5 N(#+ (1/
J(%3&1&;$'# =KLL[> 0&-%#1*/ ,"* 0903&2*1*$#$ ;#1/ !*3/ ;#," ,"* $&3#/ %&/9 (/4
8*0,#&1 ;#1/ !*3/ &1 ,"* $)"*+* ;"#0" -('*$ 7) ,"* `-&8#12 8&+,#0*$Q ,*$, 0($*5
G1$,*(/ &. ( $,(,#&1(+9 `07+3 7)Q &. ,"* 8&+,*?6 #, #$ ,+(1$)&+,*/ ($ ( $&3#/ %&/9 ($ #,
/*.&+-$ =I#25 [5K>5 H%8#&7$39 $70" ( ,*$, 0($* #$ -&+* 0"(33*12#12 (1/ -#2", ,"*+*4
.&+* %* -&+* 7$*.73 ,& /#$0+#-#1(,* %*,;**1 $0"*-*$ ,"(1 $#-)3*+ ,*$, 0($*$5 I&+
*?(-)3*6 #1 ,"* #/*(3#@*/ ,*$,$ &. ,"* !1#,*48&37-* ,+(1$)&+, $0"*-* #1 X(7+#,@*1
*, (3 =KLML> #, ;($ .&71/ ,"(, ,"* -&8#12 8&+,#0*$ ,*$, 0($* ;($ -&+* /#$0+#-#1(,4
#12 ,"(1 ,"* )7+* ,+(1$3(,#&1(3 (1/ $,(,#&1(+9 0903&2*1*$#$ ,*$, 0($*$ =(, 3*($, ;"*1
())3#*/ (1/ 0&-)(+*/ ,& ,"* <7,-(1 (1/ X#1 =KLLh> $0"*-*>5
Y*0*1,396 N(#+ (1/ X(7+#,@*1 =KLML> *?,*1/*/ X*E*A7*Q$ ,*$, 0($* =X*E*A7*6
MOOa> ,& ( 03($$ &. ,*$, 0($*$ &1 ,"* $)"*+*5 _13#'* (33 ,"* ,*$, 0($*$ 0&1$#/*+*/ $&
.(+ ,"* ;#1/ !*3/$ #1 ,"#$ ,*$, 0($* (+* ,#-* 8(+9#125 G1 ,"*$* 0($*$ ,"* ;#1/ !*3/$ (+*
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#,$ #1#,#(3 )&$#,#&1 (1/ $"()*5 T*10* ,"* (1(39,#0 $&37,#&1 #$ '1&;1 (.,*+ &1* )*+#&/
MOa
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Fig. 8.3 !"* +*0*1,39 )+&)&$*/ ,*$, 0($* %9 N(#+ (1/ X(7+#,@*1 =KLML>5 =(> !"* #1#,#(3 ;#1/ !*3/
(1/ =%> #1#,#(3 0&1/#,#&1 (1/ (1(39,#0(3 $&37,#&1 (.,*+ &1* )*+#&/5 =0> (1/ =/> $"&; ( 17-*+#0(339
0&-)7,*/ $&37,#&1 (.,*+ "(3. (1/ .733 )*+#&/6 +*$)*0,#8*395 !"* 2+#/4$0(3* 1&#$* #1 =0> (1/ =/> (+*
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The ETEX forecast experiment. !"* ;&+$, 1703*(+ )&;*+ )3(1, /#$($,*+ #1 "#$4
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!"* 3#1*(+ /#..7$#&1 (1/ /#$)*+$#&1 )+&)*+,#*$ &. ( 3#1*(+#@*/ $0"*-* 0(1 %* ($$*$$*/
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(1(39$#$>5 G, #$ ( $,(1/(+/ (1(39,#0 (1(39$#$ ,*0"1#A7* (1/ #$ /*$0+#%*/ #1 -(19 ,*?,4
%&&'$ #1 ,"* 0&1,*?, &. 2+#/4)&#1, -*,"&/$ =$**6 *5256 B7++(1 MOOO6 T(3,#1*+ (1/
C#33#(-$ MO[L> (1/ #1 ,"* 0&1,*?, &. !1#,*48&37-* -*,"&/$ #1 X(7+#,@*1 =KLLh>5
!"* (1(39$#$ 0&1$#$,$ &. ($$*$$#12 (1(39,#0(339 "&; ( $#123* I&7+#*+ -&/* #$ /(-)*/
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)"($* )+&)*+,#*$ &. ( $0"*-* (+* ($$*$$*/ %9 $7%$,#,7,#12 ,"* $&37,#&1 =[5MZ> #1,&
,"* .&+*0($, .&+-73( .&+ ,"* !1#,*48&37-* $0"*-* #1 A7*$,#&16 (1/ $7%$*A7*1,39 (14
(39@#12 ,"* 0&-)3*? (-)3#!0(,#&1 .(0,&+ K 5 !"* $,(%#3#,9 &. ( 17-*+#0(3 -*,"&/
#$ 2&8*+1*/ %9 ,"* -&/737$ &. ,"* 0&-)3*? (-)3#!0(,#&1 .(0,&+6 ,"(, #$6 ( )(+,#074
3(+ ;(8* ;#," ;(8*17-%*+ g #$ $,(%3* #. |K | ≤ M5 I&33&;#12 \(,*$ (1/ U0B&1(3/
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(-)3#!0(,#&1 .(0,&+ ($
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;* ;+#,* `#1 ,"*&+9Q $#10* #. ( ,+(1$)&+, $0"*-* #$ 1&, $,+#0,39 -&1&,&1* 3&0(3 `(/ "&0Q (/]7$,-*1,$
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[ I#1#,*4E&37-* !+(1$)&+, S0"*-*$
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;"*+* t ∗ #$ ,"* 17-*+#0(3 .+*A7*1095 B*!1* ,"* +*3(,#8* .+*A7*109 ($ ( = t ∗ /t
;"*+* t #$ ,"* *?(0, .+*A7*109 2#8*1 %9 g 5 L (1/ 5L #$ ,"* 0&1$,(1, ;#1/5 G. ( > M ,"*
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,&+$ $70" ($ 3#-#,*+$ (1/ !3,*+$ 0(11&, %* #1037/*/ #1 ,"* %($#0 (1(39$#$ ($ ;*33 ($
1&143#1*(+ "&;$5 _$7(339 ,"* $)7+#&7$ 17-*+#0(3 /#..7$#&1 (1/ /#$)*+$#&1 /*0+*($*
+()#/39 ;#," ,"* .&+-(3 &+/*+ &. ,"* $0"*-*5 S& *(0" $0"*-* )+&%(%39 "($ (1 &)4
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&+/*+ &. ,"* $0"*-* $#-)39 /&*$ 1&, )(9 &.. #1 ,*+-$ &. 3#1*(+ /#..7$#&1 (1/ /#$)*+4
$#&1 )+&)*+,#*$5 I&+ *?(-)3*6 X*&1(+/ =MOOM> (+27*/ ,"(, ,"* +*/70,#&1 #1 /#..7$#&1
%*0&-*$ ,+#8#(3 $&&1 (.,*+ ,"* &+/*+ #$ 3(+2*+ ,"(1 ,"#+/5
8.3.4 Tracer and air mass consistency
!+(0*+ (1/ (#+ -($$ 0&1$#$,*109 #$ ( $,+#0,*+ 0&10*), ,"(1 $#-)3* -($$ 0&1$*+8(,#&1
&. ,"* #1/#8#/7(3 A7(1,#,#*$5 G, %($#0(339 $,(,*$ ,"(, ,"* /#$0+*,#@*/ ,+(0*+ ,+(1$)&+,
$0"*-* $"&73/ +*/70* ,& ,"* /#$0+*,#@*/ 0&1,#17#,9 *A7(,#&1 .&+ (#+ ;"*1 4 = M
($ #$ ,"* 0($* .&+ ,"* 0&1,#17&7$ *A7(,#&1$^ =[5ML> +*/70*$ ,& =[5O> ;"*1 $*,,#12
4 = M5 !+(0*+4(#+ -($$ 0&1$#$,*109 0(16 .&+ *?(-)3*6 %* 8#&3(,*/ #. 7$#12 ( 17-*+#0(3
-*,"&/ .&+ ,+(0*+ ,+(1$)&+, ,"(, #$ /#..*+*1, .+&- ,"* $0"*-* 7$*/ .&+ )+*/#0,#12 ,"*
*8&37,#&1 &. ,"* (#+ /*1$#,9 ML 5 !& (0"#*8* ( "#2" 3*8*3 &. 0&1$#$,*109 #, #$ 7$7(339
1*0*$$(+9 ,"(, ,"* $(-* 17-*+#0(3 (32&+#,"- #$ 7$*/ .&+ ,"* /91(-#0$ ($ ;*33 ($
.&+ ,+(0*+ ,+(1$)&+,5 I&+ -&+* /#$07$$#&1 $** U(0"*1"(7*+ *, (3 =KLLO>o X** *, (3
=KLLP>o Je&0'*3 *, (3 =KLLM>o f"(12 *, (3 =KLL[>5
8.3.5 Divergence preservation
!"* ,+(1$)&+, &)*+(,&+ $"&73/ 1&, %* ( $)7+#&7$ $&7+0* &. /#8*+2*10*5 _$7(339 ,"#$
)+&)*+,9 #$ /#$07$$*/ ;#,"#1 ,"* 0&1,*?, &. 1&14/#8*+2*1, "&; !*3/$5 I&+ *?(-4
)3*6 ( 0&1$,(1, #1#,#(3 -($$ /#$,+#%7,#&1 $"&73/ +*-(#1 0&1$,(1, (, (33 ,#-* #1 (
1&14/#8*+2*1, "&; =@:-;-:#"?$3) 3< 6";;K23);?")2J>5 !"* $7%]*0, "($ +*0*#8*/ 0&14
$#/*+(%3* (,,*1,#&1 #1 ,"* -(21*,&"9/+&/91(-#0$ 3#,*+(,7+* $#10* ,"* -(21*,#0 "7?
/*1$#,9 #$ 1&14/#8*+2*1, (1/ ,"* 17-*+#0(3 $0"*-* $"&73/ #/*(339 +*,(#1 ,"(, )+&)4
ML !"#$ /#$07$$#&1 ())3#*$ ,& &13#1* ())3#0(,#&1$ ;"*+* ,+(0*+ ,+(1$)&+, #$ )*+.&+-*/ #1 0&1]710,#&1
;#," ,"* 2&8*+1#12 "7#/ (1/ ,"*+-&/91(-#0 *A7(,#&1$5 : $#-#3(+ #10&1$#$,*109 ())*(+$ ;"*1
/+#8#12 ,"* ,+(0*+ ,+(1$)&+, *A7(,#&1 #1 (1 &."#1* -&/* =)+*$0+#%*/ ;#1/$ (1/ -($$ !*3/$ .+&+*(1(39$#$6 &%$*+8(,#&1$ &+ ( /#..*+*1, -&/*3> #1 ;"#0" 0($* ,"* ,+(0*+ ,+(1$)&+, $0"*-* ;#," 4 = M
;#33 1&, *A7(3 ,"* )+*$0+#%*/ -($$4!*3/ 713*$$ (/4"&0 !?*+$ (+* ())3#*/5
KLK
X(7+#,@*1 *, (35
*+,9 =*5256 :+,*%+(1, (1/ !&++#3"&1 =KLL[> (1/ +*.*+*10*$ ,"*+*#1>5 : )+*+*A7#$#,*
.&+ 0&1,+&33#12 $)7+#&7$ 2*1*+(,#&1 &. /#8*+2*10* #$ )+*$*+8(,#&1 &. -($$40&1$,(109
($ .&+-73(,*/ (%&8* =$** ,*$, 0($* $722*$,#&1 #1 $*0,#&1 [5Z5M5M> .&+ 1&14/#8*+2*1,
"&;$5
N&,* ,"(, ,"* )+*$*+8(,#&1 &. 0&1$,(1, -#?#12 +(,#& =(1/ 1&, 0&1$,(1, ,+(0*+ -($$
!*3/> #$ ,+#8#(3 #1 -&$, 0($*$5 G. ,"* (/8*0,#8* .&+- &. ,"* (/8*0,#&1 *A7(,#&1 =[5MM>
#$ 7$*/ #, #$ ,+#8#(3 ,& -(#1,(#1 ( 0&1$,(1, -#?#12 +(,#& $#10* 4 #$ ,"* )+&21&$,#0
8(+#(%3* (1/ ,"* /#8*+2*10* /&*$ 1&, ())*(+ *?)3#0#,395 G. ,"* (#+ (1/ ,+(0*+ *A7(,#&1$
(+* $&38*/ &1 "7?4.&+- =*A7(,#&1$ =[5O> (1/ =[5ML>6 +*$)*0,#8*39> 7$#12 ,"* $(-*
17-*+#0(3 -*,"&/6 #, #$ 7$7(339 ,+#8#(3 ,& )+*$*+8* ( 0&1$,(1, -#?#12 +(,#& !*3/ $#10*
,"* -#?#12 +(,#& 4 #$ +*0&8*+*/ .+&- =[5ML> %9 /#8#/#12 ,"* )+&21&$*/ ,+(0*+ -($$
!*3/ l 4 %9 l .+&- =[5O>5 S& *8*1 #. ,"* 17-*+#0(3 $0"*-* #$ 71(%3* ,& )+*$*+8* (
0&1$,(1, -($$ !*3/ l 6 #, #$ 7$7(339 )&$$#%3* ,& /*$#21 $0"*-*$ $& ,"(, ( 0&1$,(1, 4
!*3/ #$ +*0&8*+*/ ;"*1 /#8#/#12 l 4 %9 ,"* =)&,*1,#(339> 1&14/#8*+2*10* )+*$*+8#12
.&+*0($, &. l 5
8.3.6 Physical realizability (monotone, positive-definite,
non-oscillatory, shape-preserving)
G1 ,"* (%$*10* &. $&7+0*$ (1/ $#1'$ ,"* -#?#12 +(,#& &. ( X(2+(12#(1 )(+0*3 %*4
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(%$&37,* 8(37*$ &. )+*4*?#$,#12 3&0(3 *?,+*-( #$ 1&14#10+*($#125 S,+#0, -&1&,&1#0#,9
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*+(,*$ )"9$#0(339 +*(3#@(%3* $&37,#&1$5 S#10* -#?#12 +(,#&$ 0(11&, )"9$#0(339 ,('*
1*2(,#8* 8(37*$ ,"*9 $"&73/ +*-(#1 1&141*2(,#8*5 S0"*-*$ ,"(, 0(11&, 2*1*+(,* 1*24
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3(+2* 2+(/#*1,$ (+* +*.*++*/ ,& ($ )3)K3;2$**"?3:J5 H%8#&7$39 ( $0"*-* ,"(, #$ -&1&4
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8*+$(5 G, $"&73/ %* $,+*$$*/ ,"(, #, #$ 4 ,"(, $"&73/ +*-(#1 -&1&,&1* (1/ 1&, l 45
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$#-)3* #337$,+(,#&1 &. ,"* 3(,,*+ .&+ (1 #/*(3#@*/ "&; !*3/5
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+*.*+ ,& ,"* 0(+*.73 /*!1#,#&1 2#8*1 %9 T(+,*1 =MO[Z>
[ I#1#,*4E&37-* !+(1$)&+, S0"*-*$
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8.3.7 Preservation of pre-existing functional relations between
species (correlations)
:$ /*$0+#%*/ #1 <37-% =KLLh>^ tY*3(,#&1$"#)$ %*,;**1 3&1243#8*/ $,+(,&$)"*+#0 ,+(04
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,"&7$(1/ '#3&-*,*+$6 (+* /#$)3(9*/ -&$, $,+#'#1239 #. ,"* -#?#12 +(,#& &. &1* #$ )3&,4
,*/ (2(#1$, (1&,"*+6 ;"*1 ,"* /(,( 0&33()$* &1,& +*-(+'(%39 0&-)(0, 07+8*$5u G1
&,"*+ ;&+/$6 /#..*+*1, 3&123#8*/ ,+(0* 0&1$,#,7*1,$ =$70" ($ 1#,+&7$ &?#/* . K W (1/
`,&,(3 &// 1#,+&2*1Q .W J > $**- ,& %* +*3(,*/ ,"+&72" +(,"*+ $#-)3* .710,#&1(3 +*4
3(,#&1$"#)$ #16 .&+ *?(-)3*6 ,"* )&3(+ $,+(,&$)"*+#0 8&+,*?5 S70" +*3(,#&1$"#)$ 0(1
(+#$* .+&- /#..*+*1, +*($&1$ =<37-% (1/ d&6 MOOK>6 "&;*8*+6 #, #$ ;*334'1&;1 ,"(,
,+(1$)&+, 0(1 *$,(%3#$" $70" +*3(,#&1$ =*525 !"7%7+1 (1/ U031,9+*6 MOOh>5
G1 &+/*+ ,& (007+(,*39 $#-73(,* $70" +*3(,#&1$"#)$ #1 17-*+#0(3 -&/*3$6 ,"* ,+(1$4
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MOOao !"7%7+1 (1/ U031,9+*6 MOOh>5 !"(, #$6 ,"* ,+(1$)&+, &)*+(,&+ $"&73/ -(#1,(#1
,"* +*3(,#&1$"#) #1 =[5MR> ,"+&72"&7, ,"* $#-73(,#&1
4M = a (L) + a (M) 4K ,
=[5MR>
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#1,*++*3(,*/ $)*0#*$5 : ,+(1$)&+, $0"*-* ;#33 )+*$*+8* 3#1*(+ )+*4*?#$,#12 .710,#&1(3
+*3(,#&1$ #. ,"* ,+(1$)&+, &)*+(,&+ T 6 ,"(, 7)/(,*$ 4 $ 6 $ = M, K6 #1 ,#-*6 #$ `$*-#43#1*(+Q
$
#
T (4M ) = T a (L) + a (M) 4K = a (L) T (M) + a (M) T (4K ) = a (L) + a (M) T (4K ), =[5Ma>
=X#1 (1/ Y&&/6 MOOao !"7%7+1 (1/ U031,9+*6 MOOh>5 :$ 1&,*/ %9 !"7%7+1 (1/
U031,9+* =MOOh> ,"* $700*$$.73 )+*$*+8(,#&1 &. 3#1*(+ 0&++*3(,#&1$ %9 ( ,+(1$)&+,
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&,"*+ "(1/6 #. ( -&/*3 $#21#!0(1,39 8#&3(,*$ ,"* )+*$*+8(,#&1 &. 3#1*(+ 0&++*3(,#&1$
%*,;**1 0"*-#0(3 0&1$,#,7*1,$6 ,"* -&/*3 #$ -&$, 3#'*39 1&, 2&#12 ,& )+&8#/* ,+7,".73
$#-73(,#&1$ &. ,"* +*3(,#&1 %*,;**1 ,"&$* 0&1$,#,7*1,$5
S#10* #1,*++*3(,*/ ,+(0*+$ 0(1 (3$& %* +*3(,*/ 1&143#1*(+396 #, #$ (3$& &. #1,*+*$, ,&
#18*$,#2(,* "&; ( ,+(1$)&+, &)*+(,&+ /#$,&+,$ $70" 1&143#1*(+ +*3(,#&15 I&+ *?(-)3*6
0&1$#/*+ ,;& ,+(0*+$ ,"(, (+* #1#,#(339 0&++*3(,*/ %9 ( .&7+,"4&+/*+ )&391&-#(3
4M = a (L) + a (M) (4K )P ,
=[5Mh>
=!"7%7+1 (1/ U031,9+*6 MOOh> ;"*+* ,"* 0&1$,(1,$ a (L) (1/ a (M) $"&73/ %* 0"&4
$*1 $& ,"(, ,"* .710,#&1(3 +*3(,#&1 #$ *#,"*+ 0&18*? &+ 0&10(8* #1 ,"* +(12* &. ,"*
#1#,#(3 0&1/#,#&1 8(37*$ &. 4 M (1/ 4K 5 F?0*), .&+ .7339 X(2+(12#(1 ,+(1$)&+, &)*+(4
,&+$6 $0"*-*$ (+* 7$7(339 71(%3* ,& -(#1,(#1 1&143#1*(+ .710,#&1(3 +*3(,#&1$"#)$ (1/
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0(3 -#?#12 &. ,"* 0&1$,#,7*1,$5 G1#,#(3#@#12 ,;& ,+(0*+$ ,"(, (+*6 .&+ *?(-)3*6 +*3(,*/
,"+&72" =[5Mh> (1/ 3*,,#12 ,"* ,+(0*+$ %* ,+(1$)&+,*/ %9 ( 0"(33*12#12 "&; ,"(, /*4
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8*3&)$ .*(,7+*$ ,"(, 0&33()$* ,& ,"* 1*(+ 2+#/4$0(3*6 )+&8#/*$ )"9$#0(3 #1$#2", #1,&
,"* 17-*+#0(3 -#?#12 ,"(, ,"* ,+(1$)&+, &)*+(,&+ #1,+&/70*$ =!"7%7+1 (1/ U031,9+*6
MOOh>5 N& )+(0,#0(3 F73*+#(1 (1/ $*-#4X(2+(12#(1 $0"*-* 0(1 )+*$*+8* =[5Mh> (1/
;#33 ,"*+*.&+* )+&/70* $0(,,*+ )&#1,$ ,"(, /*8#(,* .+&- ,"* )+*4*?#$,#12 .710,#&1(3
+*3(,#&1$"#) =[5Mh>5 C"*1 $0(,,*+ )&#1,$ /*8#(,* .+&- ,"* )+*4*?#$,#12 .710,#&1(3 +*4
3(,#&1 07+8* ,"* ,+(1$)&+, &)*+(,&+ #$ #1,+&/70#12 17-*+#0(3 -#?#125 !"* 17-*+#0(3
-#?#12 0(1 *#,"*+ %* $)7+#&7$ &+ +*$*-%3* `+*(3Q -#?#125 G. ,"* $0(,,*+ 8(37*$ (+*
&1 ,"* 0&10(8* $#/* &. ,"* )+*4*?#$,#12 .710,#&1(3 +*3(,#&16 ,"* 17-*+#0(3 -#?#12 #$
$#-#3(+ ,& `+*(3 -#?#12Q ,"(, #$ &%$*+8*/ #1 ,"* (,-&$)"*+* =$** I#25 [5R>5 U#?#12 #1
,"* (,-&$)"*+* &007+$6 .&+ *?(-)3*6 ;"*1 ,"* )&3(+ $,+(,&$)"*+#0 8&+,*? %+*('$ 7)
=*525 C(72" *, (36 MOOh>5 G. $0(,,*+ 8(37*$ ())*(+ &7,$#/* ,"* `0&18*? "733Q =*#,"*+ %9
)+&/70#12 $0(,,*+ )&#1,$ &1 ,"* 0&18*? $#/* &. ,"* )+*4*?#$,#12 .710,#&1(3 +*3(,#&14
$"#) 07+8* (1/V&+ &7,$#/* ,"* +(12* &. ,"* #1#,#(3 0&1/#,#&1 .&+ 4 $ 6 $ = M, K>6 ,"* -&/*3
)+&/70*$ 17-*+#0(3 71-#?#12 ;"#0" #$ 713#'* `+*(3 -#?#12Q5 !"7%7+1 (1/ U031,9+*
=MOOh> )+&8*/ ,"(, #1 &+/*+ ,& 27(+(1,** &139 `+*(3Q 17-*+#0(3 -#?#126 ,"* ,+(1$)&+,
&)*+(,&+ $"&73/ %* `$*-#43#1*(+Q (1/ -&1&,&1* (00&+/#12 ,& T(+,*1 =MO[Z>5 _14
.&+,71(,*39 &139 !+$,4&+/*+ $0"*-*$ ;#33 -**, ,"*$* +*A7#+*-*1,$5 S#10* !+$,4&+/*+
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3*8*3 &. 71-#?#125 I&+ ( -&+* 0&-)3*,* /#$07$$#&1 &. ,"#$ ,&)#0 ,"* +*(/*+ #$ +*.*++*/
,& !"7%7+1 (1/ U031,9+* =MOOh>5 Y*0*1,396 X(7+#,@*1 (1/ !"7%7+1 =KLML> )+&)&$*/
-#?#12 /#(21&$,#0$ ,"(, A7(1,#!*$ ,"* (-&71, &. 17-*+#0(3 -#?#12 ,"(, ,"* ,+(1$)&+,
&)*+(,&+ #1,+&/70*$ .&+ #1,*++*3(,*/ $)*0#*$5
:1&,"*+ $#,7(,#&1 +*3*8(1, ,& ,"* ,+(1$)&+, &. 0"*-#0(3 $)*0#*$ #$ ,"* $#,7(,#&1 #1
;"#0" -&+* ,"(1 ,;& $)*0#*$ (+* +*3(,*/ ,"+&72" $&-* 0&-)3#0(,*/ +*3(,#&1 %7, ,"*9
(// 7) ,& ( 0&1$,(1, =&+ ( $-&&," $)(,#(3 !*3/ MK >5 C#," ]7$, ,;& $)*0#*$ ,"#$ +*/70*$
,& )+*$*+8#12 ( 3#1*(+ 0&++*3(,#&1 %7, ;#," -&+* ,"(1 ,;& $)*0#*$ #, #$ 8*+9 0"(33*124
#12 ,& 27(+(1,** ,"(, ,"* ,&,(3 -#?#12 +(,#& +*-(#1$ 0&1$,(1,6 *?0*), %9 ,+(1$)&+,4
#12 ,"* ,&,(3 &+ 7$#12 ( .7339 X(2+(12#(1 $0"*-*5 !"* ,+(1$)&+, &)*+(,&+$ (%#3#,9 ,&
-(#1,(#1 ,"* 0&1$,(1, $7- #$ (1&,"*+ -*($7+* .&+ 17-*+#0(3 -#?#12 (1/ "($ %**1
*?)3&+*/ #1 &1* /#-*1$#&1 %9 H8,0"#11#'&8 (1/ F($,*+ =KLLO>5 N&,* ,"(, -(#14
,(#1#12 &+ &139 )*+,7+%#12 )+*4*?#$,#12 .710,#&1(3 +*3(,#&1$ #1 ( `)"9$#0(3 ;(9Q #$
1&, &139 #-)&+,(1, .&+ 3&1243#8*/ $,+(,&$)"*+#0 ,+(0*+$ %7, (3$& .&+ &,"*+ )(+,$ (1/
)+&0*$$*$ #1 ,"* (,-&$)"*+* $70" ($ 03&7/4(*+&$&3 #1,*+(0,#&1$ =H8,0"#11#'&8 (1/
F($,*+6 KLLO>5 G1 (336 $#123*4,+(0*+ ,*$,#12 ,"(, "($ ,+(/#,#&1(339 %**1 7$*/ ,& *8(374
(,* ,+(1$)&+, &)*+(,&+$ #1 #/*(3#@*/ $*,,#12$ /&*$ 1&, )+&8#/* #1$#2", #1,& "&; ;*33
,+(0*+ #1,*++*3(,#&1$ (+* -(#1,(#1*/ (3,"&72" #, #$ #-)&+,(1, .&+ -(19 (,-&$)"*+#0
())3#0(,#&1$5
MK
.&+ *?(-)3*6 ,&,(3 +*(0,#8* 0"3&+#1* #1 ,"* $,+(,&$)"*+*
[ I#1#,*4E&37-* !+(1$)&+, S0"*-*$
KLR
Ac6"Bd
K
AK
Ac6$)d
K
Ac6$)d
M
AM
Ac6"Bd
M
Fig. 8.5 S0"*-(,#0 8#*; &. ,"* *..*0, &. -#?#12 &1 ( $0(,,*+ )3&, ;"*+* -#?#12 +(,#& .&+ ,+(0*+ M6
4M 6 #$ )3&,,*/ (2(#1$, ,+(0*+ K6 4K 5 !"* ,;& ,+(0*+$ (+* #1#,#(339 1&143#1*(+39 0&++*3(,*/6 ,"(, #$6 ,"*
$0(,,*+ )&#1,$ (4M , 4K ) (+* &1 ,"* )+*4*?#$,#12 .710,#&1(3 +*3(,#&1 07+8* =$&3#/ ,"#0' 3#1* 07+8*>5 !"*
(6$)) (6"B)
(6$)) (6"B)
] (1/ [4K , 4K
]6 +*$)*0,#8*395 <(+,#(3 -#?#12
#1#,#(3 +(12*$ .&+ ,"* ,;& ,+(0*+$ (+* [4M , 4M
&. ,;& (#+ -($$*$ =,;& !33*/ 0#+03*$ 4 $0(,,*+ )&#1,$> ;#33 ,*1/ ,& -&8* ,"* ,;& $0(,,*+ )&#1,$ ,&;(+/$
*(0" &,"*+ (3&12 ,"* $,+(#2", =+*/> 3#1* =(3$& +*.*++*/ ,& ($ ( `-#?#12 3#1*Q>5 T*10* `+*(3 -#?#12Q
&007++#12 #1 ,"* (,-&$)"*+* ;#33 ,*1/ ,& -&8* )&#1,$ &1 ,"* $0(,,*+ )3&, ,& ,"* 0&10(8* $#/* &. ,"*
)+*4*?#$,#12 .710,#&1(3 +*3(,#&1 07+8* =(3$& +*.*++*/ ,& ($ ,"* `0&18*? "733Q 4 $"(/*/ (+*(>5
8.3.8 Robustness
!"* 17-*+#0(3 -*,"&/ $"&73/ +*-(#1 $,(%3* (1/ +*,(#1 $#-73(,#&1 8*+(0#,9 ,"+&72"4
&7, ,"* #1,*2+(,#&15 Y&%7$,1*$$ 0(1 %* ($$*$$*/ %9 ,*$,#12 ,"* (32&+#,"- .&+ -(19
/#..*+*1, "&; !*3/$6 ,*-)&+(3 (1/ $)(,#(3 +*$&37,#&1$5
8.3.9 Parallel computational efficiency
<*+.&+-(10* #-)+&8*-*1,$ (+* 3(+2*39 /7* ,& #10+*($*/ )(+(33*3#$- +(,"*+ ,"(1
#-)+&8*/ -#0+&)+&0*$$&+ 03&0' .+*A7*1095 T*10* ,"* 17-*+#0(3 (32&+#,"- $"&73/
%* (-*1(%3* .&+ *?*07,#&1 &1 -($$#8*39 )(+(33*3 0&-)7,#12 )3(,.&+-$5 : ;(9 ,&
(0"#*8* ,"#$ #$ ,& 7$* 3&0(3 -*,"&/$ ;#," -#1#-(3 23&%(3 /*)*1/*10* =.&+ -&+* /#$4
07$$#&1 $** 0"(),*+ Ma>5
KLa
X(7+#,@*1 *, (35
G, #$ ;&+," 1&,#12 ,"(, (3,"&72" 0&-)7,#12 )&;*+ "($ #10+*($*/ /+(-(,#0(339 #1
,"* 3($, KL 9*(+$ &+ $&6 ,"*$* *?,+( 0&-)7,(,#&1(3 +*$&7+0*$ "(8* 3(+2*39 %**1 7$*/
,& $(,#$.9 /*-(1/$ .&+ "#2"*+ +*$&37,#&16 -&+* (/8(10*/ )"9$#0(3 )(+(-*,*+#@(,#&1$
(1/ 0&7)3#12 ,"* (,-&$)"*+#0 0&-)&1*1, ,& &0*(16 3(1/6 (1/ #0* 0&-)&1*1,$ =#5*5
0&7)3*/ -&/*3$>5 T*10* #, #$ $,#33 /*$#+(%3* ,& /*8*3&) *.!0#*1, /91(-#0(3 0&+* (32&4
+#,"-$6 #1 )(+,#073(+6 $0"*-*$ .&+ *.!0#*1, ,+(0*+ ,+(1$)&+, =$** )(+(2+()" &1 A5*?$K
?:"2-: -<!2$-)2J %*3&;> *8*1 ,"&72" 0&-)7,#12 )&;*+ #$ #10+*($#125
8.3.10 Multi-tracer efficiency
G1 -&/*+1 (,-&$)"*+#0 -&/*3$ ,"* 17-%*+ &. ,+(0*+$ +*A7#+*/ ,& %* (/8*0,*/ 0&14
,#17* ,& #10+*($*5 I&+ *?(-)3*6 ,"* 0"*-#$,+9 8*+$#&1 &. ND:YQ$ D:U -&/*3 ,+(1$4
)&+,$ &8*+ MLL ,+(0*+$ =X(-(+A7* *, (36 KLL[>5 W#8*1 ,"(, ,"* /91(-#0(3 0&+* ,9)4
#0(339 "($ 3*$$ ,"(1 ML )+&21&$,#0 8(+#(%3*$ /*!1#12 ,"* $,(,* &. ,"* "7#/ "&; (1/
,"*+-&/91(-#0$6 ,"* 0&-)7,(,#&1(3 0&$, &. +711#12 ,"* /91(-#0(3 0&+* 0(1 )+#-(+4
#39 %* (,,+#%7,*/ ,& ,"* ,+(1$)&+, &. ,+(0*+$5 N**/3*$$ ,& $(96 #, #$ "#2"39 /*$#+(%3*
,"(, ,"* 17-*+#0(3 (32&+#,"- 7$*/ .&+ ,+(0*+ ,+(1$)&+, %* *.!0#*1, (1/ (/(),(%3*
.&+ ( 3(+2* 17-%*+ &. ,+(0*+$5 : ;(9 ,& (0"#*8* -73,#4,+(0*+ *.!0#*109 #$ ,& /*$#21
$0"*-*$ ,"(, 0(1 +*7$* #1.&+-(,#&1 .&+ *(0" (//#,#&1(3 ,+(0*+ =\(+," (1/ I+*/*+#0'4
$&16 MOOLo B7'&;#0@ (1/ \(7-2(+/1*+6 KLLL> (1/V&+ ,+(1$)&+, ,+(0*+$ ;#," 3&12*+
,#-*4$,*)$ ,"(1 7$*/ .&+ ,"* 0&1,#17#,9 *A7(,#&1 .&+ (#+ #1 ,"* /91(-#0(3 0&+* =(3$&
+*.*++*/ ,& ($ `$7)*+40903#12Q &. ,+(0*+$ ;#," +*$)*0, ,& (#+ &+6 -&+* 0&--&1396 `$7%4
0903#12Q &. (#+ ;#," +*$)*0, ,& ,+(0*+$o $** $*0,#&1 [5h5K>5
8.3.11 Geometric flexibility
G, #$ 2*1*+(339 7$*.73 ,& /*8*3&) 17-*+#0(3 -*,"&/$ ,"(, 0(1 %* 7$*/ &1 ( ;#/* +(12*
&. $)"*+#0(3 2+#/$5 N*?, 2*1*+(,#&1 /91(-#0(3 0&+*$ (+* %*#12 /*8*3&)*/ &1 $)"*+#4
0(3 2+#/$ %($*/ &1 ,+#(123*$6 A7(/+#3(,*+(36 )*1,(2&1(3 (1/V&+ "*?(2&1(3 0&1,+&3 8&34
7-*$5 G, #$ ,"*+*.&+* /*$#+(%3* ,"(, ( $0"*-* 0(1 "(1/3* (19 $)"*+#0(3 )&392&14%($*/
2+#/5 :3$& -&/*3$ 7$#12 $,(,#0 &+ (/(),#8* -*$"4+*!1*-*1, %*1*!, .+&- 2*&-*,+#4
0(339 "*?#%3* -*,"&/$5 :1 *?(-)3* &. ( 2*&-*,+#0(339 "*?#%3* (/8*0,#&1 $0"*-* #$
U<B:!: =U73,#/#-*1$#&1(3 <&$#,#8* B*!1#,* :/8*0,#&1 !+(1$)&+, :32&+#,"->o .&+
(1 &8*+8#*; $** S-&3(+'#*;#0@ =KLLa>5
8.4 Problem formulation: Discrete schemes
I#1#,*48&37-* -*,"&/$ (+* 17-*+#0(3 -*,"&/$ ;"*+* *(0" )+&21&$,#0 8(+#(%3* #$
$,&+*/ ($ (1 (8*+(2* A7(1,#,9 &8*+ ( 0*+,(#1 !1#,*39 3(+2* 0&1,+&3 8&37-* =(3$&
[ I#1#,*4E&37-* !+(1$)&+, S0"*-*$
KLh
+*.*++*/ ,& ($ 0*334#1,*2+(,*/ -*,"&/$>5 !"#$ 0"&#0* /#..*+$ .+&- -*,"&/$ ,"(, (+*
%($*/ &1 2+#/4)&#1, 8(37*$ =7$*/ #16 *5256 !1#,*4/#..*+*10* -*,"&/$> &+ ;*#2",$ .&+
*?)(1$#&1 .710,#&1$ =*5256 !1#,*4*3*-*1, &+ $)*0,+(3 -*,"&/>5 G1 &+/*+ ,& /*+#8*
!1#,*48&37-* /#$0+*,#@(,#&1 $0"*-*$ ,"* *A7(,#&1$ &. -&,#&16 #1 ,"#$ 0($* ,"* 0&14
,#17#,9 *A7(,#&16 (+* #1,*2+(,*/ &8*+ ( 0&1,+&3 8&37-*5 !"#$ (33&;$ .&+ /#$0+*,#@(4
,#&1$ ,"(, '**) ,+(0' *?(0,39 &. ,"* 3&0(3 -($$4%7/2*,$ (1/ ,"7$ )+&8#/*$ -($$4
0&1$*+8(,#&1 ,& -(0"#1* )+*0#$#&15 N&,* ,"(, (3,"&72" !1#,*48&37-* $0"*-*$ (+*
/*$#21*/ ,& 0&1$*+8* -($$ 3&0(339 ,"+&72" *?)3#0#,39 ,+(0'#12 -($$6 0&1$*+8(,#&1 &.
-($$ 0(1 (3$& %* (0"#*8*/ #1 1&1 !1#,*48&37-* -*,"&/$ =*5256 0&-)(,#%3* -*,"&/$6
$** 0"(),*+ MK>5 D&1$*+8(,#8* -*,"&/$ ,"(, (+* 1&, !1#,*48&37-* -*,"&/$ 7$7(339
/& 1&, 0&1$*+8* -($$ 3&0(3395
!9)#0(339 !1#,*48&37-* $0"*-*$ 0&-* #1 ,;& "(8&+$ 0&++*$)&1/#12 ,& ,;& .&+-$
&. /*+#8#12 ,"* *A7(,#&1$ &. -&,#&1 .+&- !+$, )+#10#)3*$^ F73*+#(1 (1/ X(2+(12#(1 MZ5
G1 -&$, ,*?,%&&'$ ,"* *A7(,#&1$ &. -&,#&1 (+* /*+#8*/ #1 F73*+#(1 .&+-6 ,"(, #$6 ($
&%$*+8*/ .+&- ( !?*/ 8&37-* #1 ,"* (,-&$)"*+* =$,(,#&1(+9 ,& ,"* F(+,"Q$ $7+.(0*>5
T*10* ,"*+* #$ ( "7? &. -($$ ,"+&72" ,"* 8&37-* %&71/(+#*$ 713*$$ ,"* 3&0(3 ;#1/
#$ @*+&5 H1* -(9 (3$& /*+#8* ,"* *A7(,#&1$ &. -&,#&1 ($ 8#*;*/ %9 ( 8&37-* 1&,
]7$, +&,(,#12 ;#," ,"* F(+,"Q$ +&,(,#&1 (?#$ %7, (3$& -&8#12 ;#," ,"* 3&0(3 "&;o (5'5(5
X(2+(12#(1 .&+-5 G1 X(2+(12#(1 .&+- ,"*+* #$ 1& "7? &. -($$ ,"+&72" ,"* `;(33$Q &.
,"* 8&37-*5 \&," &. ,"*$* .&+-$ &. ,"* !1#,*48&37-* /#$0+*,#@(,#&1 &. ,"* 0&1,#17#,9
*A7(,#&1 (+* )+*$*1,*/ 1*?, (.,*+ ,"* #1,+&/70,#&1 &. $&-* 1&,(,#&15 I&+ $#-)3#04
#,9 ;* 0&1$#/*+ ,"* ,;&4/#-*1$#&1(3 )+&%3*- #1 D(+,*$#(1 2*&-*,+9 (1/ /*.*+ ,"*
/#$07$$#&1 &. ,"* *?,*1$#&1 ,& $)"*+#0(3 2*&-*,+9 (1/ ,"+**4/#-*1$#&1$ ,& $*0,#&1
[5R5Z5Z (1/ [5a6 +*$)*0,#8*395
X*, ,"* /&-(#1 &. #1,*2+(,#&1 %* /*1&,*/ 1 =( D(+,*$#(1 )3(1* ;#," )*+#&/#0
%&71/(+9 0&1/#,#&1$ &+ 1& "7? ,"+&72" ,"* /&-(#1 %&71/(+#*$>5 !"* /&-(#1
1 #$
%
)(+,#,#&1*/ #1,& . 1&14&8*+3())#12 2+#/ 0*33$6 & C 6 C = M, .., .6 $& ,"(, .C=M &C $)(1
1 5 !"* (+*( &. 0*33 & C #$ /*1&,*/ 6 &C 5 I&+ 1&; ;* $"(33 ($$7-* ( A7(/+#3(,*+(3
-*$" #1 D(+,*$#(1 2*&-*,+96 "&;*8*+6 ,"* /#$07$$#&1 0(1 ,+#8#(339 %* *?,*1/*/ ,&
&,"*+ -*$"*$ $70" ($ ,+#(1273(+ &+ "*?(2&1(3 -*$"*$ #1 D(+,*$#(1 2*&-*,+95
:$ -*1,#&1*/ (%&8* ,"* )+&21&$,#0 8(+#(%3* 0&1$#/*+*/ #$ ,"* 0*33 (8*+(2*/ 8(37*
sC =
M
6 &C
&
&C
s (B, J) %&,
s = l &+ l 4,
=[5M[>
;"*+* s (B, J) #$ ,"* *?(0, $&37,#&15 G1 ,#-* ;* /#$0+*,#@* #1 ,*+-$ (, *A7#/#$,(1,
,#-*43*8*3$6 #5*5 $7)*+$0+#), ) +*.*+$ ,& ,"* A7(1,#,9 (, ,#-* ? = ) 6 ? ;"*+* 6 ? #$ ,"*
,#-*4$,*)5 S& ,"* $,(,* &. ( ,+(0*+ #1 0*33 & C (, ,#-*43*8*3 ) #$ /*1&,*/ s )C 5
MZ
,"* F73*+#(1 (1/ X(2+(12#(1 .&+-$ (+* 3#-#,$ &. ,"* -&+* 2*1*+(3 (+%#,+(+9 X(2+(12#(14F73*+#(1
=:XF> .&+- =T#+, *, (36 MOhP>
KL[
X(7+#,@*1 *, (35
8.4.1 (Semi-)Lagrangian schemes
D&1$#/*+ (1 (+%#,+(+9 X(2+(12#(1 (+*( &(?)5 \9 /*!1#,#&1 ,"* (+*( &(?) -&8*$ ;#,"
,"* "&; ;#,"&7, (19 "7? &. -($$ ,"+&72" #,$ $#/*$ (1/ "*10* #, (3;(9$ 0&1,(#1$ ,"*
$(-* -(,*+#(3 )(+,#03*$5 S#10* ,"*+* #$ 1& "7? &. -($$ ,"+&72" ,"* %&71/(+#*$ &.
&(?)6 ,"* -($$ #1 ,"* (+*( #$ 0&1$*+8*/5 G1 -(,"*-(,#0(3 ,*+-$ ,"#$ 0(1 %* ;+#,,*1 ($
%
%?
&
&(?)
s %& = L,
s = l &+ l 4.
=[5MO>
FA7(,#&1 =[5MO> #$ +*.*++*/ ,& ($ ,"* X(2+(12#(1 !1#,*48&37-* .&+- &. ,"* 0&1,#17#,9
*A7(,#&15 : ,*-)&+(3 /#$0+*,#@(,#&1 &. =[5MO> +*(/$
&
&(?+6 ?)
s %& =
&
&(?)
s %&.
=[5KL>
G. ,"* $(-* X(2+(12#(1 0*33 &(?) #$ ,+(0'*/ ,"+&72"&7, ,"* $#-73(,#&1 ,"* +*$73,#12
$0"*-* #$ +*.*++*/ ,& ($ .7339 X(2+(12#(15 !"* 0"(33*12* #1 $70" $0"*-*$ #$ ,"(, .&+
1&14,+#8#(3 "&;$ ,"* (+*($ A7#0'39 /*.&+- #1,& ,"#1 !3(-*1,$ $& ,"(, ,"* +*$&37,#&1
#$ 1& 3&12*+ 71#.&+- =$** I#27+* K #1 0"(),*+ h>5
G1$,*(/ &1* -(9 0&1$#/*+ ( /#..*+*1, $*, &. (+*($V)(+0*3$ (, *8*+9 ,#-*4$,*)6 .&+
*?(-)3*6 *1.&+0#12 ,"(, *#,"*+ &(? + 6 ?) &+ &(?) #$ ( +*273(+ $,(,#0 2+#/ 0*335 S70" (1
())+&(0" #$ +*.*++*/ ,& ($ $*-#4X(2+(12#(1 $#10* #, &139 ,+(0'$ ,"* $(-* X(2+(12#(1
)(+0*3$V(+*( .&+ &1* ,#-*4$,*)5 !"* (/8(1,(2* &. ,"* $*-#4X(2+(12#(1 ())+&(0"6 ($
0&-)(+*/ ,& ( .7339 X(2+(12#(1 -*,"&/6 #$ ,"(, #, +*,(#1$ ( A7($# 71#.&+- +*$&37,#&1
($ ,"* -*$" &139 /*.&+-$ .&+ &1* ,#-*4$,*)5 T&;*8*+6 ,"* 2+#/ 71#.&+-#,9 #$ #1,+&4
/70*/ (, ,"* *?)*1$* &. "(8#12 ,& #1,*+)&3(,* 8(+#(%3*$ .+&- ( +*273(+ $,(,#0 2+#/ ,&
( /*.&+-*/ X(2+(12#(1 2+#/ =&+ 8#0* 8*+$(> (, *8*+9 ,#-*4$,*)5 T&; ,"#$ #1,*+)&3(4
,#&1 0(1 %* /&1* #$ /#$07$$*/ #1 2+*(, /*,(#3 %*3&; %7, .&+ 1&; -&+* -(,"*-(,#0(3
1&,(,#&1 #$ 1**/*/5
:$$7-* ,"(, &(? + 6 ?) #$ ( +*273(+ 2+#/ 0*33 +*$73,#12 #1 ( -*,"&/ +*.*++*/ ($
7)$,+*(- $*-#4X(2+(12#(1 MP5 G. ;* 0&1$#/*+ 0*33 C #1 ,"* /#$0+*,#@*/ /&-(#1 ,"*1
,"* +*273(+ 2+#/ 0*33 (&(? + 6 ?)) #$ *?(0,39 & C ;#," (+*( 6 &C 5 !"* 0&++*$)&1/#12 7)4
$,+*(- X(2+(12#(1 (+*( (&(?)) #$ +*.*++*/ ,& ($ " C ;#," (+*( 6 "C =$** I#25 [5a(>5 C*
($$7-* ,"(, 6 #$ 0"&$*1 $70" ,"(, (33 ,"* /*.&+-*/ (+*($ " C (+* $#-)39 0&11*0,*/5
N&,* ,"(, ,"*+* *?#$,$ ( &1*4,&4&1* 0&++*$)&1/*10* %*,;**1 & C (1/ "C $70" ,"(,
,"* "C Q$ $)(1 1 ;#,"&7, 2()$ &+ &8*+3()$ %*,;**1 ,"*.
'
C=M
"C = 1 , (1/ "C ∩ "! = LV ∀ C (= !.
=[5KM>
:$$7-* ,"(, ,"* *8&37,#&1 &. ,"* X(2+(12#(1 2+#/ #$ '1&;1 (1(39,#0(339 $& ;* '1&;
,"* 0"(+(0,*+#$,#0$ &+ ,+(]*0,&+#*$ .&+ *(0" "7#/ )(+0*3 (, (33 ,#-*$5 !"* 0&-)7,(,#&1
MP
1&,* ,"(, &1* -#2", *A7(339 ;*33 0&1$#/*+ /&;1$,+*(- $0"*-*$ ;"*+* &1* 0&1$#/*+$ F73*+#(1
=+*273(+> 2+#/ 0*33$ (, ,#-*43*8*3 ) (1/ 3*, ,"*- %* ,+(1$)&+,*/ ;#," ,"* "&; .&+ &1* ,#-*4$,*)5
[ I#1#,*4E&37-* !+(1$)&+, S0"*-*$
KLO
(a)
(b)
Fig. 8.6 : 2+()"#0(3 #337$,+(,#&1 &. ,"* 7)$,+*(- $*-#4X(2+(12#(1 1&-*103(,7+*5 =(> !"* $,(,#0
F73*+#(1 0*33 &C =3#2", $"(/#12> (1/ ,"* 0&++*$)&1/#12 7)$,+*(- X(2+(12#(1 (+*( "C =/(+' $"(/#12>
,"(, *1/$ 7) (, &C (.,*+ &1* ,#-*4$,*)5 I&+ #337$,+(,#&1 ,"* ,+(]*0,&+#*$ &. ,"* 8*+,#0*$ =!33*/ 0#+03*$>
&. &C (+* /*)#0,*/ ;#," (++&;$5 !"* 0&++*$)&1/#12 7)$,+*(- 8*+,#0*$ =/*)(+,7+* )&#1,$> (+* $"&;1
;#," &)*1 0#+03*$5 =%> !"* 1&,(,#&1 7$*/ ,& /*!1* &8*+3() (+*($ %*,;**1 F73*+#(1 0*33 &! (1/
7)$,+*(- X(2+(12#(1 (+*( "C #$ "C! = &! ∩ "C =/(+' $"(/*/ (+*(>
&. "7#/ )(+0*3 ,+(]*0,&+#*$ #$ ;*33 /*8*3&)*/ #1 ,"* $*-#4X(2+(12#(1 3#,*+(,7+* =*5256
S,(1#.&+," (1/ Dj&,s* MOOM6 S,(1#.&+," *, (3 KLLZ6 T&+,(3 KLLK> (1/ #1 ,"* #1,*+*$, &.
%+*8#,9 #, #$ 1&, /#$07$$*/ .7+,"*+ #1 ,"#$ 0"(),*+6 (3,"&72" (007+(,* ,+(]*0,&+#*$ (+*
8#,(3 .&+ ,"* (007+(09 &. (19 X(2+(12#(1 -*,"&/5
C#," ,"* 1&,(,#&1 #1,+&/70*/ (%&8* ,"* .&+*0($, *A7(,#&1 =[5KL> 0(1 %* ;+#,,*1 ($
s )+M
6 &C = s )C 6 "C .
C
=[5KK>
;"*+* s )C #$ ,"* (8*+(2* ,+(0*+ /*1$#,9 &8*+ ,"* 7)$,+*(- (+*( " C
s )C =
M
6 "C
&
"C
s ) (B, J) %&,
=[5KZ>
(1/ s )+M
#$ ,"* 0*33 (8*+(2*/ 8(37* &. s &8*+ ,"* +*273(+ (+*( & C (, ,#-*43*8*3 ) + M5
C
!"* .710,#&1 s ) (B, J) #$ ,"* 0&1,#17&7$ /#$,+#%7,#&1 &. s (, ,#-*43*8*3 )5 H%8#&7$396
$#10* ,"* )+&21&$,#0 8(+#(%3*$ (+* 0*33 (8*+(2*$ s ;* /& 1&, '1&; ,"* 8(+#(,#&1
&. s (, ,"* $7%42+#/ $0(3* (1/ s ) (B, J) -7$, %* +*0&1$,+70,*/ .+&- ,"* )+&21&$,#0
0*33 (8*+(2*$MR 5 !"#$ )+&0*/7+* #$ +*.*++*/ ,& ($ $7%42+#/4$0(3* +*0&1$,+70,#&15 G1
!1#,*48&37-* $0"*-*$ ,"* +*0&1$,+70,#&1 #$ 7$7(339 3&0(3 +(,"*+ ,"(1 23&%(35 S& *(0"
0*33 C ;#33 "(8* (1 ($$&0#(,*/ $7%42+#/4$0(3* +*0&1$,+70,#&1 .710,#&1 s C (B, J) +(,"*+
,"(1 &1* 23&%(3 +*0&1$,+70,#&1 .710,#&1 &8*+ (33 0*33$ $70" ($ ,"* $)"*+#0(3 "(+-&1#0
.710,#&1$ 7$*/ #1 $)*0,+(3 ,+(1$.&+- -&/*3$5
T*10* ,"* 23&%(3 +*0&1$,+70,#&1 .710,#&1 #$ ( 0&33*0,#&1 &. 3&0(3 +*0&1$,+70,#&1
.710,#&1$
s (B, J) =
.
- X&C sC (B, J),
=[5KP>
C=M
MR
713*$$ 8(+#(%3*$ $70" ($ 2+(/#*1,$ (+* (3$& 0(++#*/ ($ )+&21&$,#0 8(+#(%3*$ =*525 c(%* *, (36 KLLM>
KML
X(7+#,@*1 *, (35
;"*+* X&C #$ ,"* #1/#0(,&+ .710,#&1
X&C =

 M, (B, J) ∈ &C ,

L, (B, J) ∈
/ &C .
=[5KR>
D&--&139 7$*/ -*,"&/$ .&+ 0&-)7,#12 s C (B, J) .+&- s C (+* /#$07$$*/ #1 $*0,#&1
[5R5K5
I#+$,6 ;* 1&,* ,"(, s (B, J) #$ 1&, 1*0*$$(+9 0&1,#17&7$ &+ /#..*+*1,#(%3* (0+&$$
0*33 %&71/(+#*$5 S& #. ,"* 7)$,+*(- (+*( " C 0&8*+$ $*8*+(3 F73*+#(1 0*33$ =*5256 I#25
[5a>6 ,"* #1,*2+(3 &1 ,"* +#2",4"(1/ $#/* &. =[5KZ> -7$, %* %+&'*1 7) #1,& &8*+3()
(+*($ %*,;**1 F73*+#(1 0*33$ (1/ " C 5 !"* /#$0+*,#@*/ $*-#4X(2+(12#(1 !1#,*48&37-*
0&1,#17#,9 *A7(,#&1 =[5KK> ,"*1 +*(/$
s )+M
6 &C =
C
GC
-
&
!=M "C!
s!) (B, J) %&.
=[5Ka>
!"* 17-%*+ &. 1&14*-),9 &8*+3() (+*($ %*,;**1 ,"* 7)$,+*(- 0*33 =/*)(+,7+* 0*33>
"C (1/ ,"* F73*+#(1 2+#/ 0*33$ #$ /*1&,*/ G C 5 N&,* ,"(, GC /*)*1/$ &1 ,"* "&; (1/
,#-*4$,*) $#@*6 (1/ .&+ ,#-*48(+9#12 "&;$ #, #$ 1&, 1*0*$$(+#39 0&1$,(1,5 !"* (+*( " C!
#$ ,"* 1&14*-),9 &8*+3() (+*( %*,;**1 ,"* 7)$,+*(- 0*33 " C (1/ ,"* F73*+#(1 2+#/
0*33 &! =$** I#25 [5a%>
"C! = "C ∩ &! ,
"C! (= Lo
V
! = M, . . . , GC , (1/ M ≤ GC ≤ .,
=[5Kh>
;"*+* . #$ ,"* 17-%*+ &. 0*33$ #1 ,"* /&-(#15
!;& 0&1/#,#&1$ -7$, %* .73!33*/ ,& 2*, 0&1$*+8(,#&1 &. -($$ #1 X(2+(12#(1 !1#,*4
8&37-* $0"*-*$^ I#+$,396 ,"* 7)$,+*(- 0*33$ " C -7$, %* $#-)39 0&11*0,*/ /&-(#1$
(1/ ,"*9 -7$, $)(1 1 ;#,"&7, 2()$ &+ &8*+3()$ =*A7(,#&1 [5KM>5 S*0&1/396 ,"* +*4
0&1$,+70,#&1 .710,#&1 #1 0*33 C6 s C (B, J)6 -7$, %* 0&1$*+8(,#8* #1 ,"* $*1$* ,"(, ,"*
#1,*2+(3 &. sC (B, J) &8*+ &C -7$, 9#*3/ ,"* 0*334(8*+(2* 8(37* ,"(, #$ 7$*/ ($ )+&24
1&$,#0 8(+#(%3*6
&
M
sC (B, J) %& = s C .
=[5K[>
6 &C &C
FA7(,#&1 =[5Ka> #$ ,"* %($#0 !1#,*48&37-* .&+- &. ,"* 0&1,#17#,9 *A7(,#&1 ;"*1
7$#12 (1 7)$,+*(- !1#,*48&37-* $*-#4X(2+(12#(1 ())+&(0"5 H%8#&7$39 ;* /& 1&,
'1&; ,"* *?(0, X(2+(12#(1 ,+(]*0,&+9 &. *8*+9 )(+0*3 #1 ,"* /&-(#1 $& $&-* ()4
)+&?#-(,#&1 ,& " C #$ 1*0*$$(+9 .&+ ,"* /*+#8(,#&1 &. (19 )+(0,#0(3 $0"*-*5 !"#$ #$
/#$07$$*/ #1 $*0,#&1 [5R5M5
G1 ,"* /#$07$$#&1 (%&8* s 2*1*+#0(339 +*.*+$ ,& %&," l (1/ l 45 G1 ,"* +*0&1$,+704
,#&1 &. l 4 &1* -(9 0"&$* ,& +*0&1$,+70, l (1/ 4 $*)(+(,*39 (1/ 0&-%#1* ,"*- ,&
)+&8#/* ( +*0&1$,+70,#&1 .&+ ,"* )+&/70, l 45 !"*+* (+* $*8*+(3 +*($&1$ .&+ 0"&&$#12
,"#$ ())+&(0"5 I#+$,6 #, #$ 4 (1/ 1&, l 4 ,"(, #$ 0&1$*+8*/ (3&12 )(+0*3 ,+(]*0,&+#*$ =$**
*A7(,#&1 [5MM> (1/ 4 $"&73/ ,"*+*.&+* &%*9 -&1&,&1#0#,9 +*A7#+*-*1,$5 T*10* &1*
0(1 (+27* ,"(, -&1&,&1* +*0&1$,+70,#&1 .710,#&1 !3,*+$ =/#$07$$*/ #1 $*0,#&1 [5R5K>
[ I#1#,*4E&37-* !+(1$)&+, S0"*-*$
KMM
$"&73/ %* ())3#*/ ,& 4 (1/ 1&, l 45 S*0&1/6 ,"* 0&1$#$,*1, 0&7)3#12 &. ,+(0*+$ (1/ (#+
/*1$#,9 *A7(,#&1$ #1 0*334#1,*2+(,*/ $*-#4X(2+(12#(1 $0"*-*$ ($ ;*33 ($ *1$7+#12
-&1&,&1* .&+*0($,$ &. 46 #$ )*+"()$ *($#*+ ;"*1 0"&&$#12 ,"#$ ())+&(0" =N(#+ (1/
X(7+#,@*16 KLML>5
!"* +*0&1$,+70,#&1$ .&+ l (1/ 4 0(1 %* 0&-%#1*/ ,& )+&8#/* ( +*0&1$,+70,#&1
.&+ l 4 %9 $#-)39 -73,#)39#12 ,"* +*0&1$,+70,#&1 .710,#&1$ .&+ l (1/ 4 ($ /&1* #1
B7'&;#0@ (1/ \(7-2(+/1*+ =KLLL>5 T&;*8*+6 #1 /&#12 $& -($$40&++*0,#&1 ,*+-$
-(9 %* 1**/*/ ,& $(,#$.9 =[5K[> .&+ "#2"*+4&+/*+ +*0&1$,+70,#&1$5 !"* /&;1$#/* &.
,"#$ ())+&(0" #$ ,"(, #.6 .&+ *?(-)3*6 ,"* +*0&1$,+70,#&1 .710,#&1 .&+ l (1/ 4 (+* )&394
1&-#(3$ &. $," (1/ L," &+/*+ ,"* )+&/70, ;#33 %* )&391&-#(3$ &. ($ + L)," &+/*+ ;"#0"
-(9 %* 0&-)7,(,#&1(339 #1,*1$#8* ,& #1,*2+(,*5 H1* -(9 $#-)3#.9 %9 +*-&8#12 $&-*
,*+-$ .+&- ,"* )+&/70, ($ /&1* #1 N(#+ (1/ X(7+#,@*1 =KLML>5 !"* 3(,,*+ (3$& .(0#3#4
,(,*$ +*1/*+#12 $0"*-*$ -&1&,&1* #1 45 G1 F73*+#(1 $0"*-*$6 /#$07$$*/ 1*?,6 ,+(0*+
-#?#12 +(,#& (1/ (#+ /*1$#,9 (+* 7$7(339 +*0&1$,+70,*/ $*)(+(,*39 .&+ $7%40903#12 =$**
$*0,#&1 [5h5K>
8.4.2 Eulerian scheme
D&1,+(+9 ,& ,"* X(2+(12#(1 /*+#8(,#&1$ #1 ,"* )+*8#&7$ $*0,#&16 ,"* *A7(,#&1$ &.
-&,#&1 (+* ,9)#0(339 /*+#8*/ #1 F73*+#(1 .&+-5 G1 ,"* 0&1,*?, &. ,"* !1#,*48&37-*
.&+- &. ,"* 0&1,#17#,9 *A7(,#&1 ,"* F73*+#(1 ())+&(0" '**)$ ,+(0' &. ,"* "7? &.
-($$ ,"+&72" ,"* F73*+#(1 0*33 ;(33$ +(,"*+ ,"(1 ,+(0'#12 ,"* -($$ #1 ( 0*33 -&8#12
;#," ,"* "&;5 : -&+* .&+-(3 /*+#8(,#&1 #$ 2#8*1 %*3&;5
I#+$,6 #1,*2+(,* =[5O> &+ =[5ML> #1 $)(0* &8*+ ( 2+#/ 0*33 & C
&
&C
,s
%& +
,?
&
&C
¢ · (s v) %& = L, ;"*+* s = l , l 4.
=[5KO>
H1 #1,*2+(,#12 ,"* !+$, ,*+- &1 ,"* 3*.,4"(1/ $#/* &. =[5KO> ,& 2*, ,"* (+*( (8*+(2*
(1/ ())39#12 ,"* /#8*+2*10* ,"*&+*- ,& ,"* $*0&1/ ,*+- ;* 2*,
%
(s 6 &C ) +
%? C
+
, &C
(s v) · n %0 = L,
=[5ZL>
;"*+* , &C #$ ,"* %&71/(+9 &. & C (1/ n ,"* &7,;(+/ 1&+-(3 8*0,&+ ,& , & C 5 !"*
$*0&1/4,*+- &1 ,"* 3*.,4"(1/ $#/* &. =[5KO> +*)+*$*1,$ ,"* #1$,(1,(1*&7$ "7? &. -($$
,"+&72" ,"* %&71/(+#*$ &. & C 5 !*-)&+(3 #1,*2+(,#&1 &. =[5ZL> &8*+ &1* ,#-*4$,*)
9#*3/$
,
)
s )+M
C 6 &C = s C 6 &C −
& ()+M) 6 ? +
)6 ?
, &C
(s v) · n %0 %?,
=[5ZM>
(.,*+ +*4(++(12#12 ,*+-$5 !"* $*0&1/ ,*+- &1 ,"* +#2",4"(1/ $#/* &. =[5ZM> #$ ,"* "7?
&. -($$ ,"+&72" ,"* ;(33$ &. & C /7+#12 &1* ,#-*4$,*)5 : 2+()"#0(3 #337$,+(,#&1 &. ,"*
"7?*$ #$ 2#8*1 #1 I#27+* [5h (1/ /#$07$$*/ #1 ,"* 1*?, )(+(2+()"5
KMK
X(7+#,@*1 *, (35
(a)
(b)
:'
( 'o!"
( 'o!%
(c)
:'
(d)
:'
:'
( 'o!#
( 'o!$
:'ïM
Fig. 8.7 : 2+()"#0(3 #337$,+(,#&1 &. ,"* `"7?4(+*($Q ($$&0#(,*/ ;#," F73*+#(1 0*33 &C =(+*( #1 ,"*
7))*+ +#2", 0&+1*+ &. *(0" )3&, %&71/*/ %9 ,"#0' 3#1*$>5 I&+ *(0" 8*+,*? &. 0*33 &C =!33*/ 0#+03*$>
,"* 7)$,+*(- ,+(]*0,&+#*$ (+* $"&;1 =07+8*/ (++&;$ /*)(+,#12 .+&- &)*1 0#+03*$>5 !"* $"(/*/ (+*($
$"&; ,"* "7?4(+*($ .&+ ,"* =(> *($, "Co =M 6 =%> 1&+," "Co =K 6 =0> ;*$, "Co =Z (1/ =/> $&7," "Co =P .(0*6
+*$)*0,#8*396 7$#12 $,(1/(+/ 0&-)($$ &+#*1,(,#&15 !"*$* (+*($ (+* $;*), ,"+&72" *(0" .(0* /7+#12
&1* ,#-*4$,*)5 S** ,*?, .&+ /*,(#3$5
X*, o /*1&,* ,"* .(0* 17-%*+ (1/ . < ,"* 17-%*+ &. .(0*$ &. ,"* 0*33$5 I&+ $#-)3#04
#,9 ;* ($$7-* ( A7(/+#3(,*+(3 -*$" . < = P6 "&;*8*+6 ,"* -*,"&/ 0(1 (00&--&/(,*
(19 '#1/ &. -*$" =.&+ *?(-)3*6 .&+ ( ,+#(1273(+ (1/ "*?(2&1(3 -*$" . < ;&73/ %* Z
(1/ a6 +*$)*0,#8*39>5 : 2+()"#0(3 #337$,+(,#&1 &. ,"* "7?*$ ,"+&72" ,"* 0*33 ;(33$ .&+
F73*+#(1 0*33 C (+* $"&;1 &1 I#25 [5h5 :$ ;#33 %*0&-* 03*(+6 ,"* I#27+* (3$& $"&;$
,"* 7)$,+*(- X(2+(12#(1 0*33 (3,"&72" #, #$ 1&, *?)3#0#,39 1**/*/ .&+ "7? 0&-)7,(4
,#&1$5 !"* $#/*$ &. ,"* F73*+#(1 0&1,+&3 8&37-* (+* 17-%*+*/ 0&71,*+403&0';#$* $&
,"(, $#/*$ o = M, K, Z, P 0&++*$)&1/ ,& ,"* *($,6 1&+,"6 ;*$, (1/ $&7," ;(33$6 +*$)*04
,#8*39 =7$#12 $,(1/(+/ 0&-)($$ 1&,(,#&1>5 !"* "7? &. -($$ ,"+&72" ,"* $#/* o = M
0&++*$)&1/$ ,& ,"* -($$ &8*+ ,"* $"(/*/ (+*( &1 I#25 [5h( ,"(, #$ `$;*),Q ,"+&72"
,"* ;(33 /7+#12 &1* ,#-* $,*)5 !"* $"(/*/ (+*(6 +*.*++*/ ,& ($ " Co =M 6 #$ %&71/*/ %9
,"* .(0* o = M6 ,"* ,;& 7)$,+*(- ,+(]*0,&+#*$ .&+ ,"* *1/ )&#1,$ &. .(0* o = M6 (1/
,"* 7)$,+*(- ,+(1$3(,#&1 &. ,"* $#/* o = M5 C* ;#33 +*.*+ ,& " Co =M ($ ,"* `"7?4(+*(Q
.&+ .(0* o = M5 S#-#3(+396 ,"* "7?*$ ,"+&72" ,"* +*-(#1#12 0*33 $#/*$ (+* #337$,+(,*/
#1 I#25 [5h%4/5
_$#12 ,"* 1&,(,#&1 #1,+&/70*/ (%&8* =[5ZM> 0(1 %* ;+#,,*1 ($
.<
s )+M
6 &C = s )C 6 &C − - SCo ,
C
o =M
;"*+* SCo #$ ,"* "7? &. -($$ ,"+&72" .(0* o /7+#12 &1* ,#-*4$,*)
=[5ZK>
[ I#1#,*4E&37-* !+(1$)&+, S0"*-*$
KMZ
SCo = ;oC
&
"oC
s ) (B, J) %&.
=[5ZZ>
!"* `"&;4/#+*0,#&1Q .710,#&1 ; oC #$ 7$*/ ,& #1/#0(,* #1"&; (1/ &7,"&;
;oC = $21(v · n),
=[5ZP>
;"*+* $21(·) #$ ,"* $#214.710,#&15 T*10* ; oC #$ M .&+ &7,"&; (1/ 4M .&+ #1"&; Ma 5 G1
I#27+* [5h ,"* "&;4/#+*0,#&1 .710,#&1 ; oC #$ M .&+ o = M, K (1/ 4M .&+ o = Z, P5
N&,* ,"(, ,"* "7? &. -($$ ,"+&72" &1* 0*33 ;(33 #$ #/*1,#0(36 %7, ;#," &))&$#,*
$#216 ,& ,"* "7? &. -($$ ,"+&72" ,"* 1*#2"%&+#12 0*33 ,"(, #, $"(+*$ ( .(0* ;#,"5 I&+
,"* *?(-)3* &1 I#27+* [5h/
o =K
,
=[5ZR>
"Co =P = "C−M
;"*+* ,"* 0*33 3&0(,*/ #--*/#(,*39 ,& ,"* $&7," &. ,"* F73*+#(1 0*33 & C #$ &C−M 5 S&
#1 ( )+(0,#0(3 #-)3*-*1,(,#&1 &. ( $0"*-* %($*/ &1 =[5ZK> &139 ,;& "7?*$ )*+ 0*33
(+* 0&-)7,*/ #. . < = P5
:3,"&72" ,"* $0"*-* &7,3#1*/ (%&8* #$ ,*+-*/ `F73*+#(1Q #, #$ 1&, F73*+#(1 #1 ,"*
03($$#0(3 $*1$* ;"*+* ,"* $)(0* (1/ ,#-* /#-*1$#&1$ (+* $*)(+(,*/5 G1 &,"*+ ;&+/$6
,"* $0"*-* &7,3#1*/ (%&8* 0&73/ (3$& %* ,*+-*/ "7?4.&+- $*-#4X(2+(12#(1 $#10*
"7?4(+*($ ,"(, -&8* ;#," ,"* "&; (+* ,+(0'*/ =`+*-()4,9)*Q $0"*-*>5 G, #$ F73*+#(1
#1 ,"* $*1$* ,"(, ;* 0&1$#/*+ ,"* "7? &. -($$ ,"+&72" ,"* =$,(,#&1(+9 &+ F73*+#(1>
0*33 ;(33$5 C"*1 $*)(+(,#12 ,"* ,*-)&+(3 (1/ $)(,#(3 /#-*1$#&1$6 ($ /&1* #1 03($$#0(3
F73*+#(1 $0"*-*$6 ,"*+* (+* 1& ,+(]*0,&+9 0(3073(,#&1$ (1/ "7?*$ (+* 0&-)7,*/ 7$#12
3&0(3 #1.&+-(,#&1 (1/ )(+,#(3 /*+#8(,#8*$ (3&12 ,"* 0&&+/#1(,* /#+*0,#&1$ (, $)*0#!0
,#-*$5 !"* ,*-)&+(3 /#$0+*,#@(,#&1 #$ 7$7(339 %($*/ &1 Y712(4d7,,( -*,"&/$ =$**
0"(),*+ a>5 H1* -(9 (+27* ,"(, ,"* 03($$#0(3 F73*+#(1 $0"*-*$ (+* (1 ())+&?#-(,#&1
&. ,"* 2*1*+(3 F73*+#(14X(2+(12#(1 0&10*), )+*$*1,*/ #1 ,"#$ 0"(),*+ ;"*+* ,+7*
=(3&12 ,"* ,+(]*0,&+#*$> "7?*$ (+* ())+&?#-(,*/ ;#," )(+,#(3 "7?*$ =#5*56 ,"* )(+,#03*
)(," 8*0,&+ 0(1 %* /*0&-)&$*/ #1,& 8*0,&+ 0&-)&1*1,$ (3&12 ,"* 0&&+/#1(,* (?*$>5
8.4.3 Equivalence between the Lagrangian and Eulerian
discretizations
G, #$ #1,*+*$,#12 ,& 1&,* ,"* *A7#8(3*10* %*,;**1 ,"* X(2+(12#(1 !1#,*48&37-* 0&1,#4
17#,9 *A7(,#&1 =[5Ka> (1/ ,"* F73*+#(1 8*+$#&1 =[5ZK>^ G. ,('#12 ,"* $7- &. ,"* "7?4
(+*($ "oC ;#," ;*#2", M .&+ &7,"&; (1/ ;*#2", 4M .&+ #1"&; ($ ;*33 ($ & C ;#," ;*#2",
M =(33 (+*($ #18&38*/ &1 ,"* +#2",4"(1/ $#/* &. =[5ZK>>6 ,"* 7)$,+*(- X(2+(12#(1 (+*(
"C +*$73,$ =$** *?(-)3* &1 I#25 [5h>5 !"(, #$6 ,"* +#2",4"(1/ $#/* &. =[5ZK> ;+#,,*1 #1
,*+-$ &. (+*($ #$
Ma
.&+ $#-)3#0#,9 ;* /& 1&, 0&1$#/*+ ,"* $#,7(,#&1 ;"*+* ;oC #$ -73,#48(37*/ (3&12 ( )(+,#073(+ .(0*5
I&+ -&+* /*,(#3$ &1 $70" ( $#,7(,#&1 $** T(++#$ *, (3 =KLML>
KMP
X(7+#,@*1 *, (35
.<
6 &C − - (;oC 6 "oC ) = 6 "C .
=[5Za>
o =M
S& ,"* X(2+(12#(1 (1/ F73*+#(1 $0"*-*$ (+* #/*1,#0(36 ($ *?)*0,*/6 $#10* 1& ())+&?4
#-(,#&1$ "(8* %**1 -(/* $& .(+ =*8*1 #. ())+&?#-(,#&1$ (+* -(/* (1/ ,"* +*$73,#12
$0"*-*$ (+* ())3#*/ ,& ,"* F73*+ *A7(,#&1$6 F73*+#(1 (1/ $*-#4X(2+(12#(1 $0"*-*$
-(9 )+&/70* 8*+9 $#-#3(+ +*$73,$ ($ $"&;1 #1 X*$3#* (1/ B#*,(0"-(9*+6 MOOh>5 G14
$#2",$ #1,& $0"*-*$ 0(1 %* &%,(#1*/ #1 ,"* 3#2", &. ,"* *A7#8(3*10* /*$0+#%*/ (%&8*5
:19 F73*+#(1 "7?4.&+- $0"*-* $"&73/ #/*(339 (1/ *..*0,#8*39 "(8* (1 ($$&0#(,*/
7)$,+*(- 0*33 .+&- ;"#0" #1.&+-(,#&1 #$ .*,0"*/ =(5'5(5 /&-(#1 &. /*)*1/*10*> ,&
)+&/70* ,"* .&+*0($,5 : -&+* /*,(#3*/ /#$07$$#&1 #$ 2#8*1 #1 $*0,#&1 [5R5M5K5
: $#21#!0(1, /#..*+*10* %*,;**1 ,"* X(2+(12#(1 (1/ F73*+#(1 .&+-73(,#&1 #$ ,"*
1*0*$$(+9 0&1/#,#&1$ .&+ -($$ 0&1$*+8(,#&15 W#8*1 ( -($$40&1$*+8(,#8* +*0&1$,+704
,#&1 .710,#&1 =[5K[>6 ( 1*0*$$(+9 0&1/#,#&1 .&+ ,"* X(2+(12#(1 $0"*-* ,& %* 0&1$*+4
8(,#8* #$ ,"(, ,"* 7)$,+*(- (+*($ " C $)(1 ,"* /&-(#1 1 ;#,"&7, &8*+3() (1/ 2()$ %*4
,;**1 ,"*- =[5KM> (1/ ,"(, ,"* +*0&1$,+70,#&1 .710,#&1 #$ -($$40&1$*+8(,#8* =[5KZ>5
I&+ ,"* F73*+#(1 $0"*-*6 "&;*8*+6 ,"* "7?4(+*($ " oC 1**/ 1&, 1*0*$$(+#39 ,& $)(1 ,"*
/&-(#1 1 (1/ ,"* +*0&1$,+70,#&1 .710,#&1 /&*$ 1&, 1**/ ,& $(,#$.9 =[5KZ> ,& )+&4
/70* ( -($$40&1$*+8(,#8* $0"*-*5 G1 .(0, (19 *$,#-(,#&1 &. ,"* "7? ;#33 )+&8#/* (1
#1"*+*1,39 -($$40&1$*+8(,#8* $0"*-* $#10* ,"* "7? 0&-)7,*/ .&+ ( )(+,#073(+ 0*33
;(33 #$ $7%,+(0,*/ #1 ,"* 1*#2"%&+#12 0*33 ;#," ;"#0" #, $"(+*$ ,"(, )(+,#073(+ .(0*5
S& ,"* X(2+(12#(1 $0"*-* "($6 #1 ,"* $*1$* /*$0+#%*/ (%&8*6 ( $,+#0,*+ +*A7#+*-*1,
.&+ -($$40&1$*+8(,#&1 ,"(1 ,"* F73*+#(1 "7?4.&+- .&+-73(,#&15
:1&,"*+ $#21#!0(1, /#..*+*10* %*,;**1 ,"* F73*+#(1 (1/ X(2+(12#(1 .&+-73(4
,#&1$ #$ ,"(, ,"* X(2+(12#(1 .&+-73(,#&1 +*A7#+*$ ,"* 7)$,+*(- (+*($ ,& %* $#-)394
0&11*0,*/ /&-(#1$5 !"* F73*+#(1 .&+-73(,#&1 /&*$ 1&, +*A7#+* ,"(,6 #1 .(0,6 *8*1 .&+
+*3(,#8*39 $#-)3* "&;$ ,"* "7?4(+*($ 0(1 %* 1&14$#-)39 0&11*0,*/ =$**6 *5256 I#25K
#1 T(++#$ *, (3 KLML>5 !"* F73*+#(1 .&+-73(,#&1 #$ ,"*+*.&+* -&+* +&%7$, #1 ,"* $*1$*
,"(, #, 0(1 "(1/3* 1&14$#-)39 0&11*0,*/ "7?4(+*($ =(1/ 0&1$*+8* -($$ $#-73,(1*4
&7$39> ;"*+*($ ,"* X(2+(12#(1 $0"*-* ;#33 %+*(' /&;1 #. (1 7)$,+*(- (+*( #$ 1&,
$#-)39 0&11*0,*/5 !"#$ /#..*+*10* 0&73/ %* #-)&+,(1, .&+ (1 &)*+(,#&1(3 ())3#0(,#&1
&. ,"* $0"*-*5
8.5 Discrete schemes: Approximations
!"* X(2+(12#(1 (1/ F73*+#(1 !1#,*48&37-* $0"*-*$6 2#8*1 #1 =[5Ka> (1/ =[5ZK> +*4
$)*0,#8*396 (+* *?(0,5 T*10* ;* ($$7-* ,"* ,+(]*0,&+9 &. *8*+9 )(+0*3 #$ '1&;1 *?4
(0,39 =,"* *?(0, 7)$,+*(- (+*( (1/ "7?4(+*($ (+* '1&;1>6 ,"* $7%42+#/40*33 +*0&14
$,+70,#&1 #$ *?(0, (1/ ,"* #1,*2+(,#&1 &. ,"* $7%42+#/40*33 +*0&1$,+70,#&1 .710,#&1
&8*+ ,"* 7)$,+*(- (+*($ (1/ "7?4(+*($ 0(1 %* /&1* (1(39,#0(3395 N&; ;* $,(+, ,&
/#$07$$ $&-* &. ,"* ())+&?#-(,#&1$ ,"(, 0(1 %* -(/* #1 &+/*+ ,& /*+#8* )+(0,#0(3
17-*+#0(3 $0"*-*$ ,"(, &139 "(8* ( !1#,* 17-%*+ &. /*2+**$ &. .+**/&-5 !"* ()4
)+&?#-(,#&1$ 0(1 %* /#8#/*/ #1,& .&7+ $,*)$^ D&-)7,(,#&1 &. )(+0*3 ,+(]*0,&+#*$6 (+*(
[ I#1#,*4E&37-* !+(1$)&+, S0"*-*$
(a)
KMR
(b)
@4(?#$
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Geometric error
Reconstruction error
Fig. 8.8 : $0"*-(,#0 #337$,+(,#&1 &. ,"* =(> 2*&-*,+#0 *++&+ (1/ =%> +*0&1$,+70,#&1 =2+(/#*1,> *++&+6
+*$)*0,#8*396 .&+ ( 0*33 #1 ,;& /#-*1$#&1$5 =(> !"* 2*&-*,+#0 *++&+ &007+$ /7* ,& ,"* *?(0, +*2#&1
&. #1,*2+(,#&1 =$"(/*/ (+*(> %*#12 ())+&?#-(,*/ %96 .&+ *?(-)3*6 $,+(#2", 3#1* $*2-*1,$ =/($"*/
3#1*$>5 =%> !"* +*0&1$,+70,#&1 *++&+ +*.*+$ ,& ,"* 17-*+#0(3 -*,"&/$ #1(%#3#,9 ,& +*0&1$,+70, ,"*
*?(0, $7%42+#/4$0(3* 8(+#(,#&1 =%3(0' 3#1* $7+.(0*>5 !"* 2+*9 3#1*$ 0&1,&7+ ,"* +*0&1$,+70,*/ $7%4
2+#/4$0(3* /#$,+#%7,#&1 =#1 ,"#$ 0($* ( 3#1*(+ ())+&?#-(,#&1>5
())+&?#-(,#&1 =*#,"*+ 7)$,+*(- X(2+(12#(1 (+*($ &+ F73*+#(1 "7?4(+*($>6 $7%42+#/4
0*33 +*0&1$,+70,#&1 (1/ #1,*2+(,#&1 &. s ) (B, J) &8*+ /*.&+-*/ (+*($5 :$ (3+*(/9 -*14
,#&1*/ ;* ;#33 1&, /#$07$$ ,"* 0&-)7,(,#&1 &. ,+(]*0,&+#*$ "*+* (1/ ,"*+*.&+* $#-)39
($$7-* ,"(, ,"*9 (+* 2#8*15
I#+$,396 ,"* ())+&?#-(,#&1 ,& (+*($ (+* /#$07$$*/5 H10* ,"* (+*($ "(8* %**1 /*4
!1*/6 ,"* ,+(1$)&+, )+&%3*- "($ %**1 +*/70*/ ,& ( +*-())#12 )+&%3*-6 ,"(, #$6 (
0&1$*+8(,#8* 2+#/4,&42+#/ #1,*+)&3(,#&1 )+&%3*-5 !"#$ +*A7#+*$ ( +*0&1$,+70,#&1 &.
,"* $7%42+#/40*33 /#$,+#%7,#&1 (1/ (1 #1,*2+(,#&1 &8*+ &8*+3() (+*($5 !"*$* ,"+** $,*)$
=(+*( ())+&?#-(,#&16 +*0&1$,+70,#&16 #1,*2+(,#&1 &8*+ &8*+3() (+*($> (+* /#$07$$*/
$*)(+(,*39 %*3&;5
8.5.1 Approximation to areas
C#," &139 ( !1#,*417-%*+ &. /*2+**$ &. .+**/&- (1/ ,"*+*.&+* &139 "(8#12 ,"* 0(4
)(%#3#,9 &. ,+(0'#12 ( !1#,* 17-%*+ &. )(+0*3$ =,9)#0(339 ,"* $(-* 17-%*+ ($ 0*33$ .>
$&-* ())+&?#-(,#&1 -7$, %* -(/* ,& ,"* *?(0, 7)$,+*(- X(2+(12#(1 (+*( &+ F73*4
+#(1 "7?4(+*(5 !"* #1(%#3#,9 &. ,"* $0"*-* ,& ())+&?#-(,* ,"* *?(0, (+*($ #$ +*.*++*/
,& ($ ,"* E-36-?:$2 -::3: =X(7+#,@*1 (1/ N(#+6 KLL[> (1/ #$ #337$,+(,*/ 2+()"#0(339
&1 I#25 [5[(5 H%8#&7$39 ,"* 2*&-*,+#0 *++&+ -(9 3*(/ ,& 3&0(3 -($$ *++&+$5 :1&,"*+
*++&+ #$ /7* ,& #1*?(0, $7%42+#/40*33 +*0&1$,+70,#&15 !"#$ *++&+6 +*.*++*/ ,& ($ ,"* :-K
23);?:52?$3) -::3:6 #$ #337$,+(,*/ &1 I#25 [5[% (1/ /#$07$$*/ .7+,"*+ #1 $*0,#&1 [5R5K5
S,+(,*2#*$ .&+ (+*( ())+&?#-(,#&1$ (+* ,"* $7%]*0, &. ,"#$ $*0,#&15
KMa
X(7+#,@*1 *, (35
(a)
(b)
(c)
(d)
Fig. 8.9 W+()"#0(3 #337$,+(,#&1 &. ())+&?#-(,#&1$ ,& ,"* 7)$,+*(- X(2+(12#(1 0*33 "C (5'5(5 ,"*
/*)(+,7+* 0*335 :$$7-* ,"* /*)(+,7+* )&#1,$ 0&++*$)&1/#12 ,& ,"* 8*+,#0*$ &. ,"* F73*+#(1 2+#/ 0*33
(+* '1&;1 =&)*1 0#+03*$>5 =(> F?(0, /*)(+,7+* 0*33 =$"(/*/ (+*(> ;#," $#/*$ /*)#0,*/ ;#," ,"#0'
3#1*$5 =%> S#/*$ &. ,"* /*)(+,7+* 0*33 ())+&?#-(,*/ ;#," $,+(#2", 3#1*$ %9 0&11*0,#12 ,"* /*)(+,7+*
)&#1,$5 =0> B*)(+,7+* 0*33 ())+&?#-(,#&1 7$*/ #1 N(#+ (1/ U(0"*1"(7*+ =KLLK> ;"*+* ,"* *($, (1/
;*$, $#/*$ (+* $,+(#2", 3#1*$ )(+(33*3 ,& ,"* F73*+#(1 3&12#,7/*$ =J4(?#$ &1 ,"* )3&,> (1/ ,"* 1&+,"
(1/ $&7," $#/*$ (+* ())+&?#-(,*/ ;#," `$,*) .710,#&1$Q5 =/> !"* X(2+(12#(1 0*33 7$*/ #1 ,"* 0($0(/*
$0"*-*$ ,"(, (+* %($*/ &1 #1,*+$*0,#&1$ =0+&$$*$> %*,;**1 ,"* X(2+(12#(1 3(,#,7/*$ =/($"*/V$&3#/
07+8*/ 3#1*$> (1/ ,"* F73*+#(1 3&12#,7/*$5 !"* `$,*)Q #1 ,"* $,*) .710,#&1$ 7$*/ #1 ,"* 0($0(/*
$0"*-*$ (3;(9$ 0&#10#/*$ ;#," ,"* F73*+#(1 3&12#,7/*$ =B4#$&3#1*$ &1 ,"* I#27+*>5
8.5.1.1 Lagrangian area approximations
S5**J ?D3K%$6-);$3)"* G"E:")E$") ":-" "@@:3B$6"?$3);
<+&%(%39 ,"* -&$, +#2&+&7$ ())+&?#-(,#&1 ,& ,"* *?(0, 7)$,+*(- 0*33 " C 6 I#27+* [5O(6
#$ ,& .&33&; ,"* ,+(]*0,&+#*$ &. ,"* 8*+,#0*$ &. & C 7)$,+*(- (1/ ,"*1 0&11*0, ,"* 7)4
$,+*(- 8*+,#0*$ ;#," $,+(#2", 3#1*$ =I#25 [5O%>o =Y(1#0#s0 MOOK6 X(7+#,@*1 *, (3 KLML>5
[ I#1#,*4E&37-* !+(1$)&+, S0"*-*$
KMh
:33 &,"*+ ())+&?#-(,#&1$ #18&38* ())+&?#-(,#12 " C ;#," 3#1* $*2-*1,$ )(+(33*3 ,&
,"* 0&&+/#1(,*$ (?#$ ;"#0"6 #1 2*1*+(36 $#-)3#!*$ ,"* &8*+3()4(+*( #1,*2+(,#&1 (32&4
+#,"-5 S&-* *?(-)3*$ (+* 2#8*1 &1 I#25 [5O5 I&+ -&+* /*,(#3$ &1 X(2+(12#(1 0*33
())+&?#-(,#&1$ .&+ &+,"&2&1(3 -*$"*$ $** ,"* 0&-)+*"*1$#8* +*8#*; %9 X(7+#,@*1
*, (3 =KLLa> (1/ U(0"*1"(7*+ *, (3 =KLLO>5
S*3DK;@*$? G"E:")E$") ":-" "@@:3B$6"?$3);
U&+* +*0*1,39 ,"* !1#,*48&37-* 0($0(/* Mh ())+&(0" ;($ $722*$,*/ %9 N(#+ *, (3
=KLLK> (1/ f*++&7'(, *, (3 =KLLK> ;"#0" 7$*$ ( 0&-%#1(,#&1 &. F73*+#(1 (1/ X(4
2+(12#(1 &)*+(,&+$6 ,"(, #$6 ,"* &1*4/#-*1$#&1(3 &)*+(,&+$ (+* $700*$$#8*39 ())3#*/
(3&12 ( 0&&+/#1(,* 3#1* (1/ ( X(2+(12#(1 3#1*6 +*$)*0,#8*395 :1 *?(-)3* #$ 2#8*1
#1 I#25 [5O/ ;"*+* ,"* !+$, &1*4/#-*1$#&1(3 &)*+(,&+ #$ ())3#*/ (3&12 ,"* F73*+#(1
3&12#,7/#1(3 /#+*0,#&1 (1/ ,"* $*0&1/ #$ ())3#*/ (3&12 ,"* /*.&+-*/ X(2+(12#(1 3(,4
#,7/* =07+8*/ $&3#/V/($" 3#1*$ &1 I#27+* [5O/>5 S& +(,"*+ ,"(1 %*#12 ( !?*/ /#+*0,#&1
%($*/ $)3#,,#12 -*,"&/ #, #$ "&;4%($*/ =.&+ ( +*8#*; $** U(0"*1"(7*+ *, (3 KLLO>5
!"* 7)$,+*(- X(2+(12#(1 0*33 .&+ ,"* 0($0(/* $0"*-* #$ #337$,+(,*/ &1 I#27+* [5O/5
!"* -(#1 /#..*+*10* %*,;**1 ,"* .7339 ,;&4/#-*1$#&1(3 (+*( ())+&?#-(,#&1 7$*/ #1
N(#+ (1/ U(0"*1"(7*+ =KLLK>6 $"&;1 &1 I#27+* [5O06 (1/ ,"* 0($0(/* $0"*-* (+*(
())+&?#-(,#&16 #$ ,"* 3&0(,#&1 &. ,"* `]7-)Q #1 ,"* 1&+," (1/ $&7," $#/*$ &. ,"* /*4
)(+,7+* 0*335 S#10* ,"* !+$, 0($0(/* `$;**)Q #$ (3&12 F73*+#(1 3&12#,7/*$ ,"* ]7-)
#1 ,"* 1&+," (1/ $&7," $#/*$ 0&#10#/* ;#," (1 F73*+#(1 3&12#,7/*5 G1 ,"* N(#+ (1/
U(0"*1"(7*+ =KLLK> ,"* ]7-) #$ 3&0(,*/ -#/;(9 %*,;**1 ,"* *($, (1/ ;*$, 0*33
$#/*$5
:))+&?#-(,#12 ,"* X(2+(12#(1 0*33 ;#," 3#1*4$*2-*1,$ )(+(33*3 ,& ,"* 0&&+/#1(,*
(?#$6 *#,"*+ ;#," .7339 ,;&4/#-*1$#&1(3 &+ 0($0(/* -*,"&/$6 #$ (,,+(0,#8* .&+ &+,"&24
&1(3 2+#/$ $70" ($ ( D(+,*$#(1 +*0,(1273(+ -*$" =*5256 f*++&7'(, *, (3 KLLK> (1/ (
+*273(+ 3(,#,7/*43&12#,7/* 2+#/ &1 ,"* $)"*+* =*5256 N(#+ (1/ U(0"*1"(7*+ KLLK6 N(#+
*, (3 KLLK6 f*++&7'(, *, (3 KLLP>5 G, #$ 3*$$ &%8#&7$ "&; ,& *?,*1/ $70" ())+&(0"*$
,& 1&14&+,"&2&1(3 2+#/$ $70" ($ ,+#(1273(+ &+ "*?(2&1(3 2+#/$ $#10* ,"* 0*33 $#/*$ (+*
1& 3&12*+ &+,"&2&1(35
8.5.1.2 Eulerian flux area approximations
!"* ())+&?#-(,#&1 ,& "7?4(+*($ #1 F73*+#(1 $0"*-*$ 0(1 %* /#8#/*/ #1,& ,;& 0(,*4
2&+#*$^ I7339 ,;&4/#-*1$#&1(3 ())+&?#-(,#&1$ ,& ,"* "7?4(+*($ (1/ /#-*1$#&1(339
$)3#, (+*( ())+&?#-(,#&1$5 C* +*-#1/ ,"* +*(/*+ ,"(, &139 -*,"&/$ ,"(, "(8* %**1
*?,*1/*/ ,& 23&%(3 $)"*+#0(3 /&-(#1$ (+* /#$07$$*/ "*+*5 C* (+* ,"*+*%9 *?037/#12
-(19 ,+(1$)&+, $0"*-*$ )7%3#$"*/ #1 ,"* -*,*&+&3&2#0(3 3#,*+(,7+*5
Mh ,"* 1&140&1$*+8(,#8* 0($0(/* #1,*+)&3(,#&1 -*,"&/ #1 D(+,*$#(1 2*&-*,+9 ;($ #1,+&/70*/ %9
<7+$*+ (1/ X*$3#* =MOOM>
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Fig. 8.10 : $0"*-(,#0 #337$,+(,#&1 &. /#..*+*1, "7? ())+&?#-(,#&1$6 )(+(33*3&2+(- =(6%6/6*> (1/
A7(/+#3(,*+(3 =26"6]6'> "7?4(+*($6 (1/ ,"* *A7#8(3*1, 7)$,+*(- X(2+(12#(1 (+*($ =.63>5 !"* *A7#8(3*1,
7)$,+*(- (+*($ (+* 0&-)7,*/ %9 ,('#12 ,"* $7- &. (33 (+*($ #18&38*/ #1 ,"* .&+*0($, =(6%606/6*>
&+ =26"6#6]6'> ;#," ())+&)+#(,* $#21$ =$** *A7(,#&1 =[5Za>>5 !"* 8*3&0#,9 8*0,&+$ 7$*/ .&+ ,"* "7?
0&-)7,(,#&1$ (+* (3$& $"&;15 !"* *?(0, 7)$,+*(- X(2+(12#(1 0*33 =&)*1 0#+03*$ 0&11*0,*/ ;#,"
07+8*/ 3#1*$> #$ (3$& $"&;1 (3,"&72" #, #$ 1&, *?)3#0#,39 7$*/ #1 ,"* "7?4.&+- $0"*-*$5
S5**J ?D3K%$6-);$3)"* #5BK":-" "@@:3B$6"?$3);
!"* .7339 ,;&4/#-*1$#&1(3 "7?4(+*( ())+&?#-(,#&1$ 0(1 %* /#8#/*/ #1,& ,;& 0(,*4
2&+#*$5 I#+$,396 &1* #1 ;"#0" &1* .(0*40*1,*+*/ 8*3&0#,9 8*0,&+ )*+ .(0* #$ 7$*/ ,&
,+(0* %(0' ,"* "7?4(+*( (1/6 $*0&1/396 ,"* ())+&(0" #1 ;"#0" ,"* 8*+,#0*$ &. ,"* .(0*
(+* ,+(0*/ 7)$,+*(- ,& 0&-)7,* ,"* "7?4(+*(5 !"* !+$, ())+&(0" &139 "($ &1* /*2+**
&. .+**/&- .&+ ,"* "7?4(+*($ ;"*+*($ ,"* 3(,,*+ ())+&(0" "($ ,;&5 D&1$*A7*1,39 ,"*
+*$73,#12 "7?4(+*($ (+* )(+(33*3&2+(-$ (1/ (+%#,+(+9 A7(/+#3(,*+(3$6 +*$)*0,#8*396 .&+
,"* ,;& ())+&(0"*$5 :1 *3(%&+(,#&1 #$ 2#8*1 %*3&;5
Y*0*1,396 U#7+( =KLLh> $722*$,*/ ,& ())+&?#-(,* ,"* "7?4(+*($ .+&- ( .(0*4
0*1,*+*/ ;#1/ 8*3&0#,95 S& ,"* ,;& 8*+,#0*$ &. ,"* .(0* ;&73/ "(8* #/*1,#0(3 7)4
[ I#1#,*4E&37-* !+(1$)&+, S0"*-*$
KMO
$,+*(- /#$)3(0*-*1,$ %($*/ &1 ,"* $(-* .(0*40*1,*+*/ 8*3&0#,9 8*0,&+5 !"* ,+(]*0,&4
+#*$ (+* ,"*+*.&+* )(+(33*3 (1/ "(8* ,"* *?(0, $(-* 3*12,"5 T*10* ,"* "7?4(+*($ " oC 6
o = M, K, Z, P6 (+* )(+(33*3&2+(-$ =I#25 [5ML(6%6/6*6 +*$)*0,#8*39>5 !"* .(0, ,"(, ,"*
7)$,+*(- (+*( #$ ( )(+(33*3&2+(- -(9 $#-)3#.9 ,"* )+(0,#0(3 #1,*2+(,#&1 &. &8*+3()
(+*($ (, ,"* *?)*1$* &. $&-* )&,*1,#(3 3&$$ &. (007+(09 #. ,"* "&; #$ "#2"39 /*.&+4
-(,#&1(35 !"#$ #$ #337$,+(,*/ %9 0&-)7,#12 ,"* *..*0,#8* 7)$,+*(- X(2+(12#(1 (+*(
.&+ ,"* U#7+( =KLLh> $0"*-* 7$#12 ,"* -*,"&/ &7,3#1*/ #1 $*0,#&1 [5P5Z5 !"(, #$6
%9 ,('#12 ,"* $7- &. ,"* "7?4(+*($ =;#," $#21$> $"&;1 &1 I#27+* [5ML(6%6/6* (1/
,"* F73*+#(1 (+*( =I#25 [5ML0>6 ,"* *..*0,#8* 7)$,+*(- (+*( " C +*$73,$ =I#27+* [5ML.>5
!"* 7)$,+*(- (+*( -&$,39 0&#10#/*$ ;#," ,"* *?(0, /*)(+,7+* 0*336 "&;*8*+6 ,"*+* (+*
-#1&+ 0&1,+#%7,#&1$ ,+(0#12 ,"* F73*+#(1 0*33 8*+,#0*$ ,"(, (+* 1&143&0(3 =1&, &8*+4
3())#12 ;#," ,"* ,+7* /*)(+,7+* 0*33>5 :3$&6 ,"* "7?4(+*($ .&+ (33 0*33$ /& 1&, $)(1
,"* /&-(#1 1 5 G. ,"* "&; #$ 0&1$,(1, =1& /*.&+-(,#&1> ,"* 1&143&0(3 )(+, &. ,"*
"7?4(+*($ /#$())*(+ ($ (33 ,"* .(0*40*1,*+*/ 8*3&0#,9 8*0,&+$ ;&73/ %* (3#21*/5
G. ,"#$ #1(%#3#,9 &. +*)+*$*1,#12 ,"* 3&0(3 "7?4(+*($ =2*&-*,+#0 *++&+> #$ ( $#21#.4
#0(1, $&7+0* &. *++&+ "($ 1&, %**1 #18*$,#2(,*/ =($ .(+ ($ ,"* (7,"&+$ (+* (;(+*> (1/
,"* *++&+ ;&73/ &139 $"&; .&+ 0"(33*12#12 ,*$, 0($*$ ;#," $,+&12 /*.&+-(,#&15 I&+
*?(-)3*6 ,"* ;#/*39 7$*/ $&3#/ %&/9 (/8*0,#&1 ,*$, &1 ,"* $)"*+* ;&73/ -&$, 3#'*39
1&, *?)&$* ,"#$ )&,*1,#(3 /*!0#*1095 :1 #337$,+(,#8* *?(-)3* &. ( "#2"39 /*.&+-(4
,#&1(3 "&; #$ 2#8*1 &1 I#27+* [5MM ,"(, $"&;$ ,"* X(2+(12#(1 =7)$,+*(-> 2+#/ .&+
*(0" 07%*/4$)"*+* )(1*3 .&+ &1* &. ,"* ,*$, 0($*$ #1 N(#+ (1/ X(7+#,@*1 =KLML>5 F8*1
.&+ ( +*3(,#8* $"&+, ,#-*4$,*) =+*$73,#12 #1 ( -(?#-7- DIX 17-%*+ #1 *(0" 0&&+/#4
1(,* /#+*0,#&1 &. ())+&?#-(,*39 L5[> ,"* 7)$,+*(- 0*33$ (+* "#2"39 /*.&+-*/ (1/ ,"*9
-#2", %* 0"(33*12#12 ,& ())+&?#-(,* (007+(,*39 7$#12 $#-)3#!*/ "7?*$ 713*$$ 8*+9
$"&+, ,#-*4$,*)$ (+* 7$*/5 G, $"&73/6 "&;*8*+6 %* 1&,*/ ,"(, ,"* 2*&-*,+#0 /#$07$$*/
(%&8* ;#33 &139 $"&; #. #, #$ 3(+2*+ ,"(1 ,"* +*0&1$,+70,#&1 *++&+5 D&1$A7*1,396 ,"*
2*&-*,+#0 *++&+ #$ -&$, 3#'*39 1&, $#21#!0(1, ;"*1 7$#12 3&;4&+/*+ +*0&1$,+70,#&1
.710,#&1$ =0&1$,(1, &+ 3#1*(+ +*0&1$,+70,#&1$>5
!"* )&,*1,#(3 1&143&0(3#,9 )+&%3*- /*$0+#%*/ (%&8* 0(1 %* +*$&38*/ %9 #1$,*(/
&. 7$#12 &1* .(0*40*1,*+*/ 8*0,&+ =.&+ ,"* ,+(]*0,&+#*$> )*+ .(0*6 ,& 7$* ,+(]*0,&+#*$
.&+ ,"* 8*+,#0*$ &. ,"* 0*33 & C =Y(1#0#s06 MOOKo X#)$0&-% (1/ Y#123*+6 KLLRo c*"6
KLLh>5 !"#$ *?,+( /*2+** &. .+**/&- (33&;$ ,"* "7?4(+*($ ,& /*.&+- #1,& (+%#,+(+9
A7(/+#3(,*+(3$5 !"* *A7#8(3*1, 7)$,+*(- (+*( 1&; *A7(3$ ,"* X(2+(12#(1 (+*( +*4
$73,#12 .+&- 0&11*0,#12 ,"* 7)$,+*(- )&#1,$ ;#," $,+(#2", 3#1*$5 !"#$ 0(1 %* $"&;1
($ (%&8* %9 ,('#12 ,"* $7- &. ,"* (+*($ #18&38*/ #1 ,"* .&+*0($, *A7(,#&1 =[5ZK>6
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G-)+&8#12 ,"* *..*0,#8* ())+&?#-(,#&1 ,& ,"* 7)$,+*(- (+*( .7+,"*+ ;&73/ #14
8&38* ,"* #1,+&/70,#&1 &. -&+* )(+0*3$ ,"(, (+* ,+(0'*/ =($ $722*$,*/ %9 X(7+#,@*1
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(//+*$$#12 ,"* )&,*1,#(3 %*1*!,$ &. ())+&?#-(,#12 ,"* 7)$,+*(- (+*($ ;#," "#2"*+4
KKL
X(7+#,@*1 *, (35
Fig. 8.11 !"* $,(,#0 F73*+#(1 2+#/ =,"#1 3#1*$ (3#21*/ ;#," 0&&+/#1(,* 3#1*$> (1/ /*)(+,7+* 2+#/
=/*.&+-*/ ,"#1 3#1*$> (, ,"* !+$, ,#-*4$,*) $"&;1 &1 ,"* 21&-&1#0 )+&]*0,#&1 &1 *(0" 07%*/4$)"*+*
)(1*3 .&+ ,*$, 0($* M &. N(#+ (1/ X(7+#,@*1 =KLML> #337$,+(,*/ &1 I#25 [5Z =,#-*4$,*) ;($ 0"&$*1 $70"
,"(, ,"* -(?#-7- DIX 17-%*+ #$ ())+&?#-(,*39 L5[>5 !"* /*)(+,7+* 2+#/ "($ %**1 0&1$,+70,*/ %9
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=2+*(,40#+03* (+0$ &1 ,"* $)"*+*>5
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G1 ,"* )+*$*1, /#$07$$#&1 &1 *..*0,#8* 7)$,+*(- (+*($6 ,"#$ &)*+(,&+ $)3#,,#12 ()4
)+&(0" ;($ (1(39@*/ %9 X(7+#,@*1 =KLLh> (1/ U(0"*1"(7*+ *, (3 =KLLO>5 C"*1 7$4
#12 /#-*1$#&1(339 $)3#, ())+&(0"*$ ,"* *..*0,#8* 7)$,+*(- (+*( #$ ())+&?#-(,*/ ;#,"
( 0&-%#1(,#&1 &. +*0,(123*$ (3#21*/ ;#," ,"* 2+#/ 3#1*$ (1/ ;#," /#..*+*1, ;*#2",$
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;#," M5 S** U(0"*1"(7*+ *, (3 =KLLO> .&+ /*,(#3$5
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X(2+(12#(1 $0"*-*$ =!"7%7+1 *, (36 KLML>5
8.5.1.3 Comment on area approximations
H1* -#2", (+27* ,"(, ,"* *++&+$ ($$&0#(,*/ ;#," $&-* &. ,"* $#-)3#!*/ "7?4 (1/
7)$,+*(-4 (+*( ())+&?#-(,#&1$ (+* 1&, $#21#!0(1, (, 3*($, .&+ &+,"&2&1(3 -*$"*$5 I&+
*?(-)3*6 .&+ $*-#4X(2+(12#(1 !1#,*48&37-* $0"*-*$ X(7+#,@*1 *, (3 =KLML> .&71/
3#,,3* /#..*+*10* %*,;**1 ,"* +#2&+&7$ 7)$,+*(- (+*( ())+&?#-(,#&1 (1/ $#-)3*+ (+*(
())+&?#-(,#&1$ 7$#12 3#1*4$*2-*1,$ )(+(33*3 ,& ,"* 0&&+/#1(,* (?#$5
H1 1&14,+(/#,#&1(3 -*$"*$ $#-)3#!*/ "7?*$ -#2", #1,+&/70* $#21#!0(1, #1(0074
+(0#*$5 I&+ *?(-)3*6 0&1$#/*+#12 ( $&3#/4%&/9 +&,(,#&1 "&; !*3/ &1 ,"* $)"*+* &1
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(+* "#2"39 /*.&+-*/ *8*1 ,"&72" ,"* "&; !*3/ #$ 1&14/#8*+2*1,6 1&14/*.&+-(,#&1(3
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( 07%*/4$)"*+* )(1*3 .+&- 1*#2"%&+#12 )(1*3$ (+* "#2"39 $'*;*/ 0&-)(+*/ ,& ,"*
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,;&4/#-*1$#&1(3 (+*( ())+&?#-(,#&1$ .&+ 1&14,+(/#,#&1(3 2+#/ ())3#0(,#&1$ $**-$
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Fig. 8.13 =(> D7%*/4$)"*+* 2+#/ $"&;1 ;#," 3#2", $"(/*/ 3#1*$ (1/ )(1*3 */2*$ ;#," %3(0' 3#1*$5 =%>
!"* 7)$,+*(-V/*)(+,7+* 2+#/ =/($"*/ 3#1*$> $"&;1 &1 ( 3&0(3 =21&-&1#0> )+&]*0,#&1 .&+ &1* &. ,"*
07%*/4$)"*+* )(1*3$ 7$#12 ,"* $&3#/4%&/9 (/8*0,#&1 "&; !*3/ =,#-*4$,*) #$ $& ,"(, &1* +*8&37,#&1
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0*33$ (+* 0*33$ *1,*+#12 .+&- 1*#2"%&+#12 )(1*3$ /7+#12 ,"* ,#-*4$,*)5 !"* )(+,$ &. ,"* /*)(+,7+*
0*33$ &7,$#/* ,"* )(1*3 "(8* %**1 `0"&))*/ &..Q5 I&+ (1 #1,+&/70,#&1 ,& ,"* 07%*/4$)"*+* 2+#/ $**6
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(+* 1**/*/ (, ,"* 0*1,*+ &. 0*33 .(0*$ =.&+ )(+(33*3&2+(- "7? (+*($>6 0*33 8*+,#0*$
=.&+ ,"* A7(/+#3(,*+(3 "7? (+*($> &+ (, -73,#)3* 3&0(,#&1$ (3&12 ,"* 0*33 $#/*$ =.&+
"#2"*+4&+/*+ )&392&1 "7?*$>5 !& (8&#/ (19 #1,*+)&3(,#&1 &. 8*3&0#,9 0&-)&1*1,$
:+('(;( \ (1/ F 2+#/ $,(22*+#12 =$** 0"(),*+ Z> $"&73/ %* 7$*/ .&+ A7(/+#3(,*+(3
(1/ )(+(33*3&2+(- "7? (+*($6 +*$)*0,#8*396 ;"*+*($ ,"* "#2"*+4&+/*+ )&392&1 "7?
#1*8#,(%39 ;#33 +*A7#+* #1,*+)&3(,#&1 &. ,"* 8*3&0#,9 0&-)&1*1,$ =(, 3*($, (, ( $7%$*,
&. ,"* )&#1,$ (3&12 ,"* 0*33 $#/*$>5 !"* #1,*+)&3(,#&1 &. 8*3&0#,9 0&-)&1*1,$ 0(1
)&,*1,#(339 /*2+(/* ,"* &8*+(33 (007+(09 &. ,"* $0"*-* =U0W+*2&+6 KLLR> (1/ ,"*
0"&#0* &. 8(+#(%3* $,(22*+#12 #-)(0,$ ;(8* )+&)(2(,#&1 =;"*1 $&38#12 ,"* (#+ -($$
0&1,#17#,9 *A7(,#&1 ;#," ,"* -&-*1,7- *A7(,#&1$> ($ /#$07$$*/ #1 0"(),*+ Z5 T*10*
,"* 0"&#0* &. "7?4(+*( ())+&?#-(,#&1 (1/ 8(+#(%3* $,(22*+#12 (+* `#1,*+,;#1*/Q (1/
,"* 0"&#0*$ #-)(0, 1&, &139 ,"* (007+(09 &. ,"* ,+(1$)&+, &)*+(,&+ %7, (3$& ;(8*
)+&)(2(,#&1 )+&)*+,#*$ #1 .733 -&/*3$ ($ ;*33 ($ &,"*+ )+&)*+,#*$ $70" ($ ,"* 1**/
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8(+#(%3* $,(22*+#12 (1/ "7?4(+*( ())+&?#-(,#&1 #$ %*9&1/ ,"* $0&)* &. ,"#$ 0"(),*+5
8.5.2 Sub-grid-scale reconstruction
G1 ,"* )+*8#&7$ $*0,#&1$ ,"* 2*&-*,+#0(3 ())+&?#-(,#&1 ,& ,"* 7)$,+*(- (+*($ (1/
"7?4(+*($ "(8* %**1 /#$07$$*/5 N*?, 0&-*$ ,"* (0,7(3 #1,*2+(,#&1 &. s (B, J) &8*+
,"*$* (+*($6 .&+ ;"#0" ( $7%42+#/4$0(3* +*0&1$,+70,#&1 &. ,"* ,+(0*+ !*3/ #$ 1**/*/5
C* $,(+, %9 /#$07$$#12 +*0&1$,+70,#&1 -*,"&/$ #1 &1* $)(,#(3 /#-*1$#&1 (1/ ,"*1
[ I#1#,*4E&37-* !+(1$)&+, S0"*-*$
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%+#*"9 /#$07$$ ,;& /#-*1$#&1(3 *?,*1$#&1$ %*.&+* 0&8*+#12 ,"* #1,*2+(,#&1 &. s (B, J)
&8*+ &8*+3() (+*($5
8.5.2.1 One-dimensional reconstruction functions
!"* $7%42+#/4$0(3* +*0&1$,+70,#&1 #$ 8#,(3 .&+ ,"* &8*+(33 (007+(09 (1/ *.!0#*109 &.
( $0"*-*6 (1/ ( ,"&+&72" /#$07$$#&1 #$ %*9&1/ ,"* $0&)* &. ,"#$ 0"(),*+5 C* ;#336
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"($ 0&1$*A7*10*$ .&+ (19 !1#,*48&37-* $0"*-*5 T*+* (+* $&-* /*$#+(%3* )+&)*+,#*$
.&+ +*0&1$,+70,#&1 .710,#&1$ ,"(, $"&73/ %* 0&1$#/*+*/^
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#$ 1&, $,+#0,39 1*0*$$(+9 .&+ F73*+#(1 "7?4.&+- $0"*-*$ %7,6 #1 2*1*+(36 3*(/$ ,&
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*..&+, ,& ,+9 (1/ +*)+&/70* #, "*+*5 D*+,(#1 +*0&1$,+70,#&1$ 0(1 (3$& %* 7$*/ ,"(,
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=[5Zh>
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+,- 1$-2-D$;- /3);?")? A-?,3% c1/3Ad
<*+"()$ ,"* $#-)3*$, $7%42+#/4$0(3* +*)+*$*1,(,#&1 #$ ,"* $&40(33*/ )#*0*;#$* 0&14
$,(1, -*,"&/ =<D&U>6 ;"#0" $#-)39 7$*$
sC (B) = s C .
=[5ZO>
!"#$ ())+&(0" #$ (,,+#%7,*/ ,& W&/71&8 =MORO> (1/ ,+#8#(339 $(,#$!*$ =[5Z[>6 /&*$
1&, 1**/ ( "(3&6 (1/ #$ (3$& #1"*+*1,39 -&1&,&1* $#10* #, 0(11&, 3*(/ ,& 1*; *?4
,+*-(5 !"#$ ())+&(0" #$ (3$& .&+-(339 !+$,4&+/*+ (007+(,* (1/ "#2"39 /#..7$#8* ;"*1
7$*/ ;#," (19 $0"*-* &8*+ $-&&," "&;$ (1/ /#$,+#%7,#&1$5 :$ ( 0&1$*A7*10*6 ,"#$
0"&#0* &. +*0&1$,+70,#&1 #$ 0&1$#/*+*/ ,&& /#..7$#8* .&+ (,-&$)"*+#0 ,+(1$)&+, )+&%4
[ I#1#,*4E&37-* !+(1$)&+, S0"*-*$
KKR
3*-$ =713*$$ ,"* "&; #$ `+&72"Q>6 (1/ $& ;* -7$, ,7+1 &7+ (,,*1,#&1 ,& "#2"*+4&+/*+
+*0&1$,+70,#&1$5
F$E,-:K3:%-: :-23);?:52?$3);
N&,* ,"(, %9 ())+&)+#(,*39 $"#.,#12 ,"* )&391&-#(3 =[5Zh>6 ;* 0(1 (3;(9$ -() & C
&1,& ,"* 1&+-(3#@*/ #1,*+8(3 B ∈ [−6 B C /K, 6 BC /K] ;#," 0*1,*+)&#1, B = L5 \9 /&#12
$&6 ,"* -(," %*"#1/ ,"* +*0&1$,+70,#&1 #$ /+(-(,#0(339 $#-)3#!*/6 (1/ $& ;* ;#33
"*+*(.,*+ ($$7-* ,"(, ;* (+* ;&+'#12 &8*+ ,"#$ /&-(#15 I7+,"*+6 ;* ;#33 ($$7-*
,"(, ,"* 2+#/ #$ 5)$<3:6 $& ,"(, 6 B L = 6 B .&+ (33 L5 Y*0&1$,+70,#&1$ %($*/ &1 1&14
71#.&+- 2+#/$ (+* 2*1*+(339 ( $,+(#2",.&+;(+/ *?,*1$#&1 &. ,"* 71#.&+- 0($*5
<*+"()$ ,"* -&$, #1,7#,#8* -*,"&/ .&+ /*,*+-#1#12 ,"* 0&*.!0#*1,$ &. =[5Zh> #$ ,&
7$* ( !(93&+ $*+#*$ *?)(1$#&1 (%&7, ,"* 0*1,*+ &. ,"* 0*33 =B = L>6
.
sC (B) = sC .
!
, sC
+
B=L
,B
"..
.
.
.
B=L
M
B+
K
!
, K sC
, BK
"..
.
.
.
! @ "..
, sC .
M
BK + . . . +
. B@
@| , B @ .
B=L
B=L
0
/
+ O (6 B) @+M . =[5PL>
\9 )(#+#12 ,*+-$ &. *A7(3 &+/*+6 ;* &%,(#1 ,"* ($$&0#(,#&1
".
!
M , $ sC ..
(L)
($)
2C = sC (L),
2C =
.
$| , B$ .
.
=[5PM>
B=L
S#10* ;* /& 1&, '1&; ,"* *?(0, 8(37* &. s C &+ #,$ /*+#8(,#8*$6 ;* -7$, ())+&?#4
-(,* ,"*$* 8(37*$ 7$#126 .&+ *?(-)3*6 #1,*+)&3(,*/ )&391&-#(3$ ,"+&72" '1&;1 0*334
(8*+(2*/ 8(37*$5
N&,* ,"(, &1* -7$, %* 0(+*.73 #1 0"&&$#12 ,"* 0&++*0, ())+&?#-(,#&1$ ,& ,"*$*
/*+#8(,#8*$ ,& )+*$*+8* "#2"4&+/*+ (007+(095 S)*0#!0(3396 .&+ =[5PL> ,& %* .&+-(339
O [(6 B) @ ] (007+(,*6 *(0" &. ,"* /*+#8(,#8*$ , ) sC /, B) -7$, %* ())+&?#-(,*/ ,& &+/*+
O [(6 B) @−) ]6 (1/ sC (L) -7$, %* ())+&?#-(,*/ ,& &+/*+ O [(6 B) @ ]5 !"* +(,#&1(3*
%*"#1/ ,"#$ 03(#- #$ ($ .&33&;$^ C"*1 *8(37(,#12 ,"* +*0&1$,+70,#&1 =[5PL>6 *(0"
&. ,"* /*+#8(,#8*$ , ) sC /, B) #$ -73,#)3#*/ %9 B @ 6 ;"#0" -7$, $(,#$.9 |B| @ ≤ (6 B) @ 5
T*10*6 #. , ) sC /, B) #$ ())+&?#-(,*/ ,& O [(6 B) )−@ ] ,"*1 *(0" ,*+- #1 ,"* $*+#*$
=[5PL> #$ ())+&?#-(,*/ ,& O [(6 B) @ ]5 T&;*8*+6 $#10* s C (L) #$ 1&, -73,#)3#*/ %9 (19
)&;*+ &. B6 #, -7$, %* ())+&?#-(,*/ ,& .733 &+/*+4&.4(007+(095
S$)$?-K%$<<-:-)2- "@@:3B$6"?$3);
H1 (8*+(2#12 ,"* !(93&+ $*+#*$ =[5PL> &8*+ ( 0*33 & C 6 ;* &%,(#1
".
!
&
0
/
M , K sC ..
M 6 B/K
sC (B)%B = sC (L) +
. (6 B)K + O (6 B)P .
K
.
6 B −6 B/K
MK , B
B=L
=[5PK>
KKa
X(7+#,@*1 *, (35
!"* 3*.,4"(1/4$#/* &. ,"#$ *?)+*$$#&1 #$ $#-)39 ,"* 0*33 (8*+(2* s C 6 ;"#0" #$ '1&;1
#1 ( !1#,*48&37-* 0&1,*?,5 !"* !+$, ,*+- &1 ,"* +#2",4"(1/4$#/* #$ ,"* 8(37* &. s C (B)
*8(37(,*/ (, ,"* 0*3340*1,*+)&#1,
(1/
0 #, #$ .&33&;*/ %9 "#2"*+4&+/*+ ,*+-$5 T*10*6 ;*
/
0(1 0&1037/* ,"(, s C #$ ( O (6 B)K ())+&?#-(,#&1 ,& ,"* 8(37* &. s C (B) *8(37(,*/ (,
,"* 0*1,*+)&#1,5 !"#$ +*$73, #-)3#*$ ,"(, #. ;* 7,#3#@* !1#,*4/#..*+*10* ())+&?#-(,#&1$
,& ())+&?#-(,*
&. (19 &+/*+ (, B = L6 $70" ())+&?#-(,#&1$ ;#33 &139 %*
0
/ /*+#8(,#8*$
8(3#/ 7) ,& O (6 B)K #1 ( !1#,*48&37-* 0&1,*?,5
!"* $#-)3*$, !1#,*4/#..*+*10* ())+&?#-(,#&1 #$ ,"* )#*0*;#$*43#1*(+ -*,"&/
=<XU>6 2#8*1 %9
!
".
, sC ..
sC (B) = s C +
=[5PZ>
. B,
,B .
B=L
=8(1 X**+6 MOhh> ;"*+* , s C /, B #$ (, 3*($, ( !+$,4&+/*+4(007+(,* ())+&?#-(,#&1 ,&
,"* /*+#8(,#8* (, B = L5 S&-* 0"&#0*$ #1037/* (1 7);#1/ /#$0+*,#@(,#&16
!
".
s − s C−M
, sC ..
+ O(6 B),
= C
=[5PP>
.
,B .
6B
B=L
&+ ( 0*1,*+*/ /#$0+*,#@(,#&1
!
".
, sC ..
.
,B .
=
B=L
/
0
s C+M − s C−M
+ O (6 B)K .
K6 B
=[5PR>
F#,"*+ 0"&#0* ;#33 3*(/ ,& ( $0"*-* ;"#0" #$ .&+-(339 $*0&1/4&+/*+ (007+(,*5 X(+2*+
$,*10#3$ 0(1 %* 0"&$*1 .&+ ,"* ())+&?#-(,#&1$ ,& ,"*$* /*+#8(,#8*$6 %7, ,"*9 0(1
&139 3*(/ ,& +*0&1$,+70,#&1$ ,"(, (+* (, -&$, $*0&1/4&+/*+4(007+(,*5 N&1*,"*3*$$6
;#," 3(+2*+ $,*10#3$ ,&,(3 (007+(09 -(9 #-)+&8* $#21#!0(1,39 *8*1 ,"&72" ,"* .&+4
-(3 &+/*+4&.4(007+(09 ;#33 1&,5
!"* 3#1*(+ +*0&1$,+70,#&1 /+($,#0(339 #-)+&8*$ ,"* *++&+ -*($7+*$ &. !1#,*48&37-*
$0"*-*$6 ;"*1 0&-)(+*/ ,& <D&U5 !"#$ +*$73, #$ #337$,+(,*/ #1 I#25 [5MP #1 ,*+-$
&. ( 8&1 N*7-(11 $,(%#3#,9 (1(39$#$ &. ( !1#,*48&37-* $0"*-* %($*/ &1 <D&U (1/
<XU =7$#12 ,"* 0*1,*+*/ ())+&?#-(,#&1 =[5PR>>5 G1 -(19 3(+2*4$0(3* (,-&$)"*+#0
-&/*3$ <XU #$ $,#33 0&1$#/*+*/ ,&& /#..7$#8* (1/ ,"*+*.&+* *8*1 "#2"*+4&+/*+ +*0&14
$,+70,#&1$ (+* &.,*1 0&1$#/*+*/5
I#1#,*4/#..*+*10* $0"*-*$ 0(1 (3$& %* 7,#3#@*/ ,& &%,(#1 ( ,"#+/4&+/*+ +*0&1$,+704
,#&16 *8*1 #1 ( !1#,*48&37-* 0&1,*?,5 Y*(++(12#12 =[5PK>6 ;* 0(1 &%,(#1 (1 *?)+*$4
$#&1 .&+ ,"* 0*1,*+)&#1, 8(37* s C (L)6
".
!
0
/
(6 B)K , K sC ..
=[5Pa>
sC (L) = s C −
. + O (6 B)P ,
K
.
KP
,B
B=L
;"#0" #$ ( .&7+,"4&+/*+4(007+(,* ())+&?#-(,#&1 ,&/ ,"* )&#1,;#$*
8(37* &. s C (L)6 ($
0
3&12 ($ , K sC /, BK #$ ())+&?#-(,*/ ,& (, 3*($, O (6 B)K 5 D&-%#1#12 ,"#$ ())+&?#4
-(,#&1 ;#," =[5PL>6 ;* &%,(#1 ( ,"#+/4&+/*+ =)(+(%&3#0> +*0&1$,+70,#&1
[ I#1#,*4E&37-* !+(1$)&+, S0"*-*$
| K |
2
R
0.9
(a)
5
0.9
5
KKh
0.6
(b)
L (6x)
L (6x)
0.3
8
0.
0.6
3
4
0.8
3
0.3
2
0
0.2
0.
6
0.6
1.0
1.0
4
PCoM
PLM
0.4
0.4
0.4
0.6
&! u0 6t / 6x
0.8
1
2
1.2
1.4
0
0.2
0.4
0.6
&! u0 6t / 6x
0.8
1
Fig. 8.14 !"* $,(%#3#,9 )+&)*+,#*$ =$** $*0,#&1 [5Z5Z (1/V&+ X(7+#,@*1 KLLh> &. ( &1*4/#-*1$#&1(3
!1#,*48&37-* $0"*-* %($*/ &1 <D&U =2+*9 3#1*> (1/ <XU =%3(0' 3#1*>6 +*$)*0,#8*395 N&,* ,"(,
#1 &1* /#-*1$#&1 (33 !1#,*48&37-* $0"*-*$ /#$07$$*/ #1 ,"#$ 0"(),*+ (+* #/*1,#0(3 ;"*1 7$#12 ,"*
$(-* +*0&1$,+70,#&1 -*,"&/5 =(> SA7(+*/ -&/737$ &. ,"* (-)3#!0(,#&1 .(0,&+ =|K |K > ($ ( .710,#&1 &.
D&7+(1, 17-%*+ =B4(?#$> (1/ ;(8*3*12," G =J4(?#$>5 T*10* =(> $"&;$ "&; -70" *(0" ;(8*3*12,"
#$ /(-)*/ #1 &1* ,#-*4$,*) ($ ( .710,#&1 &. D&7+(1, 17-%*+5 I&+ ( !?*/ D&7+(1, 17-%*+ & ,"*
/(-)#12 /*0+*($*$ -&1&,&1#0(339 ($ ( .710,#&1 &. ;(8*3*12," G (1/ 3#-G→' |K |K = M5 I&+ D&7+(1,
17-%*+ L &+ M ,"* $0"*-* #$ *?(0, (1/ "*10* |K |K = M5 =%> S(-* ($ =(> %7, .&+ ,"* +*3(,#8* )"($*
$)**/ =(>6 ,"(, #$6 "&; -70" *(0" I&7+#*+ -&/* #$ (00*3*+(,*/ &+ /*0*3*+(,*/ ($ ( .710,#&1 &.
D&7+(1, 17-%*+5
"..
.
.
.
"..
.
.
.
!
"
/
0
(6 B)K
BK −
+ O (6 B)Z ,
MK
B=L
B=L
=[5Ph>
;"*1 0&-%#1*/ ;#," $#-)3* !1#,*4/#..*+*10* ())+&?#-(,#&1$ &. ,"* .&+- =[5PR> (1/
! K "..
0
/
s
− Ks C + s C+M
, sC .
=[5P[>
= C−M
+ O (6 B)K .
.
K
K
.
,B
(6 B)
!
, sC
sC (B) = s C +
,B
M
B+
K
!
, K sC
, BK
B=L
G1 .(0,6 #, 0(1 %* A7#0'39 8*+#!*/ ,"(, =[5Ph> (3$& $(,#$!*$ ,"* 0*334(8*+(2*/ )+&)*+,9
=[5Z[>5 !"#$ -*,"&/ "($ ,"* "#2"*$, .&+-(3 &+/*+ &. (007+(09 ,"(, 0(1 %* &%,(#1*/
%9 ,+*(,#12 !1#,*48&37-* -*,"&/$ #1 ( !1#,*4/#..*+*10* 0&1,*?,5 !"#$ 0"&#0* &. +*4
0&1$,+70,#&1 ;($ 7$*/ %9 X()+#$* (1/ <3(1,* =MOOR>5
S$)$?-K#3*56- "@@:3B$6"?$3);
!& &%,(#1 ())+&?#-(,#&1$ "#2"*+ ,"(1 ,"#+/4&+/*+ #1 (007+(096 ;* -7$, !+$, ,('* (
$,*) %(0' (1/ 71/*+$,(1/ "&; !1#,*48&37-* -*,"&/$ (+* .&+-73(,*/5 I#+$,6 +*0(33
,"(, !1#,*48&37-* -*,"&/$ 7$* 0*334(8*+(2*/ 8(37*$6 ;"#0" #-)3#*$ ,"(, ,"* 71/*+4
39#12 $0(3(+ !*3/ #$ 1&, '1&;1 )&#1,4%94)&#1,5 G1$,*(/6 #, #$ 0*334(8*+(2*/ 8(37*$ ,"(,
(+* '1&;1 *?(0,39
&
M 6 B/K
sC =
sC (B)%B.
=[5PO>
6 B −6 B/K
KK[
X(7+#,@*1 *, (35
T*10*6 #1 ,"* 0&1,*?, &. !1#,*48&37-* -*,"&/$6 "#2"4&+/*+ $7%42+#/4$0(3* +*0&14
$,+70,#&1$ 0(11&, %* #1,*+)&3(,*/ ,"+&72" $)*0#!0 )&#1,$ =($ ;#," !1#,*4/#..*+*10*
-*,"&/$>6 %7, -7$, #1$,*(/ $(,#$.9 0*+,(#1 0*334(8*+(2*/ )+&)*+,#*$5
!& %7#3/ ( +*0&1$,+70,#&1 ,"(, 7,#3#@*$ 0*334(8*+(2*$6 &1* 2*1*+(339 /*!1*$ ( 25K
65*"?$#- 6";; <5)2?$3) Q (B) 8#(
Q (B) =
& B
BC− L−M/K
sC (B)%
r B,
r
=[5RL>
;"*+* BC− L−M/K /*1&,*$ ,"* 3*.,4$#/* */2* &. 0*33 & C− L 5 N&;6 &%$*+8*
Q (BC− L−M/K ) = L,
#
$
Q (BC− L+M/K ) = 6 B s C− L ,
#
$
Q (BC− L+Z/K ) = 6 B s C− L + s C− L+M ,
55
5
H8*+ $70" ( $*, &. 0&1$*07,#8* 0*33$ &1* 0(1 ,"*1 /*!1* (1 #1,*+)&3(,#12 )&391&4
-#(3 &. /*2+** 6 ,"(, ())+&?#-(,*$ ,"* *?(0, 07-73(,#8* -($$ .710,#&1 Q (B)5 C*
/*1&,* ,"#$ ())+&?#-(,#&1 %9 Qr (B)5 I#1(3396 ;* &%$*+8* ,"(, #1 (00&+/(10* ;#," ,"*
.71/(-*1,(3 ,"*&+*- &. D(30737$6 /#..*+*1,#(,#12 =[5RL> 2#8*$
%Q
(B) = sC (B).
%B
=[5RM>
r
\9/ *8(37(,#12
0 ,"* !+$, /*+#8(,#8* &. Q (B) (, ( 2#8*1 )&#1,6 ;* (0,7(339 &%,(#1 (
O (6 B)6−M ())+&?#-(,#&1 ,& ,"* 71/*+39#12 !*3/ s C (B) .+&- #,$ 0*334(8*+(2*$5
!"#$ -*,"&/ 0(1 ,"*1 %* 7$*/ ,& +*0&1$,+70, s C (B) (, (19 )&#1, (1/6 %9 ,('#12 (//#4
,#&1(3 /*+#8(,#8*$ &. Qr (B)6 #,$ 0&++*$)&1/#12 /*+#8(,#8*$5
:3,*+1(,#8*396 &1* 0(1 &%,(#1 (1 #/*1,#0(3 +*0&1$,+70,#&1 %9 *1.&+0#12 ,"* 0*334
(8*+(2*/ 0&1$,+(#1, &1 (1 #1,*+)&3(,#12 )&391&-#(3 #1 1*#2"%&+#12 0*33$ =f*++&7'(,
1C (B) &. /*2+** @ ,"(, *?(0,39 $(,#$!*$ ,"* -($$4
*, (36 KLLK>5 !"(, #$6 ( )&391&-#(3 s
0&1$*+8(,#&1 0&1$,+(#1, 1&, &139 #1 0*33 C %7, (3$& #1 @ (/](0*1, 0*33$^
& B
#
L+M/K
@$
@$ #
1C (B) %B = s L 6 B, L = C −
.. C +
,
=[5RK>
s
K
K
B L−M/K
.&+ @ *8*1 (1/
& B
L+M/K
B L−M/K
1C (B) %B = s L 6 B,
s
!
" !
"
@+M
@−M
L = C−
.. C +
,
K
K
=[5RZ>
.&+ @ &//5
r (B) = s
1C (B)>6 (3,"&72"
F#,"*+ -*,"&/ ;#33 9#*3/ (1 #/*1,#0(3 +*0&1$,+70,#&1 = Q
,"* 3(,,*+ #$ -&+* (/(),(%3* ,& ,;& /#-*1$#&1$ (1/ %*9&1/5
[ I#1#,*4E&37-* !+(1$)&+, S0"*-*$
KKO
G. ;* 7,#3#@* ,"* (.&+*-*1,#&1*/ )+&0*/7+* &8*+ ( Z40*33 $,*10#3 =0&1$#$,#12 &.
0*33$ C − M6 C (1/ C + M>6 ;* ;#33 *?(0,39 &%,(#1 =[5PR>6 =[5Pa> (1/ =[5P[>5 T&;*8*+6
%*9&1/ Z 0*33$6 ,"* !1#,*4/#..*+*10* (1/ !1#,*48&37-* +*0&1$,+70,#&1$ ;#33 /#..*+
$7%$,(1,#(3395 I&+ #1$,(10*6 &8*+ ( 0*1,*+*/ R40*33 $,*10#36 ;* &%,(#1 ())+&?#-(,#&1$
3
2
Os C−K − MMas C−M + KMZPs C − MMas C+M + Os C+K
+ O (6 B)a ,
sC (L) =
MOKL
!
"..
2
3
Rs C−K − ZPs C−M + ZPs C+M − Rs C+K
, sC .
+ O (6 B)R ,
=
.
,B .
P[ 6 B
B=L
*,05
T#2"4&+/*+ +*0&1$,+70,#&1$ &. ,"#$ ,9)* ;*+* (/&),*/ .&+ ( $"(33&;4;(,*+ -&/*3 %9
_33+#0" *, (3 =KLML>5
0J66-?:$2 !)$?-K#3*56- ;2,-6-;
G1 (33 ,"* -*,"&/$ $& .(+ /#$07$$*/6 ;* "(8* 1&, ,&70"*/ &1 ,"* #$$7* &. 0&1,#17#,9
%*,;**1 0*33$5 G1 .(0,6 (33 &. ,"* -*,"&/$ ;* "(8* /*$0+#%*/ $& .(+ /& 1&, *1.&+0*
(19 $&+, &. 0&1,#17#,9 %*,;**1 +*0&1$,+70,#&1$ #1 1*#2"%&+#12 0*33$5
:$ ;* "(8* $**1 $& .(+6 ($ ,"* &+/*+ &. ,"* +*0&1$,+70,#&1 )&391&-#(3 #$ #10+*($*/6
($)
-&+* &),#&1$ .&+ "&; ,& ())+&?#-(,* ,"* 0&*.!0#*1,$ 2 C (+* (8(#3(%3*5 D&1,#17#,9
(, */2*$ 0(1 %* *1.&+0*/ =&8*+ (1 (+%#,+(+9 $0(3(+ !*3/> #. ;* (/&), ( +*0&1$,+70,#&1
,"(, #$ (, 3*($, )(+(%&3#06 $'-'
(L)
(M)
(K)
sC (B) = 2C + 2C B + 2C BK .
=[5RP>
S#10* ;* "(8* ,"+** /*2+**$ &. .+**/&- #1 ,"#$ )&391&-#(36 ;* 0(1 0"&&$* ,& *14
.&+0* sC (−6 B/K) = sCG (1/ sC (6 B/K) = sC( 6 ;"*+* sCG (1/ sC( (+* +*0&1$,+70,*/
8(37*$ (, ,"* 3*.,4 (1/ +#2",4 */2*$6 +*$)*0,#8*395 !"*$* (+* )7+)&$*39 0"&$*1 ,& %*
0&1$#$,*1, %*,;**1 1*#2"%&+#12 0*33$6 ;"#0" 2#8*$ 7$ ,"* /*$#+*/ 0&1,#17#,9 +*$,+#04
,#&15 C#," &7+ +*-(#1#12 /*2+** &. .+**/&- ;* *1.&+0* ,"* 0*334(8*+(2*/ 0&1/#,#&1
=[5Z[>5
!"#$ $0"*-* #$ ,"* ;*334'1&;1 )#*0*;#$*4)(+(%&3#0 -*,"&/ =<<U> &. D&3*33(
(1/ C&&/;(+/ =MO[P>5 !& &%,(#1 */2*)&#1, 8(37*$ s CG (1/ sC( 6 <<U -('*$ 7$* &.
,"* !1#,*48&37-* .&+-73(,#&1 /#$07$$*/ *(+3#*+6 ,('*1 &8*+ .&7+ 0*33$ (1/ *8(37(,*/
(, ,"* 0*33 */2*)&#1,6 ;"#0" 2#8*$
sC( =
0
/
h
M
(s C + s C+M ) − (s C+K + s C−M ) + O (6 B)P ,
MK
MK
=[5RR>
($)
G = s ( 5 G1 ,*+-$ &. ,"* 0&*.!0#*1,$ 2 6
=(3$& $** f*++&7'(, *, (3 =KLLK>> (1/ s C+M
C
C
,"#$ +*0&1$,+70,#&1 0(1 %* ;+#,,*1 ($
KZL
X(7+#,@*1 *, (35
(a) Picewise-linear reconstruction
exact
PLM
PLM+filter
Density
4
3
2
1
0.1
0.3
0.5
0.7
x-axis
(b) Picewise-parabolic reconstruction
exact
PPM
PPM+filter
Density
4
3
2
1
0.1
0.3
0.5
0.7
x-axis
Fig. 8.15 Y*0&1$,+70,#&1$ .&+ ,"* #++*273(+ $#21(3 &. S-&3(+'#*;#0@ (1/ W+(%&;$'# =MOOL6 %37*
3#1*> 7$#12 ,"* =(> )#*0*;#$*43#1*(+ -*,"&/ =<XU> (1/ =%> ,"* )#*0*;#$* )(+(%&3#0 -*,"&/ =<<U>
;#," +*0&1$,+70,#&1 .710,#&1 !3,*+ =2+*9> (1/ ;#,"&7, =+*/>5 !"* !3,*+ .&+ <XU #$ ,"* UGNUHB
3#-#,*+ =$** ,*?,> ;#," ,"*,(wM (1/ ,"* <<U 3#-#,*+ #$ ,"* &+#2#1(3 !3,*+ )+*$*1,*/ #1 D&3*33( (1/
C&&/;(+/ =MO[P>5
(K)
2
(L)
2C = s C − (6 B)K C ,
MK
,
5 M 4
5
M K4
(M)
s
− s C−M −
s
− s C−K ,
2C =
6 B Z C+M
MK C+K
,
4
5 M4
5
M
(K)
−Rs C + Z s C+M + s C−M − s C+K + s C−K .
2C =
K
K (6 B)K
[ I#1#,*4E&37-* !+(1$)&+, S0"*-*$
| K |
2
R
5
0.9
(a)
0.9
4
L (6x)
L (6x)
5
KZM
0.9
0.6
3
4
3
(b)
PPM-s
PPM2
PPM3
PPM4
PPM5
0.8
0.8
0.6
0.3
1.2
2
0
0.2
0.4
0.6
&! u0 6t / 6x
0.8
1
2
1.4
0
0.2
0.4
0.6
&! u0 6t / 6x
0.8
1
Fig. 8.16 S(-* ($ I#25 [5MP %7, .&+ <<U 7$#12 /#..*+*1, *$,#-(,*$ .&+ ,"* */2* 8(37*$ =$&3#/ 3#1*$>
($ ;*33 ($ <<U4$ =/($"*/ 3#1*>5 <<U4$ #$ ,"* $7%42+#/40*33 +*0&1$,+70,#&1 -*,"&/ %($*/ &1 ,"*
-*,"&/ &. X()+#$* (1/ <3(1,* =MOOR>6 ,"(, #$6 7$#12 =[5RK> ;#," @ = K ,& /*,*+-#1* ,"* $7%42+#/40*33
+*0&1$,+70,#&1 .710,#&15
($)
:33/ &. ,"*0 0&*.!0#*1,$ 2 C #1 ,"#$ 0($* ())+&?#-(,* s C (B) (1/ #,$ /*+#8(,#8*$ ,&
O (6 B)P 5
!"* ())+&?#-(,#&1 ,& s CG #1 =[5RR> (1/ s C( (+* .&7+,"4&+/*+ (007+(,* .&+ 71#.&+2+#/$5 H%8#&7$396 &1* 0&73/ (3$& /*+#8* $*0&1/6 ,"#+/6 .&7+,"6 !.," &+ $#?,"4&+/*+
(007+(,* *$,#-(,#&1$ %9 !,,#12 ( 3#1*(+6 )(+(%&3#06 07%#06 A7(+,#0 (1/ A7#1,#0 )&394
1&-#(3 $& ,"(, =[5RL> &+ =[5RK4[5RZ> #$ $(,#$!*/ #1 K6 Z6 P6 R (1/ a (/](0*1, 0*33$6
+*$)*0,#8*395 C* ;#33 +*.*+ ,& <<U %($*/ &1 $*0&1/6 ,"#+/6 .&7+,"6 !.,"4&+/*+ */2*
8(37* *$,#-(,*$ ($ <<UK6 <<UZ6 <<UP6 <<UR6 +*$)*0,#8*395 G1 ,"#$ 0&1,*?, <<U
(1/ <<UP +*.*+ ,& ,"* $(-* +*0&1$,+70,#&15
G1 ,*+-$ &. @4*?(0,1*$$ ,"* */2* *$,#-(,*$ -7$, %* (, 3*($, ,"#+/4&+/*+ .&+ ,"*
<<U ,& *?(0,39 +*0&1$,+70, ( 23&%(3 )(+(%&3(5 T*10* <<UK #$ 1&, @4*?(0, =)wK>
;"*+*($ <<UZ6 <<UP6 (1/ <<UR (+*5 G, #$ 1&,*/ ,"(, <<UP #$ $#21#!0(1,39 -&+*
(007+(,* ,"(1 <<UZ #1 ,*+-$ &. ( E&1 N*7-(1 $,(%#3#,9 (1(39$#$ =I#25 [5Ma> ;"*+*($
<<UR &139 2#8*$ -&/*$, #10+*($*$ #1 (007+(095 H%8#&7$39 <<UR 1**/$ ( 3(+2*+ "(3&
,"(1 <<UP5 :$ ( 0&1$*A7*10*6 ,"* )&,*1,#(3 #10+*($* #1 0&$, ($$&0#(,*/ ;#," ,"* 7$*
&. 3(+2*+ $,*10#3$ "($ 3#'*39 %**1 ( $#21#!0(1, .(0,&+ #1 /*,*+-#1#12 ,"* ;#/*$)+*(/
(/&),#&1 &. <<UP &8*+ ,"*$* &,"*+ $0"*-*$5 U&+* /#$07$$#&1 &1 */2*48(37* *$,#4
-(,*$ #$ 2#8*1 #1 C"#,* (1/ :/0+&., =KLL[>5
H1* 0&73/ 1(,7+(339 ($' ,"* A7*$,#&1 ;"9 $"&73/ &1* 1&, 7$* ,"* "#2"*$,4&+/*+
)&391&-#(3 ,"(, 0(1 %* ())+&?#-(,*/ ;#," ( 2#8*1 "(3& =$,*1$#3>k I&+ *?(-)3*6 ,"*
07%#0 )&391&-#(3 7$*/ ,& *$,#-(,* ,"* */2* 8(37* #1 <<UP 0&73/ %* 7$*/ ($ ,"*
1C (B)5 C"#3* ,"#$ -#2", #-)+&8* ,"* (007+(09 &.
+*0&1$,+70,#&1 .710,#&16 s C (B) = s
,"* $0"*-*6 #, ;#33 -('* !3,*+#12 (1/ #1,*2+(,#&1 &8*+ &8*+3()4(+*($ -&+* 07-%*+4
$&-* (1/ 0&-)7,(,#&1(339 *?)*1$#8*6 ($ 0&-)(+*/ ,& $,#0'#12 ,& ( )(+(%&3( ;#,"
"#2"4&+/*+ */2*48(37* *$,#-(,*$ =<<UP>5
Y*0&1$,+70,#&1$ %($*/ &1 )&391&-#(3$ &. /*2+** "#2"*+ ,"(1 K "(8* %**1 )+&4
)&$*/ #1 ,"* 3#,*+(,7+* %7, "(8* 1&, %**1 ;#/*39 (/&),*/ #1 ,+(1$)&+, $0"*-*$ ($ &.
,"* ,#-* &. ;+#,#125 f*++&7'(, *, (3 =KLLK> #1,+&/70*/ ( $9--*,+#0 )#*0*;#$*407%#0
-*,"&/ =<DU>6 (3&12 ;#," (/8(10*/ !3,*+#12 ,*0"1#A7*$ =f*++&7'(, *, (36 KLLR>6
KZK
X(7+#,@*1 *, (35
| K |
R
5
(a)
0.9
4
L (6x)
L (6x)
5
2
0.9
0.6
3
4
3
(b)
PQM3
PQM4
PQM5
PQM6
0.8
0.8
0.6
0.3
1.2
2
0
0.2
0.4
0.6
&! u0 6t / 6x
0.8
1
2
1.4
0
0.2
0.4
0.6
&! u0 6t / 6x
0.8
1
Fig. 8.17 S(-* ($ I#25 [5Ma %7, .&+ <bU5
(1/ C"#,* (1/ :/0+&., =KLL[> )+&)&$*/ ( $9--*,+#0 )#*0*;#$*4A7(+,#0 -*,"&/
=<bU> %($*/ &1 )&391&-#(3$ &. /*2+** P5 :$ .&+ ,"* <<U ,"* */2*48(37* *$,#-(,*
#$ )(+(-&71, .&+ ,"* (007+(09 &. ,"* $0"*-* =$**6 *5256 I#25 [5Mh>5 T&;*8*+6 *8*1
.&+ ,"* -&$, (007+(,* <bUa ,"* #10+*($* #1 (007+(09 =#1 ,*+-$ &. ( E&1 N*7-(11
(1(39$#$> #$ -&/*$, 0&-)(+*/ ,& <<UP 2#8*1 ,"* #10+*($* #1 ,"* "(3& $#@*5 :3$& ,&
0&1$#/*+ #$ ,"(, )&391&-#(3$ &. /*2+** Z =<DU> (1/ P =<bU> 0(1 "(8* K (1/ Z *?4
,+*-( ;#,"#1 ( 2+#/ 0*33 -('#12 #, "(+/*+ ,& !3,*+ $70" )&391&-#(3$ 0&-)(+*/ ,& (
=+*3(,#8*39 3&;4&+/*+> )(+(%&3(5
1$-2-D$;- 45"%:"?$2 ;@*$)-;'
:1 #1,*+*$,#12 8(+#(1, &1 ,"* +*0&1$,+70,#&1 -*,"&/$ /#$07$$*/ $& .(+6 (1/ (3$& %($*/
&1 )(+(%&3($6 #$ ,"* )#*0*;#$* A7(/+(,#0 $)3#1* -*,"&/ =f*++&7'(, *, (3 KLLa> (1/
"#2"*+4&+/*+ *?,*1$#&1$ $70" ($ ,"&$* )+*$*1,*/ #1 f*++&7'(, *, (3 =KLML>5 G1$,*(/
&. &139 *1.&+0#12 / L 0&1,#17#,9 (0+&$$ 0*33 */2*$ (3$& ,"* !+$, /*+#8(,#8* &. ,"*
+*0&1$,+70,#&1 #$ 0&1$,+(#1*/ ,& %* 0&1,#17&7$6 #5*5 ,"* +*0&1$,+70,#&1 #$ / M (0+&$$
0*33 */2*$5 F1.&+0#12 0&1,#17#,9 #1 ,"* /*+#8(,#8*$ &. ,"* +*0&1$,+70,#&1 .710,#&1$ (,
0*33 %&71/(+#*$ +*$73,$ #1 (1 #-)3#0#, $9$,*- &. *A7(,#&1$ .&+ ,"* )&391&-#(3 0&*.!4
0#*1,$5 C"*1 ;+#,,*1 #1 -(,+#? .&+-6 "&;*8*+6 ,"* -(,+#? ,"(, 1**/$ ,& %* #18*+,*/
"($ ( ,+#4/#(2&1(3 .&+-5
G1 #/*(3#@*/ ,*$, 0($*$ 7$#12 ,"* $0"*-* &. f*++&7'(, *, (3 =KLLK> ,"* )#*0*;#$*
$)3#1* +*0&1$,+70,#&1 -*,"&/ #$ $7)*+#&+ ,& <<U ;"#3* %*#12 PLv -&+* *.!0#*1, #1
,*+-$ &. 17-%*+ &. &)*+(,#&1$ =f*++&7'(, *, (36 KLLh>5 !"* )+#0* ,& )(96 #1 ,*+-$ &.
( )(+(33*3 0&-)7,(,#&1(3 *18#+&1-*1,6 #$ ,"(, $)3#1*$ (+* #1"*+*1,39 23&%(3 $#10* ,"*
#18*+$#&1 &. ( 23&%(3 ,+#4/#(2&1(3 -(,+#? #$ 1*0*$$(+95
M;;-)?$"**J )3)K3;2$**"?3:J cM.Wd :-23);?:52?$3);'
F$$*1,#(339 1&14&$0#33(,&+9 +*0&1$,+70,#&1$ ;*+* &+#2#1(339 /*8*3&)*/ %9 T(+,*1
*, (3 =MO[h> .&+ $"&0' "9/+&/91(-#0$ )+&%3*-$5 !"#$ ())+&(0" #$ )(+,#073(+39 #14
[ I#1#,*4E&37-* !+(1$)&+, S0"*-*$
KZZ
,*+*$,#12 $#10* #, 3*(/$ ,& ( +*0&1$,+70,#&1 ,"(, #$ =71/*+ -&$, 0#+07-$,(10*$> -&1&4
,&1#0 (1/ )&$#,#8*5 !"* FNH $0"*-* ;&+'$ %9 ())39#12 *#,"*+ ( !1#,*4/#..*+*10*
&+ !1#,*48&37-* ())+&(0" =($ /#$07$$*/ *(+3#*+> &1 ( 8(+#*,9 &. $,*10#3$5 !"* +*0&14
$,+70,#&1 ,"(, $(,#$!*$ $&-* 3*($,4&$0#33(,&+9 )+&)*+,96 (-&12 (33 )&$$#%3* $,*10#3$6
#$ ,"*1 0"&$*1 ,& 2#8* ,"* `,+7*Q +*0&1$,+70,#&15 !"* -(#1 /+(;%(0' &. ,"#$ ())+&(0"
#$ ,"(, #, +*A7#+*$ ( 3(+2* $,*10#3 #1 &+/*+ ,& &%,(#1 ,"* $(-* &+/*+ &. (007+(09 ($
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())+&?#-(,* +*0&1$,+70,#&1$ &1 71$,+70,7+*/ 2+#/$5 _1/*+ ,"#$ -*,"&/6 ;* #1,+&4
/70* $&-* A7(1,#!0(,#&1 &. ,"* -#$!, %*,;**1 ,"* +*0&1$,+70,#&1 =[5Zh> (1/ ,"*
'1&;1 0*334(8*+(2*$6 7$7(339 2#8*1 %9 ,"* $A7(+* &. ,"* /#..*+*10* %*,;**1 ,"* 0*334
(8*+(2*$ &. ,"* +*0&1$,+70,#&1 (1/ ,"* '1&;1 0*334(8*+(2*$5 !"* -#$!, #$ ,"*1 -#14
#-#@*/ &8*+ (33 )&$$#%3* +*0&1$,+70,#&1$ #1 &+/*+ ,& 2#8* ,"* t%*$, )&$$#%3*u +*0&14
$,+70,#&15 :1 *?(-)3* &. ( D(+,*$#(1 !1#,*48&37-* $0"*-* &1 (1 71$,+70,7+*/ 2+#/
%($*/ &1 ,"* 3*($,4$A7(+*$ ,*0"1#A7* 0(1 %* .&71/ #1 \(+," (1/ I+*/*+#0'$&1 =MOOL>5
W)-K%$6-);$3)"* :-23);?:52?$3) *$6$?-:;f!*?-:;
:$ /#$07$$*/ #1 $*0,#&1 [5Z5a6 #, #$ /*$#+(%3* ,"(, ( ,+(1$)&+, $0"*-* 7,#3#@*$ )"9$4
#0(339 +*(3#@(%3* +*0&1$,+70,#&1$5 !"*+* (+* ,;& ;(9$ ,& (0"#*8* ,"#$ 2&(36 *#,"*+
" @:$3:$ %9 !3,*+#12 ,"* $7%42+#/40*33 +*0&1$,+70,#&1 .710,#&1 $& ,"(, #, &139 ,('*$
)"9$#0(339 +*(3#@(%3* 8(37*$6 &+ " @3;?-:$3:$ %9 3#-#,#12 )+&21&$*/ 0*33 (8*+(2*$ &+
%9 (3,*+#12 ,"* "7?*$ #1/#8#/7(3395 G1 ,"* 0&1,*?, &. (1 7)$,+*(- $*-#4X(2+(12#(1
$0"*-* "7?43#-#,#12 #$ &%8#&7$39 1&, (1 &),#&15 I&+ F73*+#(1 $0"*-*$ &1* -(9 ()4
)39 " @:$3:$ !3,*+$ &+ "7?43#-#,*+$ ,& )+&8#/* )"9$#0(339 +*(3#@(%3* $&37,#&1$5 & @:$3:$
!3,*+$ (+* (3$& +*.*++*/ ,& ($ $3&)*43#-#,*+$ ($ ,"*9 (0, /#+*0,39 &1 ,"* $7%42+#/4$0(3*
+*0&1$,+70,#&1 .710,#&15
!"* <XU6 7$7(339 %($*/ &1 =[5PP> &+ =[5PR>6 -(9 8#&3(,* -&1&,&1#0#,9 ($ #337$4
,+(,*/ #1 I#25 [5MR=(>5 U&1&,&1#0#,9 0(1 %* *1.&+0*/ %9 +*)3(0#12 ,"* +*0&1$,+70,*/
/*+#8(,#8* ;#," $&-* ;*#2",*/ (8*+(2* &. ,"* 7);#1/ (1/ /&;1;#1/ ())+&?#-(4
,#&1$5 U(19 $70" 0&-%#1(,#&1$ *?#$,6 #1037/#12 UGNUHB6 S7)*+%** =Y&*6 MO[R>6
(1/ -&1&,&1* 0*1,+(3 =8(1 X**+6 MOhh>6 ,& 1(-* ( .*; =$**6 .&+ *?(-)3*6 !&+& MOOO>5
G, #$ %*9&1/ ,"* $0&)* &. ,"#$ 0"(),*+ ,& )+&8#/* ( 0&-)+*"*1$#8* +*8#*; &. ,"*$*
!3,*+$ %7, ,"*9 (33 $**' ,& %3*1/ ,"* /*+#8(,#8* *$,#-(,*$6 ($ "#1,*/ (%&8*6 ,& &%,(#1
,"* 3*($, /#..7$#8* -&1&,&1* $&37,#&15
:$ #337$,+(,*/ &1 I#25 [5MR=%> <<UP #$ (3$& 1&14-&1&,&1* ;#,"&7, ,"* ())3#0(4
,#&1 &. !3,*+$5 !"* $*-#1(3 )()*+ &. D&3*33( (1/ C&&/;(+/ =MO[P> 0&1$,+(#1$ ,"*
+*0&1$,+70,#&1 $& ,"(, ,"* *1,#+* $7%42+#/4$0(3* +*0&1$,+70,#&1 #$ %&71/*/ %9 ,"*
0*334(8*+(2*$ &. ,"* 1*#2"%&+#12 0*33$ =&+ #$ +*/70*/ ,& ( 0&1$,(1, ;"*1 ,"* +*0&14
$,+70,#&1 #$ ( 3&0(3 *?,+*-(>5 S** I#27+* [5MR=%> .&+ (1 *?(-)3*5 !"#$ ,*0"1#A7* .&+
KZP
X(7+#,@*1 *, (35
!3,*+#12 ,"* +*0&1$,+70,#&1 "($ ,"* ,*1/*109 ,& t07, &..u &+ "(,,*1 $-&&,"6 )"9$#4
0(3 -(?#-( (1/ -#1#-(5 S*8*+(3 ())+&(0"*$ "(8* %**1 )+&)&$*/ ,& +*,(#1 )"9$#0(3
*?,+*-( ;"#3* !3,*+#12 &7, $)7+#&7$ 2+#/4$0(3* 1&#$* =$**6 .&+ *?(-)3*6 8(1 :3%(/(
*, (3 =MO[K>6 f*++&7'(, *, (3 =KLLR>6 X#7 *, (3 =KLLh> (1/ D&3*33( (1/ S*'&+( =KLL[>>5
G. -#1#$073* &8*+4 (1/ 71/*+$"&&,$ 0(1 %* ,&3*+(,*/6 3*$$ #18($#8* !3,*+$ 0(1 %* /*4
$#21*/ 7$#12 =C>FNH4,9)* -*,"&/$ ;"*+* ,"* 7$*+4$)*0#!*/ !3,*+ #$ ())3#*/ &139
;"*1 ( $-&&,"1*$$ -*,+#0 *?0**/$ ( 0*+,(#1 ,"+*$"&3/ =\3&$$*9 (1/ B7++(16 KLL[>5
:0"#*8#12 "#2"4&+/*+ (007+(09 (1/ )"9$#0(3 +*(3#@(%3* )+&21&$*/ 8(37*$ =-&1&4
,&1#0#,9> #$ 8*+9 0"(33*12#12 (1/ /*$*+8*$ ( 0"(),*+ &1 #,$ &;1 .&+ ( 0&-)+*"*1$#8*
/#$07$$#&15 C* ;#33 1&, /#$07$$ +*0&1$,+70,#&1 !3,*+#12 .7+,"*+6 (3,"&72" #, "($ )+&4
.&71/ #-)(0, &1 ,"* /#..7$#&1 (1/ /#$)*+$#&1 )+&)*+,#*$ &. ( $0"*-* (, $-(33 $0(3*$5
8.5.2.2 Two-dimensional reconstruction functions
!;&4/#-*1$#&1(3 +*0&1$,+70,#&1$ 0(1 %* &%,(#1*/ 7$#12 1*(+39 /#+*0, 2*1*+(3#@(4
,#&1$ &. ,"* -*,"&/$ )+*$*1,*/ #1 $*0,#&1 [5R5K5M5 G1 .(0,6 .&+ $*0&1/4&+/*+ (007+(,*
$0"*-*$ ,"(, 7$* ( 3#1*(+ +*0&1$,+70,#&1 ,"* 3#1*(+ /*+#8(,#8*$ 0(1 %* 0(3073(,*/ #1
*(0" /#+*0,#&1 #1/*)*1/*1,39 =/#-*1$#&14$)3#,,#12>5
(-23);?:52?$3) @:3>*-6 <3:65*"?$3)
!"* ,;&4/#-*1$#&1(3 +*0&1$,+70,#&1 )+&%3*- .&+ ( !1#,*48&37-* $0"*-* 7,#3#@#12
( )&391&-#(3 %($#$ #$ (1(3&2&7$ ,& ,"* &1*4/#-*1$#&1(3 0($*^ W#8*1 /#$0+*,* 0*334
(8*+(2*/ 8(37*$ s C &8*+ 0*33$ &C 6 /*,*+-#1* 0&*.!0#*1,$ 2 ($, L) 6 $ + L ≤ @ =$ (1/ L (+*
L &+ )&$#,#8* #1,*2*+$>6 $& ,"(,
sC (B, J) =
-
$+ L≤@
($, L) $ L
2C
BJ ,
=[5Ra>
#$ (1 ())+&?#-(,#&1 ,& ,"* 71/*+39#12 !*3/ s #1 0*33 & C 5 !"* 0*334(8*+(2*/ )+&)*+,9
#1 ,;& /#-*1$#&1$ ,"*1 +*(/$6
&
&C
sC (B, J) %& = s C 6 &C .
=[5Rh>
:2(#1 ;* 0(1 0"&&$* ,& $"#., ,"* +*0&1$,+70,#&1 $& ,"(,6 .&+ $#-)3#0#,96 & C "($ (
0*1,+&#/ 3&0(,*/ (, (B, J) = (L, L)5
1$-2-D$;- /3);?")? A-?,3% c?D3 %$6-);$3);d
!"* *?,*1$#&1 &. ,"* <D&U ,& ,;& /#-*1$#&1$ #$ ,+#8#(36 2#8*1 %9
sC (B, J) = s C .
=[5R[>
[ I#1#,*4E&37-* !+(1$)&+, S0"*-*$
KZR
!"#$ $0"*-* $7..*+$ .+&- ,"* $(-* /*!0#*10#*$ ($ /#$07$$*/ #1 ,"* &1*4/#-*1$#&1(3
0($*6 (1/ $& #$ 1&, /#$07$$*/ .7+,"*+ "*+*5
1$-2-D$;- G$)-": A-?,3% c?D3 %$6-);$3);d
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(M,L)
sC (B, J) = s C + 2C
(M,L)
:19 0"&#0* &. 2 C
(L,M)
(1/ 2C
(L,M)
B + 2C
J.
=[5RO>
;#33 9#*3/ ( -($$40&1$*+8(,#8* +*0&1$,+70,#&1 (1/6 ($
(M,L)
(L,M)
(1/ 2C
0&++*$)&1/ ,& ,"* 0&-)&1*1,$ &.
;#," ,"* &1*4/#-*1$#&1(3 <XU6 2 C
,"* 2+(/#*1, (3&12 *(0" 0&&+/#1(,* /#+*0,#&1 (, ,"* 0*33 0*1,+&#/6
!
".
, sC ..
(M,L)
=
,
2C
, B .(B,J)=(L,L)
!
".
, sC ..
(L,M)
2C
=
.
,J .
(B,J)=(L,L)
I&+ ,"#$ -*,"&/6 ,"* 2+(/#*1, 0(1 %* 3#-#,*/ ($ #1 ,"* &1*4/#-*1$#&1(3 0($* =$**6
*5256 B7'&;#0@ (1/ \(7-2(+/1*+6 KLLL>5
F$E,K3:%-: :-23);?:52?$3); c?D3 %$6-);$3);d
!+7* ,"#+/4&+/*+ (1/ "#2"*+ $0"*-*$ +*A7#+* $&-* -*,"&/ &. #10&+)&+(,#12 0+&$$4
/*+#8(,#8*$ #1 &+/*+ ,& %* .&+-(339 ,"#+/4&+/*+ (007+(,*5 I&+ *?(-)3*6 ( ,+7* ,"#+/4
&+/*+ )(+(%&3#0 +*0&1$,+70,#&1 0&73/ -('* 7$* &. ( +*0&1$,+70,#&1 &. ,"* .&+(M,L)
(L,M)
sC (B, J) = s C + 2C B + 2C J
=[5aL>
!
!
"
"
K
K
(6 B)
(6 J)
(K,L)
(M,M)
(L,K)
+2C
BK −
JK −
+ 2C BJ + 2C
,
MK
MK
(M,L)
(L,M)
(K,L)
(L,K)
(M,M)
(+* &%,(#1*/ %9 (2(#1 ())+&?#-(,#12 ,"*
;"*+* 2C 6 2C 6 2C 6 2C 6 (1/ 2C
/*+#8(,#8*$ &. s 5 N&,* ,"(, 2 (L.L) /&*$ 1&, *A7(3 ,"* 0*33 (8*+(2* s %7, #1037/*$
-&+* ,*+-$ ,& *1$7+* ,"* -($$40&1$*+8(,#&1 )+&)*+,95 F?,*1$#&1$ &. ,"#$ .&+- (+*
/*$0+#%*/ #1 N(#+ (1/ U(0"*1"(7*+ =KLLK> (1/ _33+#0" *, (3 =KLML>5 G, "($ %**1
$"&;1 ,"(, ,"* 3&$$ &. (007+(09 (,,+#%7,*/ ,& 1*23*0,#12 ,"* 0+&$$4/*+#8(,#8* ,*+(M,M)
0(1 %* 3(+2*6 %7, #$ 3*$$ $#21#!0(1, &1 2+#/$ &. 3&; +*$&37,#&1 =X(7+#,@*1 *, (36
2C
KLML>5
KZa
X(7+#,@*1 *, (35
1$-2-D$;- 1":">3*$2 A-?,3% c?D3 %$6-);$3);d
: +#2&+&7$ *?,*1$#&1 &. ,"* <<U -*,"&/ #1,+&/70*/ %9 D&3*33( (1/ C&&/;(+/
=MO[P> ;&73/ +*A7#+* ,"* .7339 ,;& /#-*1$#&1(3 %#)(+(%&3#0 )&391&-#(3$ ,& %* 0&14
,#17&7$ (0+&$$ 0*334%&+/*+$ (, $*3*0,*/ )&#1,$ (1/V&+ #1 $&-* (8*+(2* $*1$*5
H1* $70" *?,*1$#&1 ;($ /*8*3&)*/ %9 Y(1#0#s0 =MOOK>6 ;"& 0"&$*
s (B, J) = qK (J) BK + qM (J) B + qL (J),
=[5aM>
qL (J) = qLK JK + qLM J + qLL ,
qM (J) = qMK JK + qMM J + qML ,
qK (J) = qKK JK + qKM J + qKL .
=[5aK>
;"*+*
=[5aZ>
=[5aP>
!"#$ +*0&1$,+70,#&1 "($ 1#1* /*2+**$ &. .+**/&-6 ;"#0" (+* +*$,+#0,*/ %9 $(,#$.9#12
=#> ,"* 0*334(8*+(2*/ 0&1$,+(#1, =[5Rh> #1 0*33 '6 =##> *A7(3 (8*+(2* 8(37*$ (3&12 *(0"
&. ,"* .&7+ */2*$ &. ,"* A7(/+#3(,*+(3 0*33$6 (1/ =###> 0&1,#17#,9 (, ,"* 0&+1*+ )&#1,$
&. *(0" 0*335 !"*$* +*$,+#0,#&1$ 3*(/ ,& O 0&1$,+(#1,$6 (1/ "*10* /*!1* ( 71#A7*
+*0&1$,+70,#&15 N&,* ,"(, ,"* +*0&1$,+70,#&1 #$ 1&, 23&%(339 0&1,#17&7$5 C* +*.*+ ,&
Y(1#0#s0 =MOOK> .&+ .7+,"*+ /*,(#3$ &1 ,"#$ (32&+#,"-5
MB?-);$3); ?3 $::-E5*": E:$%;
:33 &. ,"* -*,"&/$ /*$0+#%*/ (%&8* (+* ,#*/ ,& A7(/+#3(,*+(3 =&+,"&2&1(3> -*$"*$
(1/ ,"* *?,*1$#&1 ,& ,+#(1273(+6 "*?(2&1(3 (1/ &,"*+ 2+#/$ ;"*+* ,"* 0*33$ /& 1&,
"(8* *?(0,39 .&7+ 8*+,#0*$6 #$ 1&, &%8#&7$5 !"* (7,"&+$ (+* 1&, (;(+* &. (19 +#2&+&7$
*?,*1$#&1$ &. <<U ,& $70" 2+#/$ ;"*+* 0&1,#17#,9 (0+&$$ 0*33 %&+/*+$ #$ *1.&+0*/5
G1 ,"#$ 0($* *1.&+0*-*1, &. ,"* 0*334(8*+(2* )+&)*+,9 #$ -&+* /#.!073,6 (1/ +*A7#+*$
$)*0#(3 ,+*(,-*1, &. ,"* )(+(%&3#0 ,*+-$5 I&+ #1$,(10*6 ;* -7$, "(8*
2
3
2
3
2
3
(L,L)
(K,L) K
(K,L)
(L,K) K
(L,K)
(M,M)
B − 6C
+ 2C
J − 6C
+ 2(M,M) B J − 6C
, =[5aR>
2C = s C + 2C
($, L)
=_33+#0" *, (36 KLLO> ;"*+* 6 C
(+* ,"* (+*(4(8*+(2*/ "#2"*+4&+/*+ -&-*1,$
($, L)
6C
=
M
6 &C
&
&C
B$ J L %&.
:))+&?#-(,#&1 &. ,"* /*+#8(,#8* ,*+-$ -(9 %* /#.!073, &1 #++*273(+ 2+#/$5 I&+
2+#/$ ;"*+* !1#,*4/#..*+*10* ())+&?#-(,#&1$ ,& ,"* /*+#8(,#8*$ (+* 1&, &%8#&7$ ,&
0&-)7,*6 ($ #$ ,"* 0($* .&+ 0&-)3*,*39 71$,+70,7+*/ 2+#/$6 &1* -#2", 7$* ( ,;&4
/#-*1$#&1(3 *?,*1$#&1 &. ,"* X()+#$* (1/ <3(1,* =MOOR> -*,"&/5 !"(, #$6 *1.&+0* ,"*
-($$40&1$*+8(,#&1 0&1$,+(#1, 1&, &139 #1 0*33 C %7, #1 ( $*, &. (/](0*1, 0*33$5 I&+ 2+#/$
#1 ;"#0" 0*33 C "($ ( 8(+#(%3* 17-%*+ &. (/](0*1, 1*#2"%&+$ ,"#$ ())+&(0" -(9 1&,
[ I#1#,*4E&37-* !+(1$)&+, S0"*-*$
KZh
%* &),#-(35 G1 $70" 0($*$ ( 3*($,4$A7(+*$ ())+&(0" -#2", %* ( -&+* 1(,7+(3 0"&#0*
,& (8&#/ %#($*$ #1,+&/70*/ %9 *?037/#12 $&-* (/](0*1, 0*33$ (1/ 1&, &,"*+$5 C"*1
7$#12 ( 3*($, $A7(+*$ -*,"&/ &1* -(9 0"&$* ,& &),#-#@* ,"* ())+&?#-(,#&1 ,& ,"*
0&*.!0#*1,$ 1&, ]7$, ,& -($$40&1$*+8(,#&1 #1 (/](0*1, 0*33$ %7, (3$& ,& @4*?(0,1*$$
.&+ *?(-)3* =$**6 *5256 \(+," (1/ I+*/*+#0'$&16 MOOL>5
+D3K%$6-);$3)"* *$6$?-:; f !*?-:;
Y*0&1$,+70,#&1 .710,#&1 !3,*+#12 #1 ,;&4/#-*1$#&1$ #$ $#21#!0(1,39 -&+* 0&-)3#4
0(,*/ ,"(1 #1 &1* /#-*1$#&16 $#-)39 %*0(7$* ( ,;&4/#-*1$#&1(3 )&391&-#(3 &. /*4
2+** K =( )(+(%&3#0 +*0&1$,+70,#&1> 0(1 )&$$*$$ (1 *?,+*-( ;#,"#1 ( 0*336 (3&12 (
0*33 %&71/(+96 &+ (, ( 0&+1*+4)&#1, =(33 &. ;"#0" -7$, %* 0"*0'*/>5 T*10*6 !3,*+4
#12 0&-*$ #1 ,;& "(8&+$^ B#-*1$#&1(339 $)3#, !3,*+#12 (1/ .7339 ,;&4/#-*1$#&1(3
())+&(0"*$5
!"* /#-*1$#&1(3 $)3#, ())+&(0" $#-)39 ())3#*$ ,"* &1*4/#-*1$#&1(3 !3,*+$ )+*4
$*1,*/ #1 $*0,#&1 [5R5K5M #1 *(0" 0&&+/#1(,* /#+*0,#&15 T&;*8*+6 %9 /&#12 $& ,"*
*1,#+* +*0&1$,+70,#&1 s C #$ 1&, 27(+(1,**/ ,& %* -&1&,&1#0 ;#,"#1 & C 5 S)*0#!0(3396
,"*+* (+* 1& 27(+(1,**$ &. -&1&,&1#0#,9 (, 0*33 0&+1*+4)&#1,$ ;"*+* ,"* +*0&1$,+704
,#&1 #1 *(0" 0&&+/#1(,* /#+*0,#&1 #$ (//#,#8* =X(7+#,@*1 *, (36 KLLa>5
S,+#0, -&1&,&1#0#,9 (, (33 )&#1,$ ;#,"#1 ( 0*33 0(1 %* 27(+(1,**/ 7$#12 ,"* .7339
,;&4/#-*1$#&1(3 ())+&(0" &. \(+," (1/ J*$)*+$*1 =MO[O>6 ;"#0" 0(1 (3$& %* ())3#*/
,& 71$,+70,7+*/ 2+#/$5 !"#$ !3,*+ 27(+(1,**$ $,+#0, -&1&,&1#0#,9 &. 3#1*(+ +*0&1$,+704
,#&1$ %9 !+$, /*,*+-#1#12 ;"*+* ( 2#8*1 3#1*(+ +*0&1$,+70,#&1 "($ *?,+*-( =,"#$ #$
,9)#0(339 ,"* 0*33 0&+1*+4)&#1,$>6 (1/ ,"*1 +*$0(3#12 ,"* 3#1*(+ /*+#8(,#8*$ $& ,"(,
,"* 3#1*(+ +*0&1$,+70,#&1 #$ -&1&,&1#0 ;#," +*$)*0, ,& #,$ 1*#2"%&+$5 !"#$ ())+&(0"
;($ (3$& *?,*1/*/ ,& )(+(%&3#0 =,"#+/4&+/*+> +*0&1$,+70,#&1$ %9 _33+#0" *, (3 =KLLO>6
;"#0" ())3#*$ +*$0(3#12 ,& %&," 3#1*(+ (1/ "#2"4&+/*+ /*+#8(,#8*$5 G. $,+#0, -&1&4
,&1#0#,9 #$ 1&, 1*0*$$(+96 ( CFNH4,9)* 0+#,*+#&1 0(1 %* 7$*/ ,& #/*1,#.9 )3(0*$
#1 ;"#0" ( !3,*+ $"&73/ %* ())3#*/5 :1 *?,*1$#&1 &. ,"* &1*4/#-*1$#&1(3 CFNH4
!3,*+#12 #1 \3&$$*9 (1/ B7++(1 =KLL[> 0(1 %* .&71/ #1 T(++#$ *, (3 =KLML>5
I&+ "7?43#-#,#12 ,"* -&$, ;#/*39 7$*/ -*,"&/ #$ "7?40&++*0,*/ ,+(1$)&+, =ID!>
#1,+&/70*/ %9 f(3*$(' =MOhO>5 :$ #1 &1* /#-*1$#&1 #, $**'$ ,& !1/ ,"* &),#-(3
t%3*1/#12u &. ( -&1&,&1* "7? (1/ ( "#2"4&+/*+ 1&14-&1&,&1* "7?5 ID! #$ /*4
$0+#%*/ #1 /*,(#3 #1 B7++(1 =MOOO> (1/ "*10* 1&, +*)*(,*/ "*+*5
8.5.3 Practical integration over areas
I&+ ,"* ())+&?#-(,#&1 &. ,"* &8*+3() #1,*2+(3$ #1 =[5Ka> (1/ =[5ZZ> ;* "(8* &139 /#$4
07$$*/ "&; ,& ())+&?#-(,* ,"* &8*+3() (+*($ (1/ "&; ,& /& +*0&1$,+70,#&1$ $& .(+5
G, +*-(#1$ ,& %* $"&;1 "&; ,& 2& (%&7, #1,*2+(,#12 ,"* $7%42+#/4$0(3* +*0&1$,+704
,#&1 .710,#&1 &8*+ ,"(, (+*(5 G. ,"* $#/*$ &. ,"* &8*+3() (+*($ (+* (3#21*/ ;#," ,"*
0&&+/#1(,* 3#1*$6 /#+*0, #1,*2+(,#&1 #$ 7$7(339 $,+(#2",.&+;(+/ $#10* ,"* #1,*2+(3$ *.4
KZ[
X(7+#,@*1 *, (35
.*0,#8*39 +*/70* ,& &1* /#-*1$#&1 =$**6 .&+ *?(-)3*6 N(#+ (1/ U(0"*1"(7*+ =KLLK>>5
T&;*8*+6 #. ,"* &8*+3() (+*( #$ (33&;*/ ,& %* (1 (+%#,+(+9 )&392&1 ,"* #1,*2+(,#&1 #$
-&+* #18&38*/5 !"*+* (+* %($#0(339 ,;& ())+&(0"*$ ,"(, *?(0,39 #1,*2+(,* )&391&-#(3
.710,#&1$ &8*+ )&392&1$5 I#+$,396 /#+*0, #1,*2+(,#&1 &8*+ &8*+3() (+*($ 7$#12 W(7$4
$#(1 A7(/+(,7+*5 S*0&1/396 ,"* (+*(4#1,*2+(3$ 0(1 %* 0&18*+,*/ #1,& 3#1*4#1,*2+(3$ 8#(
W(7$$4W+**1Q$ ,"*&+*-5
\&," &. ,"*$* ())+&(0"*$ (+* /#$07$$*/ %*3&;5 C* ($$7-* ,"(, ,"* &8*+3() 0*33
$#/*$ (+* $,+(#2", 3#1*$ ;#," (+%#,+(+9 &+#*1,(,#&1 (1/ ,"(, ,"* &8*+3() (+*( " C! #$
$#-)39 0&11*0,*/5 !"#$ #$ ,"* -&$, 2*1*+(3 0($*5 U(,"*-(,#0(339 ,"* )+&%3*- #$
$,(,*/ ($ .&33&;$^ W#8*1 ( +*0&1$,+70,#&1 .710,#&1 #1 0*33 & ! ;"#0" #$ ( )&391&-#(36
$(9 &. &+/*+ Z6
($, L)
s! (B, J) = - 2! B$ J L ,
=[5aa>
$+ L≤K
($, L)
;"*+* 2!
(+* +*0&1$,+70,#&1 0&*.!0#*1,$6 0&-)7,* ,"* #1,*2+(3
&
"C!
s! (B, J)%&.
=[5ah>
8.5.3.1 Direct integration using Gaussian quadrature
I&+ ,"* /#+*0, #1,*2+(,#&1 7$#12 W(7$$#(1 A7(/+(,7+* #, #$ &.,*1 0&18*1#*1, ,& %+*('
7) "C! #1,& ,+#(123*$ ;"#0" #$ ,"* 0($* ;* ;#33 /#$07$$ "*+*5 S&6 .&+ $#-)3#0#,96 $7)4
)&$* ,"* &8*+3() (+*( #$ (3+*(/9 (1 (+%#,+(+9 ,+#(123* M[ ;#," 8*+,#0*$ 3&0(,*/ (, B C!,, 6
JC!,, 6 , = M, K, Z6 (1/ 17-%*+*/ 0&71,*+403&0';#$*5 F?(0, #1,*2+(,#&1 &. ,"* )&394
1&-#(3 =[5aa> 0(1 %* (0"#*8*/ 7$#12 W(7$$#(1 A7(/+(,7+* ;"#0" ())+&?#-(,*$ ,"*
#1,*2+(3 #1 ,*+-$ &. ( ;*#2",*/ $7- &. .710,#&1(3 *8(37(,#&1$ (, A7(/+(,7+* )&#1,$5
!"* A7(/+(,7+* )&#1,$ (+*
(")
(")
(>)
(>)
(BC! , JC! ) =
(BC! , JC! ) =
(2) (2)
(BC! , JC! )
=
4
5
M
a PBC!,M + BC!,K + BC!,Z , PJC!,M + JC!,K + JC!,Z ,
4
5
M
a BC!,M + PBC!,K + BC!,Z , JC!,M + PJC!,K + JC!,Z ,
4
5
M
a BC!,M + BC!,K + PBC!,Z , JC!,M + JC!,K + PJC!,Z .
=[5a[>
=[5aO>
=[5hL>
=B7'&;#0@ (1/ \(7-2(+/1*+6 KLLL> (1/ ,"* #1,*2+(3 &. s ! (B, J) &8*+ ,"* &8*+3()
,+#(123* "C! #$ 2#8*1 %9
&
"C!
s! (B, J) %& =
3
6 "C! 2
(") (")
(>) (>)
(2) (2)
s! (BC! , JC! ) + s! (BC! , JC! ) + s! (BC! , JC! ) ,
Z
=[5hM>
;"*+* 6 "C! #$ ,"* (+*( &. "C!
6 "C! =
M[
0
M/
(BC!,K − BC!,M)(JC!,Z − JC!,M) − (JC!,K − JC!,M)(BC!,Z − BC!,M ) .
K
1&,* ,"(, (19 (+*( ;#," $,+(#2", 3#1* $#/*$ 0(1 %* %+&'*1 7) #1,& ,+#(123*$
=[5hK>
[ I#1#,*4E&37-* !+(1$)&+, S0"*-*$
KZO
N&,* ,"(, ,"* A7(/+(,7+* )&#1,$ &139 "(8* ,& %* 0&-)7,*/ &10* .&+ *(0" &8*+3()
(+*( (1/ 0(1 ,"*1 %* +*47$*/ .&+ *(0" (//#,#&1(3 ,+(0*+ =$#10* (33 ,+(0*+$ .&33&; ,"*
$(-* ,+(]*0,&+#*$V(+*($>5 I&+ "7?4.&+- ,+(1$)&+, $0"*-*$ *.!0#*1, (32&+#,"-$ 0(1
%* /*$#21*/ ,"(, /*0&-)&$* ,"* &8*+3() (+*($ #1,& ,+#(123*$ #. ,"* "7?4(+*($ (+*
0&1!1*/ ,& 1*(+*$, 1*#2"%&+$ =B7'&;#0@ (1/ \(7-2(+/1*+6 KLLL>5 I&+ 3&12*+ ,#-*4
$,*)$ ;"*+* ,"* "7?4(+*($ -(9 $)(1 $*8*+(3 F73*+#(1 0*33$ 1&, $"(+#12 ( .(0* ;#," ,"*
"7?4.(0*6 ,"* /*0&-)&$#,#&1 #1,& ,+#(123*$ #$ -&+* 0&-)3#0(,*/5 G1 $70" $#,7(,#&1$
#, -(9 %* -&+* 0&18*1#*1, ,& 7$* ,"* 3#1*4#1,*2+(3 ())+&(0" /*$0+#%*/ 1*?,5
8.5.3.2 Converting area-integrals into line-integrals
!"#$ ())+&(0" ;($ &+#2#1(339 #1,+&/70*/ %9 B7'&;#0@ =MO[P> (1/ B7'&;#0@ (1/
d&/#$ =MO[h> #1 17-*+#0(3 $0"*-*$^ I&+ ,"* $#-)39 0&11*0,*/ &8*+3() (+*( " C! =1&,
1*0*$$(+#39 ( ,+#(123* %7, (19 )&392&1> ,"* .&33&;#12 #1,*2+(3 *A7(,#&1 "&3/$6
&&
"C!
s! (B, J) %& =
+
, "C!
[1 %B + N %J],
=[5hZ>
;"*+* , "C! #$ ,"* %&71/(+9 &. " C! 5 !"* )&,*1,#(3$ 1 = 1(B, J) (1/ N = N(B, J) (+*
0"&$*1 $70" ,"(, ,"*9 $(,#$.9
−
,1 ,N
+
= s! (B, J).
,J ,B
!"* #1,*2+(3 &. ,"* )&391&-#(3 +*0&1$,+70,#&1 .710,#&1 s ! (B, J) #1 =[5aa> 0(1 %*
;+#,,*1 ($
&
($, L) ($, L)
s! (B, J) %& = - 2! DC! ,
=[5hP>
"C!
($, L)
;"*+* 2!
%9
$+ L≤K
($, L)
(+* ,"* +*0&1$,+70,#&1 .710,#&1 0&*.!0#*1,$ (1/ D C! (+* ;*#2",$ 2#8*1
KPL
X(7+#,@*1 *, (35
54
5
M ., 4
BC!,, + BC!,,−M JC!,, − JC!,,−M ,
K ,=M
(L,L)
=
(M,L)
=
(L,M)
=−
(K,L)
=
(L,K)
=−
DC!
DC!
DC!
DC!
DC!
(M,M)
DC!
=[5hR>
54
5
M ., 4 K
BC!,, + BC!,, BC!,,−M + BKC!,,−M JC!,, − JC!,,−M ,
a ,=M
54
5
M ., 4 K
- JC!,, + JC!,, JC!,,−M + JKC!,,−M BC!,, − BC!,,−M ,
a ,=M
54
54
5
M ., 4
- BC!,, + BC!,,−M BKC!,, + BKC!,,−M JC!,, − JC!,,−M ,
MK ,=M
=[5ha>
=[5hh>
=[5h[>
54
54
5
M ., 4
JC!,, + JC!,,−M JKC!,, + JKC!,,−M BC!,, − BC!,,−M , =[5hO>
MK ,=M
6,
4
5
M .,
=
JC!,, Z BKC!,, + K BC!,, BC!,,−M + BKC!,,−M +
KP ,=M
4 K
5
K
JC!,,−M BC!,, + K BC!,, BC!,,−M + Z BC!,,−M
7
(JC!,, − JC!,,−M) , =[5[L>
(1/
(BC!,, , JC!,, ),
, = M, . . . , .,
=[5[M>
(+* ,"* 0&&+/#1(,*$ .&+ ,"* $#/*$ &. ,"* &8*+3() (+*( " C! 17-%*+*/ 0&71,*+403&0';#$*5
S& ., = Z .&+ ,+#(1273(+ &8*+3() (+*($6 . , = P .&+ A7(/+#3(,*+(3 " C! Q$ *,05 N&,* ,"(,
=BC!,,−M 6 JC!,,−M > (1/ =BC!,, 6 JC!,, > (+* 0&1,#27&7$ )&#1,$ =/*!1#12 ( 3#1* $*2-*1,> (1/
,"* #1/*? , #$ 0903#0 $& ,"(, , = L *A7(3$ , = . , 5
($, L)
!"* ;*#2",$ DC! 2#8*1 #1 *A7(,#&1$ =[5hR4[5[L> "(8* %**1 /*+#8*/ %9 7$#12
=[5hZ> ;#," ,"* .&33&;#12 )(#+$ (1 ($, L) , N($, L) )
!
"
1(L,L) = L, N(L,L) = B ,
"
!
K
1(M,L) = L, N(M,L) = BK ,
"
!
JK
(L,M)
(L,M)
=− , N
=L ,
1
K
"
!
(K,L)
BZ
(K,L)
= L, N
= Z ,
1
"
!
JZ
(L,K)
(L,K)
=− , N
=L ,
1
Z
"
!
BK J
(M,M)
(M,M)
= L, N
= K .
1
[ I#1#,*4E&37-* !+(1$)&+, S0"*-*$
KPM
!"* 0"&#0* &. 1 (1/ N #$ 1&, 71#A7* (1/ 0(1 &.,*1 %* 0"&$*1 .&+ 0&18*1#*10*5 T*+*
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KPZ
pt o p
Topography
ps
Fig. 8.18 : 2+()"#0(3 #337$,+(,#&1 ,"* "&(,#12 X(2+(12#(1 0&&+/#1(,*5 !"* 8*+,#0(3 0&&+/#1(,* #$
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8.6.1 Floating Lagrangian vertical coordinate
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