High-order well-balanced schemes

High-order Well-balanced Schemes
Sebastian Noelle, Yulong Xing, Chi-Wang Shu
Contents
!"#%$&('()*!+,-.$/$&0-12-!'-%$&34
6571("#!'8,%$&'-(0:9;! <,%$&.$&(!>=% -2 ?@(!!/AB!-$C
'DFE
G 6571("#!'8,%$&'-(0:9;! <,%$&.$&(!:+%$&
$C->
4HI"#(($&JK$&"#A!'8LM;
12'
1%$&(9N98$C-(O!+$C
'PFQ7
R6ST!'
FR7
"!#$%&'!(*)+,$-'&.!#)/)012134)5761'!$'898:34;589-'<6
*-7+,=?>@- 575-''A-CBD<
E1F)BG&HI';)89-AHIA+,J)BG;895-F8K!L
1M N1A1-O)B)13P*%&K58?<QRE8;1E+S*)89EI)TJ!L M1M UV-'
)B$-)&%W- 9%FXEINIY6Z*)E1-C +,?-<1-8"<-7[)5'<=F\]88:34;58K5-6
*-7+,G5AB^H_`6I9*- 'a5589)2E1?)BVG;895-a6,
b1''BD)5a5;8
)+cR7_c<QRE8;1E+dA75=.\[a6I*-E*#e!$)`-89**'f)Bg*- 1+,T&)1
;5*<6Z)Z12134)5761'`5--EI75Tg)134)*-8895)5*HZh:C6I:UV'-'0)'7
)
!L-70
G!$'898:34;589-'<6FBD)5"J2 78I-89*.)B<QRE8;19ITW6YG)1'.)1
;5*<6,)0!#8834;891-<6cQRE6&75EIT!L19-7,-<5c3g,1-'918L3;J1896
)i588"Q&E189;IK&=,>J8-<
9)Y-'89E61F<QRE89;I9ij&kLl mnk*Tg!L1'CX)<!
%W8)R-:eHO%5*To56p589*)_`+,)-788122_q_rstuIvwrxy<QRE8;19I=
zaE1+,'-<8o'NI'+,Rf*)<!{'NI-88RL*)8E1)p)BE1'*E1;6|L!$'898
J*82R8:H|'*EI;<6Q&E189;IK&=
}~._gRG1y7oIK^R?R'5io& ÎoLPooV&i1&Y1oK^Ko DVaVÎAI5
VYKoZoo9',II^Vi, o5ibKKAo&D&o5Z Vi
1. Introduction
*[p&¡O&oVKD5& ÎobaFI¢ 5[5g
Io1'5a¡5'I^^Fo&D&g
DW`Yo^ZK
IV,î5o K1[I 0^[ V0P1
£¤I¥^¤¦
U + f (U, x) = s(U, x)
t
x
YV5' U ^G'VY71K'K1_1<'I f (U, x) D# oJ§oV¨&o s(U, x) ^G'VY71V'5L'5' ¥
©?V,71V'5`'5'ªpW¡O
IiCP'I«IIi
' D5Ie'1< ÎFI?&'V5`5Io D5R'K1o ¥
¬f¨V&iV^fI¡5'g1KDao&D&o5Y^W`$ô5KoVY VF7oIK^R®?R'5f1gR ÎiYK O
VI§g&`gI7 1'IIÎ1¡IV^1'^,?WI01gR Îy¯:°W±AgVi1 5o 9'K1,iRI5i1
¯ ¤W² ±oVI³³5^F§gR´¯ ¤ ±Ao&gOaiVoI C§oR´¯:° ± ¥
µ 1io&'KoY9'¶ VZ&oVI'·¡g I^DO
1o7 R& Î·^W`Gg&i5^¡ £7¤I¥^¤¦
Y9' s(U, x) = 0 b oo| D5I`&Vo W¨î& Î® y oo&D&o5]^W` £7¤I¥^¤¦ D
o7g&^K¡¸V&,']oy|1 OV9¹O
o9ºcbO ^|oK¡¸V5· V,'VOI^cWIKo
£ PICoo9'iV9»5'5o5|'V| ¦ IF oi55^fWI5&1 £ PICoo9'1I^Vi|'V5i ¦
&'VC Io'5Y s(U, x) ^ <'K¡| oCD'
'
^³& ÎOI. V0 oR'Îg1g'& I ¥
©?o5'JD5RVRa51551VJVI D
&V^L
¨V55'K1 ¥ ©?Vbo&D&g
JDW £¤1¥K¤W¦ &¢'5_1i9
7 I¡]7'R'_ I^ Îo`K Yo^]'V_ IV
0'5' s(U, x) ^
¨VI
^¡MoIÎo
[¡
'V§oÏI^5 f (U, x) ¥ o(7 1¡]R |71KV ÎoC&'co o&^^¡ZV1 'KD&
£ V¡{I [o o&^K¡®V&pIx^¡oIiÎ^ ¦ &o® V¡®I¢ 5@5I '¡yKiI 'I1pV¡ ^&
i&V^V £ P1,
¨V&iVÎ.'Vp'K^L?R'50I,7 1¡ iRKoZ?R'5,71KV Î·&J'V
oIK^R ?R'5Y1gR Î"V p_7 oVKKi1 ,5'&^"DR'5`^] oô&5 ¦<¥ ©?V
1
Î,I$a5^KoIÎo
] ViYD? MV 7 10
¨VI
^¡Z Ii,&$ o _'1¡
7'& ?71KV Îo ¥ ©?Vai17G^|I 'I#IVWI'&1L&Va5^KoIÎo
, o5iG^" gR
'V5¡&I
VR'5^¡M' IÎ,7pIKGg7'V gR Îg? M o 7 I¡[R 71K'K1o
YK ]'5DR'K15^¡O
1I' i7V ¥ *]
1|g&'^ IVIVa5^9AoIÎo
O o5i0YK^
^ 'o
?' oo5& Î|5' 1'" 0'V`'1¡_7'R'Y71K'K1&Vo
YKJ5IVVI$' IÎ
p&ô5 V'o& ÎoL _ oO7 1¡iR LVoK L V 'Vo5& Îp 'IfDbIK'1¡
p&^K$ oIp7o|5 V'oR'K1oR o$'V^ ^V00 5oV_i ¥ *
Î ² b
Y^ô RDJ oc
¨VI|oK ¥ RbI9#^$VK a9¹O5V9# C Îca5^9AoIÎo

'V|0YVD·I |VÎAI50I
VR'|Io¸VIA1'
^^^& I'¡]K¸ opV 7o
|&
V^'
11'KVK ^?^[ Vc I^'K1 ¥
*Z o^?oIg?bCo 'V,7oIK^R ?R'5aoR'K1Z1aV'&'&¡gF i V 15¡O
Pd'55 ^¡O515ÎMbKKg&D&o5OoK11'V5boVK ,K» o
1oVK 0I1KVi
Io@D 5I ^VIo0IK
^ oVK Ki5p|5 VV ¥ ]&7'5ip ¶5¨VÎK
'VMpIK¶^VI'VK1c^¶ o M&Î1 K Vp,Yo^ IK^R gc'(IV^51i VZbKK
g&D&o5[V'I5 ¡MYK V1^1 KViI V`VD
cV'I5 ^?I$ Vc1 ÎôI ViI
oO1fVÎAI5L155V'1
¡_IoiV1 5K^DR 1 ¡,5 PI'p&o5YKi'V`V' 5g
Y&
I^'K1[D'
I'KVK ^ ¥
©?ooIgbDJ1 &V^³5pILPI^KR` ¥ *
ÎZ°cbo?^'
g bcV_5a&
^ 'KV[V^^KV'^V 7'R' ¥ ©?o5·b|K' o5_'Vp' ^g&$YVD(V(
,a5^9Ao&D&g
ZV&YR'K1o&'¡OR' ¥
Y'VDYg1KY'VcoIg`7V^K'Y^ iaiIVV'11Vº#©?o0o'7`&Vo IV 5
<'K1 IVV^KL'coVK 0K» o
1IgV9'F1I^ViFIoOD 5I ^VIg0&^5K
'V| ¥ Ap' RiV^K^V'^¸PIpYo^{ V Io'5Z'5' 5I@g
1|1
^ ( Vp,I?o o
'_&a VpP1 £ ¥ ¦¥ ©?VMoIK^5oIi^c 5Io7 'o<_oVK
V9»5'5o5_1g'& IYYVD(I coK11'V515
o'& c&gVIV 5K^^& 1 ¡pP1'V
5Io 5'RR'K1c§o¨(V9»5'5o5Io'Vp71V'5, ]Io YVD¸5Iô
D_PI0gI
'5'pYKZ oc5I 0&$V^K^V'Ko71KV Îo ¥
²
©?o 7
Ig IVV'11CV K1VdPI15V'I,V^K^V'^@&gdgV9'¶I1KVi
ViD?
R15'i^ < Î ¥ ©?V I¡i'I
i^b'_ogMa5^9AoIÎo
p1gIR
V'GPI$'V?K'5I&Io oa' Do&& 5?o& Î £ ¥^¤¦
¥ oo7<'K1 ¥ °V' 5'
IV5&JP'Ii5aI ·5Ii1 p VD|K'5I&bK'¸ V9&V^Mg& ' ¥ Vg7<
Î ¥ &^K³,'VDIVV'1IyP1iR^V ?R _1oK^Ko DP1| o&^^R ?R'5
§oR` ¥
* <'K1 ² aiV' 5,o| D5If 7o9c7oRYKVZ oMI
VI5¡ &o¸bKK
o&D&o50V'I5 ^"&V&'_
DI'7.&o'V|.P1G`Vg"Ioo&^KVI^VY§gR` ¥

Î
I&ô?71|,5Io5KoVKVi'5p& ¥
2. Preliminaries: equilibria and the residual
* VD_
Î¶aOK' o5
cYV^ g&I<'5'K³pR
ÎoI ¡¶7'R'J&g®D 5o'c'V !
bYVD®i1V9'Ic'V5DR'K1 &`'V
7¡7 5 P'Iª7'& ÎoI ¡p7'& ¥ *Zg& D
VD&b_PI'iaI" oC' Do&&'07^
I^ZI`YVD]&''V,oI ?I"'V0oVK cK»5'5o5c&ÎI'K VpaK < Î I
IV0oIo]&oM V0gV9',I1KVi,&Î1 K o«K < Î IZ o,& o5YoIo ¥
"$#%&#('*)$+(,.-,/ ,0
1"
_g_&&^¶
Ig7D50'VM ¡'5 &?o&D&o5p^W` £¤I¥^¤¦
¥2 Ic5¨V&iVÎGPI
Vc o&^^R a& YoR'K1o
£ ° ¥^¤¦ U = (h, m)T , f (U ) = (m, m2/h + gh2/2)T , s(U, x) = −ghb (x),
x
YV5' h ^#'VYa& fV5Î m ^$'V?iIi5 o £ V^'o&'Ia^¡&V^^ ¦ b(x)
^F oV''
'^g &7'I 'III&o¡]&RI|M1K15 '
P o
V5Î&o g D
V,1'WK'R'K1o&I
K'& Î ¥
3(&¡M o7¡7 5pY&[g,'5Y'9 Z^[ V0P1
£ ° ¥° ¦
Vt + c(V, x)Vx = 0
PI` Ii0WI D&oK
£° ¥ ¦
V = V (U, x),
Yo^|baaIVD,^ 1J'0&^ V 5R Kg
?
Ig& V ^|oK^f
7'& ÎoI ¡c7'& ¥ & a'oRJ5Io7'I1 V $o&fKiV^¡c'oR U ^J
Ig&& Kg

55oo_&D I x o 1VI· oOR&'ÎVK|PVg< Î b ¥ ©?V 5PI'IGIVO VIo^
V
5¨g<`V1e' ^Î.V^K^V'^ ¥
2 Ic o&^^R´?R Vp1oK^Ko ^V RI D&V^,I V (U, x) = (m, E) #YV5'
'V,V^^KV'^VV5'I¡ E DYIÎM¡
£° ¥ ¦
m2
E(U, b) = 2 + g(h + b).
2h
*i'VYP1K^RYKo,b` 5 ^`R&'K1of
DI'7 E &.V^K^V'ôI1LR Îg&'¡7'R' ¥
¥
| £7¤¦ ©?Vc5^1 ?I#&^.V^^KV'DV
Etot
£ ° ¦ i&']V^K^V'^ E
¥
smooth
£ ¦ µ Io 5'RR'K1^W`º s(U, b) ≡ 0 Io f (U ) ≡ const ¥ 'R'K1o&'¡R'
ô5KoV
5Io7'I1`7'& ¥
• E0
ap
Ig&?7'& `7o&R'M¡Z|7'& ÎoI ¡O V O1`
II< ¥
• E1 1

"¡oIi^YK c³5'ÌK
K¡IW
1o7'&"o V'IW&ocI¡FIVg
• E2 i17V&oKFPVo
Î[PI?'Vc5g7K¡ ¥
£ ¦ ¬JV^^KV'D|PI ¤ '5IÎ?o&D&o50^W` ¥
£²I¦C¤ 7g&^KR ?R'5?o& Îoº
©?V,D 1C&` E YV m ≡ 0 IoZV5o5 E = g(h + b) ≡ const ¥
•
i&'['Î5a§oR`LaR
E VYV m DYV1V³5 ¥
•
river
£ D 5I ^VIg? Îb§gR` ¦
R

•
Ewaterf all
£ ¦ o&&oK071V
C pY7 o^[^y¯ 1 ± ¥
£ &¦ C17 'IVo^ 5'
P1C° 7g&^KR a& 5VYV (u, v) ≡ (u(y), 0) g(h +
Ejet
&g f ^?'Vc
1 Î^DJP1'50KZ V,oVg`V|D VV5' ¥
b)y = f u
£ 1¦ 3o9'9A^W¡15Y o&^KRda& 5
º C
&oY&?'7`&o[iRKo
&o ¥
| £¤W¦ ©?V5',&'CpI¡M|1 ,5^1 a&#V^K^V'ô g
D&^K¡MK ° ¥
£ ° ¦ AD_îg17&c ¸V&'O'oR||bKKg&D&o5y o5i&'M ÎV·
o 7 1C1V^¡[p55 '&^7o5^1 YIfV^K^V'î5¨1<'K¡ ¥ 'V5FV^K^V'îpW¡M
o 7 1MIVV'W¨KpR'5^¡pY9'V^]i55 '&^[I5?&fI
VI5¡ ¥
<'K1 '&'FV^K^V'ÔPICYV^( Vi IV
_ 5'pFI |7o&&oKc^ 'V
7o7J& £ ¥ ¦<¥ ©?o^GKg
ô Vb^ IJ&"'7GIGYV'& I¡g ¥ ©?VLpIK,'IID.'V

Ig' o
ÎpI",bKKAo&D&o5p5^1 J&.oVK CK» o
FI5R 1' ¥ *
Î
b[I'K^V] a5^9AoIÎo
yoVK ]I1Ko|[&Vo I ¥ V^K[ oMP&i5aI·K
Vo
Î ¥^¤ 5RI'bK]V'ô
^V^0IK.VK^^V DVa,5I '¡iIY'V,759g,7 o
PIYiRKV|?R ?§oR`?^ oo7<'K1 ¥ ¥
"$#"$# ~ W~ , + 0 1"
ioi&&^®
1o7D_'V[ ¡'5 Io&D&g
MDW` £¤I¥^¤¦<¥ &'OoI7'^5
Î ^¡M^ 5'7 [K( I^'K1o`
^1 0 MV^K^V'KV]gYV5' U = 0 ¥ ©?V 5PI'IVa
t
K 'o5F'V,' ^g&
£ ° ¥:²I¦
R := −f (U )x + s(U, x).
`&'0 o&
£ ° ¥ ¦
Ut = R,
&oM Vc I^'K1 U DR'aP'I7 I¡ZR 0K$Io[IV^¡OK R 6= 0 ¥
*
Î aOY^K?7 o¡¶ 5^1 _&` 5oI'IVKpVK^^V D 'R'^7P¡KV £ ¥ ¦<¥
©?V^Y1 Vi'K1ZîVK^Y o&
£ ° ¥ &¦
R = (−f (U ) + t(U, x))x
¥ 7^Vy'V^
PIM7'& ÎoI ¡® I^'K1o5?YV t(U, x) DO
'5'iKV@¡ s(U, x) ' o
V'Ia
1o7 'o<CVK1AI5155V'& ca5^KoIÎo
ZoVK i9»5'5g
_1
5R 1' ¥
*
Î a P5oM1@oVK ·I1KVi('V|MIo@V5g
¸5Io D5[
5^
WI5&1 R &.'VF' Do& ¥ KKAo&D&o5Oo1R o L&'F5Io7 'o<'iP1J'V
IVD&Y&gZi KoIVD&?oI7?&# V70^1'51'I^ ¥
3. Schemes based on well-balanced finite difference operators
* O VDb
Î1b`P5obIOc5^1 fIV^K^V'^CP1JYVDp'V IV
?'5' D
7o&&V^?Ki'VF 5g7YI Vi'K1 ¥ ° ¥ V5IK1ibKKg&D&o5|VÎAI5
155V'& ,gV9'K»5'5o5cI5R'I?PI V 7DoI ¥ I [1 V79»5'5g

1 g'& 1'.bi Îia5^9Ao&D&g
oVK OK» o
1.oVK p1I^Vi|Io·^'
1
'KV1o 0IK
^Z'V5i ¥ ©?V,V^^KV'Doo5`5Io D5R'K1Mô
ô0'V,^ I
&` ?P1? Vc o&^^R a& YoR'K1o ¥
©?o[1V
*î5o K1o&?¡g I^D[ ¡'5 IC5Io 5'RR Î{^WìYK @ IV

'5'p$VoL
Ig7D5R'K1cD$IÎ5¡ £¤1¥K¤W¦<¥ `7'& # VYD 5o'7Î¡_V 77
^VM VbK#o&D&g
[gV9'K»5'5o5'V5i ¥ ©?V_5¨'5o Î'poVK |I1KVi
Io «'V5i`DC VRY^ 'V_P1K^RY^VM Vo <'K1o ¥ CV5&^K³R'K1Z'M'V
VK KAî5o ÎoI5I
£ ¥^¤¦
Ut + f1 (U, x, y)x + f2 (U, x, y)y = s(U, x, y)
& (1VK Ii 9'oR'K1o ¥ 2 1O
¨VI|oK1aa 5 I@I ^K¡{I5o5&^K³Z'V
o 1g7] VVD1oc M V_bIAVKi5g7ÎoIG o&^^R ?R '5FoR'K1o`YK ¸^ I
&` Y1oK^Ko ^V ¥
#.% # ,(, ~ , ~1R~ ~ ~ ~
po7_
Ig7D50'VMI 'oR £¤I¥^¤¦ Dc'5IÎCg&D&o5p^W ¥ ©?oM517i&
¡'5pYY^K.c5öK1 ODR ¥ c&'0K'5'7 Z^[o 7 ^V5¨VI<'K¡Z
5 'IK
7 I¡ZR c I^'K1o
U
I £7¤I¥^¤¦ º
£ ¥° ¦
f (U )x = s(U, x).
,p 1apI'7Vi'K1oaIZ V,oR'K1 £¤1¥K¤W¦ &oM Vc7 I¡ZR , I^ Î
I £ ¥ ° ¦ o&YbcI CK'5'7 M po 7 1
¨V1< ^¡ ¥
+ , #.% # * . U !#" $% '& () * U
. -, +
£ ¥ ¦
.
/0&12 354*.
V (U, x) = constant
V (U, x)
¯ ¤ ° I ±? oMV^^KV'^V R&'ÎVK,oW1M55{5VI ¶¡
Ko7 I[& V (U, x) ¥
a(U, x)
&'O gRiK
+ , # " # 4 £ ¥ ¦
. 3 4*# $ 4
s(U, x)
X
si (V (U, x)) t0i (x)
s(U, x) =
si
ti
4 i
pY^KL7Î·MVi D5I$'V|iYVD¸&·V' 5'I_5¨VI<'K¡]'Vp'1¡
R O I^'K1o U YVD & DP¡ '7Vi'K1 ¥^¤ P1_oIÎo
iDW £7¤I¥^¤¦ YK
¸ IV
Z & DP¡^V Vi'K1 ¥ ° ¥ iI
yV Z gRp o] oIK^R
a& 5C7¡7 YK (M^ I_&'7C7 1¡]R & DoY'V 1 Vi Îg5&o
YK^#5Iii51 VDF^& 5FK VD Vo
Î ¥ ©?o I¡ZDi [IV^51cpbKK
o&D&o5i'V5iIDf 5Iig7?'V IV
?'5' 1L^ '7o|V Î ¥ °0Io|
o'7L ÎC^Ko&f'V5i?YK p&^ D5&Vi D5IIVV'W¨KpR'K1cI5R'I
PI'Vf§oV¨0 ^RR'K1J&oC'VJ ^RR'K1.^, oJ5Ii1 , IV
$'5'iYV5
&VV^^M i o,V^K^V'KV 71K'K1M'oRà0b1VDZ^ IC ioIÎo
¥
0
ooFc^^VILoVK 0K» o
FI5R'I D _1VF'R'^7P¡Ko D(af +
PI`
1o7'& a b Io]&'V9''I ¡p1 DMPVo<'K1o f &1o
bf2 ) = aD(f1 ) + bD(f2 )
¥ o5i0PI £7¤I¥^¤¦ YK p IV
C IÎZ¡ £ ¥ ¦ D'&DM OciK1^V&
f2
Vi`9"IKg oF oR'Îg ^RR ÎL&'IVV'W¨^pR'|¡|^^VILoVK CK» o

I5R 1' ¥ o]^^VI` ViCaIo^ZoWI, 'VoR'K1Z 'I
D0 (f (U )) −
X
si (V (U, x)) Di (ti (x)),
i
YV5' D I 0^KV&aoVK cV9»5'5o50I5R 1'bo M p&oV W¨^pR F o0 o& D&
5'KR& Îi ¥ 0Po7'V5Y'7 '^
YIV`&7 ÎZ iK^V&`'V5i?Yo^] & DP¡
£ ¥:²I¦
D0 = D 1 = · · · = D
PI0 oO'IV¡ R O I^'K1 ¥ `&'^5| o&cbiIoK¡ 'V^'_'oR,'V|oVK OK¢
P5'5o5Z1g'& Ic5Ii[^ D5&aP1'V]'IV¡ R ] I^ Î¶'oRpa[I
K ' ,o 7 1IWPIJI5o5&71KV Îo$'V ?oVK F9»5'5g
YI5R'I$5&
gcK»5'5 ¥ 2 I` oZ^^VI` Vi?aCgWI
0W J , , #.% # 3 4 & $ & 4 #" $ ( 4 & . " $ . * * . ( " " $
4 ** * .& 4*
¤
©?o,V IG&G VDY' VK?DYR'V5Y7 &Î7P1 ?&O&gZ&[g0P1Vo[Ky¯ °± ¥
YVR &^ I¡cgWIaVK1AI5$a5^9Ao&D&g
|'V5i#P1$ VYg&D&o5aDW`
oo5`5Io D5R'K1 ¥ RbI1 V70'V5i?I CK^V&o5o5'V5¡OY^^.g017
5K^DR 1 ¡_YV5| o71KV Îi
I&ôfD 5I ^V9'K ¥ Yb1V^V c5Io ^V5
oIV^Ko&0'V5io&i5^¡ 'V5iYVD¸&'cV1V^KV&C515 KL o_§oV¨ f (U )
Io® V IV

^ £7¤I¥^¤¦ &'[gI @K^V&|PVo
ÎoiI bPIp5ÏiVK1
oK11'V5FoVK ps(U,
K»x) o
¬«'V5i|¯ ¤ ¤ ± ¥
Üa|YK^fo 'V
o¢ yI5FoVK pK»5'5o5 ¬«'V5iiIC&¸
¨V&iV^_'[V5iIo7 R'_'V
gI ^F^VI ¥ 0Y^KVIa1K1` oC5'IKDa&. oCgI ¬ 'V5i&oM 5P5
'¯ ¤ g° ±.PI` o
&^^ ¥
©G|V 7J oCgI ^ÎbFo?
Ig7D5b o0 K o& ÎMYo5O'V ¬ 'V|Zô ®Y9'VIp]§oV¨®7V^K7 ^V £ ¥ ¥ 'V ¬ `]'V5iZ1_5
'
'^g¸^ ¯ ¤ ± ¦<¥ pVI D
i'oRc'V ¬ IVV'W¨KpR'K1 ' d YV d =
x
5IZ,515 oIK^¡OY'9 Z1I
f (U )
£ ¥ ¦
dx |x=xj ≈
r
X
ak dk+j ≡ Dd (d)j
Yo5' r = 3 PIC'V_o¢ ·I5 ¬ ª&Vo W¨î& Î Io'Vp

¹O
^5 a
V55o®VIV^^VI ^¡¸1y V[7i&'VV _KoV^R 1'_K1I^^V]'VZ1 D·PVo
Îk
¥ ©?V I¡MDiVR D? po 0 o,oVK _V9»5'5o5,I5R'I D YK d = f (U )
d
, ooI|K¡|'o F V0 IiC55¹O
^5' a IV'&^Vi V'IoIOd'V07i&'VV'
ôD5& 1p& d bIo &VV^¡ Ki ¶&Vo W¨î&k t0 (x) K 'V71V'5M'5'p £ ¥ ¦
¥
i
©?o
t0i (xj ) ≈
r
X
k=−r
ak ti (xk+j ) = Dd (ti (x))j .
µ Î ^¡II V?gV9'K» o
I5R 1 D 1I&^VP'I 'V`VÎV1' ¬ o
V',&o[YV d = f (U ) ^YV¨.gd^`|VÎ]1'Ì
VR' `&VV'W¨
^pR'K1¶ ( VZoV5'KRR'K1pPIi&¡·I'^¶PVo<'K1 ¥ ©?o5'
PI'p'VM' VK&
¥K¤ ^C'K^$R&^^ Ioa5Io5KoVc gRC VVK1AI5oVK V9»5
J 1g7K Î
o
¬ « Vi_IFR &RI1oYK V1'Vc§g¨ 7V^K7 ^VgIo]YK 'V

D&Yg&o^^V¸I` o] Io'5M'5'ii 5 ^®IgR1ILpIK'IKg5¨VI<'K¡y'V
7 I¡ZR ¥
R,Ia`
1o7D ¬ 'V5iLY9'i 1GR¨ 2 'K DoG§o¨p VKK7'Koo7o
10 o ¬ 1 2 &o ¬ 1 1 2 Vi, 5 ^g·K@¯ ¤ ± ¥ 5'i Vi§oV¨
k=−r
¤I¤
f (U )
£ ¥ &¦
^`Y K7'5IYi Vª&
f + (U )
&o
f ± (U ) =
f − (U )
o5oVZ¡
1
[f (U ) ± αU ]
2
YV5' α = max ∂f (U ) YK M VCpR¨KV«5^Vc IORIJ9'V5?_^5Io'

IÎ £ ¬ 1 1U 2f¦ ∂U
I#YIÎoII'51K1 £ ¬ 1 2$¦ R7`¯ ¤ R° ±PI#iI'J
&^^ ¥
VR i I_O|KgR Î p'V^§o¨ V^9 ^Vo¡Z'5o^1
^V ±αU ^ £ ¥ &¦
Y9' ±α sign ∂V (U,x) V (U, x) ¥ iaIVDV ]I'7Vi|V5'c'oR ∂V (U,x)
YVIFoIVIc∂UK1 ¥ ©?V_5Io7'I1 α 7oIVD]g7VK'IV^¡[1 o7 ]¡Z V∂U7^³5
& ∂V (U,x) ^yI50'[pIK'IKyVIV1·I7'9
D&f^'
7K¡ ¥ ©?Vi'5' V (U, x)
5&I∂U
^ igc'5o^1
[¡ p(V (U, x)) P1&¡MPVo<'K1 p gYV1 ,VID
c VIo^[
7o]'oR p(V (U, x)) D1
^1 ,' U IF1'7^V^cK(I5Y'M_VDR', VI'KI
KoI 1 2 §g¨¸ VKK7'Ko[YK ±αU ¥ ©?o^0iKg5& ÎyCV&c&»<cI
VI5¡I
YV^® KÎV^¡yI¶'VO41< f (U ) = f +(U ) + f −(U ) ¥ 2 1c'V[7 I¡¶7'&
71K'K1p & DP¡^V £ ¥ ¦ &'V& Kg5ÎD'
1 K¡, ±α sign ∂V (U,x) V (U, x)
£ I ±α sign ∂p(V (U,x)) p(V (U, x))¦ ^d'V 1"&¨ 2 ^'DoO§g¨ ∂U VKK7'Ko g5

Ii_ 5Io7'I1∂Uf&oy¡¸ V[
Ig7D'5o5¡¸IY V ¬ IVV'W¨KpR'K1'V

»<[&c'V (D 51 9¡® pM R?&Vi o·IVV'W¨^pR'K1 I f (U ) DM³5' ¥
©?V¶§o¨ VKK7'Ko ¬ &VV'W¨Kp& Î K 'V^· 9'oR'K1 g
Iix( KiV^¡
Vo
( Vy7 1¡@R y I^ Î D[V 7 1@IZ
PI'I9
f ± (U ) = 21 f (U )
b, KiV^¡Z V^9Y'Vc5'^W& Îb^[ Vc Io'5'5' Iº
£ ¥ 1¦
t0i (x) =
1
1 0
t (x) + t0i (x),
2 i
2
&opIVV^¡_'VF'&iY§oV¨p V^9 ^V ¬ V'
VV ?'_&oV W¨^pR ?'V5 YK
VYV1V^KV&f
5¹p5K' a
Ii^VCP 1 o ¬ d&Vo W¨î& Îo" f ±(U )
7
Î5^¡ ¥ ©?VD?Y^K1o&k&'5 £ ¥:²I¦
¥ 0'gYI&^
0W J , , # " # % 3$
4 ) ! 4*5. * * # . " " $ 4 () ( 44 4 pVR ^'
g C'VO7¡7 517 ¥ ©?V|P'I|b1
V'
'K(PI0'VO &D&
5I &OCIVV^Ki |7¡7 iJV'RD| o&aaoW1`
7&^ VRYKIF&I
V,7 1¡pR'0 I^'K1oL'0V' 5'IOKM VCPI'«& £ ¥ ¦
¥ ©J¡VD5IK^¡IPIY
¤°
¡'5«YK o& Îo V D?_1<'IIoOaCb1V^MoWI m ^& Îo VôbK
'V0PI'I £ m¥ ¦ º
£ ¥ ¦
V1 (U, x) = constant,
···
Vm (U, x) = constant
P1L'VF7 I¡iR'F I^ ÎoL o&baàIVDiK I` _o 7 1a5¨VI<'K¡ ¥ Fb1V^
'V57 ^KIKªPI`
Ii1 KoIZ5IiIV1YI" oc71V'5'5'ª^M'V0PI'
I £ ¥ ¦ "YV s
1V^·O&'VK &'¡ZPoo<'K1o,& V (U, x), · · · , V (U, x) GIo
'V`Poo<'K1o s Iio t 5IVDiK» 1JPIbV9»5'5b1 5Iig1V5Lm&'V IV

1
I ¥ ©?V| iiIKoKViZV'5o ^F o5 'Vi I|iI gRCPIF Vp'5&D&C517
Io@a &1IK IV'&^@bKFo&D&g
VÎV1' ¬ 'V5i ¥ ( VIo^
I^ Oi5'K1 o&F^5&#o&I
D'^0
1|1 K ÎD`¡VD5IK^¡Zg7K(VK1
1'V5 ¬ S ViK{I5 ·I&^
7'5pV1A1'
^KDR'I'¡(V'I5 ¡·PI
7 'IoMD 5I ^V9'K ¥ o5¸5IiV'Kop VVi5'^&§g¨ R x VK&
g&I<'5'^7 D`p& '9¨ R V5Io ^7 ^V|&G V,'K1aK15I< 1'b&G Vi+I5IVD&ZR
D?5Io7'I1?pR' K¨pPIaV¨ i ¥ 5o5C VD?o&I
5'D'^F5Iig7K Î
Ui+
o
V'bVIa&K L'VFI 1Vi5JV' 5 p&RIYP1L'VF'5IÎb517 ¥
'
P? ¯ ¤ ±.P1YiI'0V
'IKD ¥
©?oc7oIK^R ?R'5?o& Îo £7¤I¥^¤¦.£ ° ¥K¤W¦ IC'V0PI'
1
2
1
2
£ ¥^¤ ¦

 ht + (hu)
x = 0
1
 (hu)t + hu + gh2
2
2
= −ghbx ,
©?o,^ I,RY'7Y I^'K1['R'^7o £ ¥ ¦ ^Z'V0PI'
£ ¥^¤I¤W¦
V1 ≡ hu = 0,
V2 ≡ h + b = constant, .
©?oio'7_
1ig1V5,&a VO Io'5'5' D V¥ V
1ig7K Î·&a VO 5Io
5IiIV1Ì"'Vc71V
F 5'^[ V0P1 & £ ¥ ¦ D
£ ¥^¤ ° ¦
x
1
−ghbx = −g (h + b) bx + g b2 x
2
¥ ¥ s = s (V ) = −g (h + b) s = 1 g t (x) = b(x) gIo t (x) = b2(x) gYVD
'R'^7o1 '17o|2 V Î ¥ ° ¥ o
12V'V,2 V1VDV_7ÎV&2RI,5I]go7Z
1'IK¸VÎAI5FbKKAo&D&o5[oo9'|K»5'5o5|'V5i_PIC Vi o&^KR ?R'5
oR'K1oiY9'{D 1Rp p I^ Î £ ¥K¤1¤¦<¥ ©Gby^|o K1o&Y157Î{&'V
¤
7oIK^R´?R'5,1gR Îgc5IyI^ ]gMo&oVKy¡( VZ I||'VVDV ¯ ¤ I ±A
&o I (V&]7VRY o5' ¥ Ii(Vi5'D5&C' V9OY^KC· VRY ^ < Î
² '[V5iIo7 R'| ViI o 1g7'KFIJ'V ibKKg&D&o5 oK11'V5FoVK
9»5'5g
, o5i ¥
~ -+ ®~[ ~ ~
#"$# ,(, 2 I^KRY^V| o,î&$I&^V^VibKKg&D&o5[ Vi?¡M
Ii1 Ko'V
71V'5 '5'iF1]7VRYdK
Î ¥^¤ à¸15V'IK^³5(oVK ·I1KVi ¬ Vii'¶I&^ VK1AI5pa5^9AoIÎo
{'V5i ¥ ©?V(
'o5ÎK» o

g5b5 'V oo9'¸1I^Vi(&o V(oVK y9»5'5g
¬ 'V5iZDp gR
V ¬
1o' g< ÎdV 5V' P1¶oVK y1I^Vi¸'V5iy&VV^^Z
Vy71KV ÎdIo VIZ' 'V·§o¨@PVo
Î WIKo ¥ ]®
1o V5g
IòVK
II^Vic'V5i&',|1 7VK'IV^,PI5IiVR'K1oY^(
1|oK5¨ZIIi
' ¡[&o
PIco ^V(Io&'K1p|7o ¥ ©?oM
&^^_&? oOoVK MII^Vi ¬ Vi
5&®gMPIVo K ¯ ¤ $° f° ± ¥ RbI5#g5&g7Z& K»5'5|5IiVR Îg&
P'I|b1
$'V]iIK'5oIo
[&F V]a5^9Ao&D&g
V 1g7¡y'V^ i9»5'5
oV^&IVV'1Io ¥
©?oOp&^·DZ^y opV K1o,7oo7<'K1¸' Î¶[bKKg&D&o5¸VK1
I5FoVK pK» o
¬ «'V5i|^F ]
Ii1 _'Vp71V
c'5' K'[
7V &L 515& poI]&fYVD]^F^'
'
'K³]ôgo5'K¡[o KVZ|oVK
9»5'5g
]PI'_o^y
1o ^7 1OYK @'oROI0IVV'W¨KpR'Ko¸'V]§o¨ ^RR Î
5'p0K·'VO
1o7 R& Î(DW ¥ iPI^KR [ ^|^D&,D]V pIo¸V
1ig7
VCK'5I&&" V, IV
5'«^1'|7o«&G 515&g 5'p5'V5[
IiVV CI
&C V ^@·?W¡
1o7D7 5OYK gRO&0
1|o ^Vy o
I'' g1o^V(§g¨
5'p ¥ o0
1o7D` o|I , o& £7¤I¥^¤¦ DCM'5&D&Fo&D&g
cDW ¥ ©?Vi517
&# ¡'5p?Y^K,
¨VÎ'ZDR ¥
Ki^Î ^¡IaFp I071|0I' Vi ÎobIZ oCoR'K1 £¤1¥K¤W¦ Iop'V,'1¡
R , I^ Î U & £ ¥ ° ¦ 'oRYac&'F^ 'M pV' 5'IF
¨VI
^¡º
+ , # # . U #" $% '& () * . -, £ ¥^¤ ¦
. V (U, x) ≡
) 0&1 354*.&
p(x)
$
U + p(x)
= constant
q(x)
q(x)
¤
+ , # # 4 £ ¥^¤ ¦
s(U, x) =
s(U, x)
X
$ 4
4 sj (V (U, x)) t0j (x)
j
. /0&1 3 4 . sj
tj
' Vi Î ¥ 1K15_o5'bD#|1 a'7 '^
ÎL'o& gR$K
Î
¥K¤ VV0''VcIo9'K1o&V9¹O
o9'KY ^& O'i V0oo9',II^ViCPI'_VDR'K1 ¥
c
1o â Vc 5iKAD'
'
CP1 V^& ÎZ&# V,oIÎo
0DW
&'J'oR £ ¥^¤²1¦
d
1
1
Ūi (t) = −
(f (U (xi+ 21 ), t) − f (U (xi− 12 ), t)) +
dt
4xi
4xi
Z
s(U, x)dx.
Ii
©?oY îFD 5 5 ^³& Îi^Lo7g&^K¡|g7P1 i¡ VF
DI'7D5IVK1pI5 YVV1

0V7'O|5 V ¥
PI'c7'R'KopIVYVi5'D5&G ViIVa0o`V' 5Y'V,V'&
5o Z
Ig' o
'VI^Y^ _RIKV¡]'V ¬ '5Io7 'o<'K1V'&
5o 1Ioi'V5]5Ii1 Y VC^1'51'IgI. V071V
`'5' K i 5I'I p
YK ['V0I< ÎC& I5V^V V,5¨1<?o&D&g
CV 1g7¡pYK oI`'g
^V_'V
oK11'V5b155V'1
¡|&'V0 Vi ¥ ©?VC'V|F^b'V5Moo&^^¡|^ 'o
OYK
ii^VI`g&V1F1M'V0§o¨Z 5']V
1|g&'ZY9'Z'V,I'Îô& ¬ 'V5i ¥
©?oZg'7i7 {^ VV^^^V· o]IK1I'9'V ^|'y
Ig' o
U ± P'I 'V
1K15[
5^.WI'II Ū ¡i V ¬ '
1o7 'o<'K1OV'5o 1Yi+o^[&'VÎ
1'V5$I55VR a&Vo Wï î& Îo F Va
¨V1<GRIKV U (x ) ¥ A$&_a5I oIK^¡
i+
Y'K7 ]I1
1
2
1
2
£ ¥^¤ ¦
+
Ui+
1 =
2
r
X
k=−r+1
−
wk Ūi+k ≡ SŪ+ (Ū )i , Ui+
1 =
2
r−1
X
k=−r
w̃k Ūi+k ≡ SŪ− (Ū )i .
Yo5' r = 3 P1O'V V¢'d1' ¬ &Vo W¨î& Î &g Vy

¹O5K1
&g w̃ 55gZoIV^Ko&'^¡pIZ V, |I VV ?ôD5& 1'a^1I^KV o055^
wk
W15&1 Ūk¥ 5'(a·I&^d ^Ko&[I5R'I S ±(v) £ K^V&[K v ¦ YVD D
1'IKo®P 1 ¬ S
Ig' o
Î YK VǕ{55¹O
^5' w 5&D
o^&
P'I Vc55^"WI'II Ū ¥ I¡O^ViV 0D? po 0 o Ì5kR'I S ±(v)
Ū
¤²
&o[&VoK¡M Vª i'5Io7 'o<a o0PVo<'K1o
r
X
p+
= SŪ+ (p̄)i =
i+ 1
2
£ ¥^¤ I¦ q+
i+ 12
p̄i
r
X
¥ ©?o
2
2
k=−r+1
w̃k p̄i+k
r−1
X
w̃k q̄i+k .
k=−r
# bp1'IK· opg1KYD7pR&^V
& V (U, x) ¡ V (U, x)± = U +p ¥ µ Î ^¡I p± Io q± I ¸VK1
i+
i+
i+
q
I5pI
VR'MI^Y^ &VV'W¨Kp& Î¶'¸'V[PVo
Î@& p(x) &o q(x) &
VO55^fg1VoVI ¡ x ¥ o
1 V (U, x)± D,[VÎAI50&oV W¨^pR Î(
i+
¥ i+
V (U (xi+ ), xi+ )
R 1 ViF o& Da oc'IV¡MR'0 I^'K1['R'^7P¡KV £ ¥ ¦ oo&i5^¡
U
p±
i+ 1
&o
r−1
X
k=−r
−
−
qi+
1 = S (q̄)i =
Ū
wk q̄i+k ,
9'y VO'5Io7 'o
¸R&^V
q̄i
p−
= SŪ− (p̄)i =
i+ 1
wk p̄i+k ,
k=−r+1
= SŪ+ (q̄)i =
&o
2
±
±
i+ 1
i+ 1
2
2
±
i+ 1
2
1
2
±
qi+
1
2
1
2
1
2
1
2
1
2
1
2
1
2
V (U, x) = c
⇔
U + p(x) = c q(x)
PIJ Ii`
1o7'& ¥ A VF
5ôW15&I Io &'?5IiV'i^| V'&i
i
4I VK1 £ ¥ ¥ &^L
1c iV ·
¨VI
^¡I10IŪKLi 5p̄IiiV'q̄·
YK · VO'&iiVi5'^&
1gIR'V ¦ P'I U p(x) &g q(x) V Vb,
Î ^¡O&D |gWI
Ūi + p̄i = c q̄i
PI'V? I|b
Ig& c ¥ ô
J Vb
Ig' o
0WIKo U ± p± &o q± I

IiVV [P'I V
5^GW15&1 Ū p̄ &o q̄ YK ]'V i+ Ci+^Ko&Ii+5R 1'
j
j
j
V

O
¡
o

±
S (v)
1
2
1
2
1
2
ū
±
±
±
= c qi+
Ui+
1 + p
1
i+ 1
PI? oc I|,5Io7'I1 c 'oR`D
£ ¥^¤¦
2
2
2
=c
V (U, x)±
i+ 1
2
PI? o0'&i,
1o7'& c ¥ ©?VDYDY&[KiI '&?4I
? O7^IM V,a5^9AoI^Io

Vi ¥
¤
µ Î ^¡IP1ì'IV¡MR'0 I^'K1

U

& DP¡^V '7o|V Îo ¥ Io ¥
X
X
d 
sj (V (U, x)) t0j (x)
sj (V (U, x)) tj (x) = f (U )x −
f (U ) −
dx
j
j
= f (U )x − s(U, x) = 0.
©?o5'
PI'I ) − P s (V (U, x)) t (x) DJ,
1o7'& ¥ àIVD|V5| cV1
j
j j
VK'IVK¡ (t f)(U
7o
± YVD 7oIVD]gVÎAI5F&VV'W¨Kp& Îoa'
tj (xi+ )
j i+
'oR
X
£ ¥^¤ ¦
f (U ± ) −
s (V (U, x)± ) (t )± = constant
1
2
1
2
j
i+ 21
j
i+ 12
j i+ 1
2
P1[®'IV¡ 7'& ¸ I^'K1 U & DP¡^V Vi Îg ¥ Io ¥ ¥ ¸YK^

KP¡i oFV1^5LI (t )± P1L oC o&^^R ?R'5boR'K1oJ&J'VoOI. VD
j i+
Vo
Î ¥
2 ô&^^¡Ibò5p _
1|1 Y'V^ 51'IV&.'VF Io'5? 5' Ki'V`P1K^R?
^Vp?W¡pK]1'a pIV'&^]a5^KoIÎo
M'V5i
1
2
Z
=
=
£ ¥ ° ¦
s(U, x)dx =
Ii
X 1 XZ
j
Ii
sj (V (U, x))t0j (x)dx
(U, x)+
)
i− 21
(U, x)−
)
i+ 12
Z
sj (V
+ sj (V
t0j (x)dx
2
I
i
j
Z 1
+
−
0
sj (V (U, x)i− 1 ) + sj (V (U, x)i+ 1 ) tj (x)dx
+
sj (V (U, x)) −
2
2
2
Ii
X 1
sj (V (U, x)+
) + sj (V (U, x)−
) (tj (xi+ 12 ) − tj (xi− 21 ))
i− 21
i+ 12
2
j
Z 1
−
0
+
sj (V (U, x)+
t
(x)dx
.
sj (V (U, x)) −
1 ) + sj (V (U, x)
1)
j
i− 2
i+ 2
2
Ii
©?ocVV'g7,I$ VD
1|1 K Î]^` Oo V 0'oR Vc^ I&.&$'V_ IV

'5' o&D.'Vbo'7'5' &"'VJ'K1"o&gc7Db& £ ¥ ° 1¦ Yo5 V (U, x) = const
1a V,DI7Y ^'&og&b^[ VDÌ ¥
Rda,&'C'1¡i OV'
'KC V0og&PI'ªIG'Vc&Î1 K V
£ ¥ ° ¤W¦
d
1
1 ˆ
Ūi (t) = −
(f 1 − fî− 21 ) +
ŝi ,
dt
4xi i+ 2
4xi
¤
Y9'
£ ¥ °I° ¦
X1 +
−
ŝi =
sj (V (U, x)i− 1 ) + sj (V (U, x)i+ 1 )
(t̂j )i+ 21 − (t̂j )i− 12 + si,j
2
2
2
j
YV5' (t̂ ) ^`|VÎV1'Y&VV'W¨Kp& ÎO' t (x ) gYV1 ,
gV9'K1]YK^
gc 5 ^ji+
ZgKR,VIo s ^YI¡MVK1AI5?&Vjo Wi+¨î& ÎM | V,^ I&
1
2
1
2
i,j
£ ¥° ¦
Z Ii
sj (V (U, x)) −
1
−
)
+
s
(V
(U,
x)
)
sj (V (U, x)+
t0j (x) dx.
j
i+ 21
i− 12
2
?© oZVi5'D5&b§o¨ fˆ î5oV¶¡ iIVI 1Vp§o¨® o®1c'V1"R¨
i+
2 'K Dob§o¨
i
£ ¥° ¦
1h
F (U − , U + ) =
f (U − ) + f (U + ) − α(U + − U − ) .
1
2
i+ 12
i+ 12
2
i+ 12
i+ 21
i+ 21
V{ yp 1¸iKg5& Î{ y'V^p§o¨.b¡ '5o^1
^V
K £ ¥ ° ¦ Y9'
g
Ii
£ ¥ ° ²1¦
α sign(q(x))(V (U, x)+
− V (U, x)−
i+ 1
i+ 1
2
i+ 21
α(U +
− U− 1 )
¥ ©?V0o| Di+5I§o¨Mi+oR
)
1
2
2
2
i
1h
−
+
+
−
f (Ui+
fî+ 21 =
1 ) + f (U
1 ) − α sign(q(x))(V (U, x)
1 − V (U, x)
1) .
i+ 2
i+ 2
i+ 2
2
2
paIVD¸V5 ']1 ViiV5'| o& q(x) ^ £ ¥ ¦ V0V&_g&V1| K1 ¥ ©?V

Ig& α VIVDO07VK'IV^¡OI o7 O¡| V, K³I 1 ^M1'V5L'pIK'IK
5V1VI]& Kg
D&.D 51 9¡ ¥ ©?V^?iKg5& Î?o&q(x)
R»
I
VI5¡ ¥ 2 I?'V
'IV¡MR'0 I^'K1 £ ¥^¤ ¦
− V (U, x)−
)=0
α sign(q(x))(V (U, x)+
i+ 1
i+ 1
g5&g7,I £ ¥^¤ ¦<¥ 5o5Ig V_
»<F&# V7cD 51 9¡M pY5IiY³5'O&o
V,Vi5'D5&§gÖ o oÌ? p,^[p KiV^CPI'
2
£ ¥° ¦
i
1h
−
+
f (Ui+
fî+ 21 =
1 ) + f (U
1) .
i+ 2
2
2
2 I^^RYKV o^VaF' R? oc&VV'W¨Kp& Î
£ ¥ ° I¦
2
(t̂j )i+ 21 =
(t̂j )i+ 21
^ £ ¥ °I° ¦ K]p7îKD&??W¡º
i
1h
+
(tj )−
1 + (tj )
1
i+ 2
i+ 2
2
¤
Yo5'IIJ| ÎVpg5PI'I (t )± I VK1OIb&oV W¨^pR Îg$ t (x )
'R'^7P¡Ko £ ¥^¤ ¦
¥ `&''oR?aFj î+
iVKi5 £ ¥ ° I¦ P1b'V0IV5&51711VjIYi+IV^¡
P1# V?'1¡,71K'K1 ¥ ©?o5'bD"oFD 51 9¡F ^c'VY71V'5f ]&5Iio&'
YK ['V,Vi5'^&§o¨ £ ¥ ° ²I¦<¥
2 Ip'V '5p&^V^V IV
]'5' s Ya( KiV^¡®g7(¶ VK'IVK(VK1AI5
0Io'_oIV'& V'| WIKgR i'VM^ i,jI& ¥ ©?VMIVV'W¨KpR'K1¸I? VZWIKo
&c o1 0&o',I^1_I iIV'&^V¸¡¸ V ¬ '
1o7 'o<'K1¸V'
V' ¥
A|^17¡¸ (Io 5'Ii'oR|VK1¶I5|I
VI5¡ Îg&& ¸PIIV| ViI
IoZ5IMK#D'
I'KVK ^?5¨D7a^M'V,71KV Î'V0VIA1'
^^^& I'¡|V 1g7¡iD
pIK'IKo ¥
1
2
1
2
0R J , , # # 4 $
) !!& " $ " $ " $ 3$ " $ 4* . * * . . #"" $
3 $ 4 3 )( ! &
4*4 4- . .&
©?o,V I#IG'V^ 7o9`DYR o5`7 &Î7P1 ?&O&g[&],PIVgZ^¶¯ I ± ¥
©?oO
¨ o K1¸'] VZ7¡7 5 517iP1K^R`C VZ I|pDI0'oRcP1, oObKK
g&D&o5MoVK cK» o
c'V| ¥
2 IZ Vy o&^KR ?R'5[oR'K1o £ ¥^¤ ¦ YK ®^ I·RZ'7[7 I¡ 7'&
I^'K1 £ ¥K¤1¤¦ aL 1L oa'&ibV
1ig7K ÎcIV oa 5Ioc5IiIV1#&V'V
Io'50 5' 1Y^ £ ¥^¤ ° ¦¥ _IVV^¡O'V ¬ '5Io7 'o
ÎZ p'V,PVo
Î
Y9'i5
¹O
^5'J
IiVV P'I (h, hu)T &',I&^ b± &oi5oV
(b(x), 0)T
i+ 21
2
±
(t2 )±
b
.
1 =
1
i+
i+
(t1 )±
= b±
,
i+ 1
i+ 1
2
2
2
oO o
oo9'K1oO&g@9C V¸'IV¡{7'R'
5Io7'I c D? IV.aCgWI
−
f (Ui+
1) −
2
=
=
=
X
j
(tj )−
sj V (U, x)−
i+ 1
i+ 1
2
2

h+b = c u = 0
P1M71i
2
1 −
1 − 2 1 − 2
+
+
g hi+ 1 − g bi+ 1 + g
hi+ 1 + b−
b−
1 + h
1 + b
1
i+ 2
i− 2
i− 2
i+ 21
2
2
2
2
2
2
1
g h−
+ b−
− b−
h−
+ g c b−
i+ 21
i+ 21
i+ 21
i+ 12
i+ 21
2
1
1
−
−
= g c2 ,
g c h−
1 + 2b
1
1 − b
i+
i+
i+
2
2
2
2
2
¤
YV^[Dì
Ig& ¥ Ki^ÎYiIV^VVDR Î[Îoa
+
f (Ui+
1) −
2
X
1
= g c2 .
(tj )+
sj V (U, x)+
i+ 21
i+ 12
2
5g
0 ocVÎAI5aoo9'cI1KVi ¬ ViY5I]g_V K1VZPI^^RYKV
j
VcIgR1FDP1? Vc o&^^Rd?R ?oR'K1o ¥
0- ~ , ~ ~
# #(' ~ , ¶ , ),$+ + òW1a o
'7PVK^¡_7Îo|VÎAI5$a5^9AoIÎo
,oVK `V9»5'5o5`&o
oVK [II^Vi ¬ a5^9AoIÎo
¶ ViOPIi(5^1 _&¡5'g1KDpoIÎo

^W` ¥ *· VD_ Vo
^IGbp
1o ^0 VMI5o5&^K³R ^I¸&a V7pDIC ]'V
¦ i
'VV ¥ KKg&D&o5¸VK1
YVV1
07[V^'
11'KV1o 0&^5K £ I5 o5i#Y^K?V K1VcPI$F5^1 G&
Ig7 RR'K1,^W`#'R D7P¡^V
'7Vi'K1o ¥ Io ¥ ¥ ©?VJoI ^$D?^ VJ'&iL1 o&PI.'Vfoo9'LI1KVi
Vig o I?'Vc'Vo^VcIL5Iig7^Vi'V IV
0'5' Io]'5VDI5KV
VC^'
7K¡_'5'«KM VCVi5'^&o§o¨g5&g7 V i
ooa&O

Ig7D5'¸I0[IV5&^K³R'K1 &boVK iI1KVii o5i515( V1VI¸ V¡(
V&c'V^ i]'
1o7 'o<'K1¸&g¸I1K1_'VO
1|oK5 pI^¡oIiÎfKyIy
5^
PI'aI'y^ 'Ki ¥ ©?o i
'VVi&'O'V5'
P1 ZI ^5c'¸g7MPIi_VK K
Kio7Îg&oo 1VKpL^M
1|oK5¨|15Ii5 '¡I&'o&O VoVK CII^Vi'V5i5I
V,
1iVKD5& M'5Io7 'o<'K1OV'
VV 05IMcWI1^ ¥ 0'
P5a'[¯
¤ ¤1¤ ±.PIYi1 ,5'&^D?& í
'VV ¥
©?oc7|K*^'
'
' o5iaP1 £¤I¥^¤¦ IC'V0PI'
£ ¥ ° ¦
R
Ij
f (Uh (x, t))∂x vh (x)dx + fˆj+ 21 vh (x−
)
j+ 12
R
) = Ij s(Uh (x, t), t)vh (x)dx
−fˆj− 21 vh (x+
j− 1
∂t Uh (x, t)vh (x)dx −
R
Ij
2
£ ¥° ¦
Z
Uh (x, 0)vh (x)dx =
Z
U0 (x)vh (x)dx.
2 ^#aM5oVM VK1AI5cIVV'W¨^pR'K1
Ij
Ij
' V (U , x)
YV5' p Io q I L2 o <'K1oC& p &o K G 5 £ ¥ ° ¦ P1c7oh¶
V
Îh ¥ R h 1 Vii gR U ^c V['IV¡·7'& M I^ Î 'R'^7P¡Ko £ ¥ ¦
o&i5^¡
Vh (Uh , x) =
q
Vh
U (x) + p(x) = c q(x)
Uh +ph
qh
°
P1F Iic
1o7'& c &o U DY o L2 V' <'K1[I$ VD7 I¡[R 71K'K1 ¥
µ Î ^¡IV ô
C o L2 V' <h ÎZ^^^VIYI5R 1
Uh (x) + ph (x) = c qh (x)
P1? Vc'&i,
1o7'& c RY515'¡pg1K x ¥ ©?VD?KiV^K
Vh (Uh , x) =
2 Ii7g{7 1¡¶R'[ I^ Î
gWI
Uh (x) + ph (x)
= c.
qh (x)
U
'R D7P¡^V '7Vi'K1o ¥ Io ¥ Jb


X
d 
sj (V (U, x)) tj (x) = 0.
f (U ) −
dx
j
ab1V^0o5, VK'&oK¡,V1 J?Poo<'K1
W YVD_ VIo^0a`VK1AI5
&gZ VIo^]'R D7P¡p Vc5IoV9'K1
IVV'W¨KpR'K1O' t
j
X
£ ¥ ¦
s (V
f (U (x)) −
j
h
(tj )h
h (Uh (x), x))(tj )h (x)
= constant
j
P1GIK x ¥ ©?oJ
1o' g< Î,I (t ) YK^gJ VRYCPI" Vb7oIK^R·a& 5"oR'K1o
^[ V,o[&G VD` Vo <'K1 ¥ j h
Ki^Î? i oc5Iig7K Î]IG'Vc71V
F 5'^[ V_bK.oIÎo
ZoVK
1I^Vip Vi £ ¥ ° 1¦ "aiV
1ig7| opK I&$&a oO71V'5| 5'S1¸'V
'ÎYo&o] ^VCI £ ¥ ° ¦ 15º
Z
=
=
s(Uh , x)vh dx
Z
X 1 −
+
t0j (x)vh dx
sj (V (Uh , x)i− 1 ) + sj (V (Uh , x)i+ 1 )
2
2
2
I
i
j
Z 1
+
−
0
sj (V (Uh , x)) −
+
sj (V (Uh , x)i− 1 ) + sj (V (Uh , x)i+ 1 )
tj (x)vh dx
2
2
2
Ii
X 1
−
sj (V (Uh , x)+
1)
1 ) + sj (V (Uh , x)
i+ 2
i− 2
2
j
Z
+
0
· tj (xi+ 12 )vh (x−
t
(x)v
(x)dx
1 ) − tj (xi− 1 )vh (x
1) −
j
h
i+ 2
i− 2
2
Ii
Z 1
−
0
sj (V (Uh , x)) −
sj (V (Uh , x)+
+
)
+
s
(V
(U
,
x)
)
t
(x)v
dx
.
j
h
h
j
i− 12
i+ 21
2
Ii
Ii
°¤
_ V VDI
c VDF Io'5, Y9'¸poK11'V5IVV'W¨^pR'K1[If9CIÎ5
¡
X 1 ) + sj (Vh (Uh , x)−
)
sj (Vh (Uh , x)+
i+ 1
i− 1
2
2
2
Z
+
0
1
· (t̂j )h,i+ 12 vh (x−
)
−
(
t̂
)
)
−
v
(x
(t
)
(x)v
(x)dx
j h,i− 2 h i− 1
j h
h
i+ 21
2
Ii
Z 1
−
0
+
sj (Vh (Uh , x)) −
)
+
s
(V
(U
,
x)
)
sj (Vh (Uh , x)+
t
(x)v
dx
j
h
h
h
j
i+ 12
i− 12
2
Ii
j
Y V5' (t̂ )
ÔyoK11'V5O&oV W¨^pR Î{
j h,i+
PI^KR` £ ¥ ° ²I¦ &g £ ¥ ° I¦ P'I
Î ¥ °
1
2
tj (xi+ 21 )
?YV7 5oVK Î
i
1h
+
+
−
f ((Uh )−
fî+ 12 =
)
+
f
((U
)
)
−
α
sign(q(x))(V
(U
,
x)
−
V
(U
,
x)
)
,
h i+ 1
h
h
h
h
i+ 12
i+ 21
i+ 12
2
2
i
1h
+
(t̂j )h,i+ 21 =
(tj )h (x−
)
+
(t
)
(x
)
.
j h i+ 1
i+ 21
2
2
Uh
U
V (U, x) = constant
7g&^K¡1a5 PI' VC^KiK 5aIM VPVo
Î R¢ bI YoVI5 0 '
&I ¥ R,IIofVV'g7?D$ 0p&^'&^| V`7 I¡R ` I^'K1 YVDp & D
¥ ©?VIgR1YKî9'5Jo
V'`
1Vî' R¡_ VV 77
o
W& Î&g o_7 I¡c7'& I& ô
a9o VaKî9'5fD"oI
.R ob' VK ^ViKo
71K'K1 U pW¡iV|K1VIa'R'^7P¡ V (U , x) = constant ¥ C V 5PI'V'I1
Zo'7Coh ]YV
'V5,I¡]^KiK ^V[h^CVh o17I( VPVo
Î V (U , x)
K 1 YVoI
0 '·&I1JYV Z o
5^`WI'II,& V (U , x) £ o5hVh{
KiV^5i5i o]©
^KiK 5 ¦ &'[5IiV'{¡ · VK'IVK h 0&hg ioIV'& V' ¥
AbM55 '&^(55^#D§g&1I ¡['V^CV'
V'_V^VM^^|K ^Vg V o|1< g&
Kî9'5DYîVKi5 ]I U oVIÌ V (U , x) ¥ V5]'Vc^KiK ^VOV'
VV
â^|oK| O VDaaW¡11K"'hV0'IV¡OhR hU 'R'^7P¡KV V (U, x) = constant ^
IV.1o|55^.YK^gC§gIIIOIa'V^ ^Vc^^|K ^Vp Ko5 V (U , x) â1g&
V0 IiC5Io7'I11o5o5 U Y^ô&?g0^KiK Z&oO V 5PIh'`'hVC7 I¡O7'&
h
^?V' 5'I ¥
©?o^JoVD7o$'VF 5 ^'K1|I. V 'V5i ¥ C5IO
^&'^¡cIo 5'I
o&L VFI55VI
¡_^fpIK'IKo £ 5`©#&oK ^ < Î ²1¦<¥ F&D7c7'& Y5^R
VV'I1 K Î@
D&^|^Vy'V
¨VI
pV 7 R& Î &C V 7 I¡®7'& I^ Î
£ ¥ ¦
¥ ©?V,V'IGD`7 &Î7P1 ?&p&oZD? V 5PI'F1i9 ¥
°1°
0R J , , # #2 4 ! 4*!. * . . - " " $ $24 () ! &
44 4- . $ # ©?oJ
¨'5o K1c&o VbbKKg&D&o50VÎAI 'V5i F Va7¡7
I P1K^R`J V,'&iD|Ib'oRaPIa oCa5^KoIÎo
|oVK 0I1Ko|0'V5i ¥
2 IZ Vy o&^KR ?R'5[oR'K1o £ ¥^¤ ¦ YK ®^ I·RZ'7[7 I¡ 7'&
I^'K1 £ ¥^¤I¤¦ b,&]17^K¡p15'9P¡O o&Y Vc5oVK Îo`& (t )
i h
Yo5'
bh (x)
â'V
(t2 )h (x) = (bh (x))2
(t1 )h (x) = bh (x),
L2
V' < ÎM&
f (Uh ) −
X
b(x)
|'VCoVK ,5^5i1Y oI5
sj (Vh (Uh , x))(tj )h =
j
Vh
KIO
1 2
gc
2
Yo5[ Vc7 I¡ZR h + b = c u = 0 ^Y'1V'R'^7P¡Ko|1V? VK'5i1 ¥
V5@&Vo I ,IV'&îbKgo&D&g
i
'VV5¡&D
VDR'KV0'V` IV

'5'S
¨V1< ^¡ RC oi7 I¡ 7'& 1.a1F^ 'o
P1 ª|5 VVF']'WI
5IiVR'K1o&f
,^{¯ ± ¥ ©?V| IK ÎoI ªi
ooCI i VRY '[
&oIV^ZIFp&^'&^V^Vy
5 'IK '1¡ R'|
¨VI
^¡Ib90¸ p&âiKg5& Î
15K VY ocKo9'Î"
1oK Î]IY'V,§oV¨ZDYV'RD ¥ , 5P5Y'Vc^ 5'7 ^V
'15a'¯ ±
4. Schemes based on well-balanced quadrature
* ['V^F7<'K1b0P1K^R ¯ ° ¤ ±G&g[V5IK1[I&K 5'o& Î,&Vo IO'pbKK
g&D&o5KV ¥ A|ôI y1 a5^9Ao&D&g
¶o1R o p'V^_PI55âW15&10&
'Vp' ^g&f
gV·^ £ ¥^¤¦ &g(^1VC [VÎAI50I
VR'_gV9'pI1KVi
'V|0YVD]^?a5^KoIÎo
OP1YiR^V|a& YV^K^V'^ ¥
©?oM7<'K1·D,1 &V^³5¸10PI^KR`º I ·I·'VO' ^g&LIb'VOoIÎo

DW K 'o5ZK oo7<'K1° ¥ °VbcV
oV £ K Vo <'K1 ¥K¤W¦ iIV5&.
DI'
I# 5iD^'
'
'òo9'C1I^ViF'V5iaIoO1K1F|
gV9'K1OP1? o[ ViJ
|bKKg&D&o5[PI0&(V^K^V'KV 7'& V ¥ 2 IFI(VV^^^VZVK ]&f'V
'V|`?'VoK
5YD |I '5Io7 'o<'K1&L'ViVRV VioIV'& V'c&
°
Vb IVD&.oI7G& Vb 7DoII^0 VbK'5'K1&'VJ
KDWIoC oJ'
1o7 'o<'K1
&a VO KoIVD&0oI7,&a op 7DoILR0 VZ
5^f^1'5 4I5CàO
ooOZo& Î
&YbKKAo&D&o5Ko £ Vo
Î ¥ ° ¦<¥ ©?V51 ¥K¤ 7'& , o&c V7Z
Ig9'K1o
IoI'I1'5J o&f VYR15&^'V5i?D$bKKg&D&o5 ¥ * oo7<'K1 ¥ aa'IK^³5
VDJ15V'IoV'I1'Ió ^VcV^^KV'^V
Ig' o
Îo ¥ *Mo& D
VD&bY' R
V ¤ oIK^R ?R 0oR'K1o0I0ZV I I¡i&o·
1o7DF'Vi^ IiR,'7
Î.§oRÌo,?R 74IKDIKÎV0Y9', oLI'^ ¥ ¬f¨V
"P1GD'
I'KVK ^"YV^
&'VI?IKÎVpYK Z VCI'â05&M V 5PI'oIÎo
FIV5& ¤ V^^KV'^V
71K'K1o0P1, VZ7oIK^R´?R'5,1gR Îg ¥ 3 &¡(Ib'Vp'VVDVcV 7
V5'c5I[1V&V Z'O&'V55^1 ?Ifo&D&o5,^W` ¥ * Vo <'K1o ¥ ¥ ²
b,D 5o'Y71|0^ 5'7 ^VpI <?&$ ^& M'V5i ¥
~ (, ~ g -.+ ~] , R~ ,!0 , #%&# 0 ~_g { 5',b,5Io ^V5Y|I5o5&.oIÎo
C^W K['V0PI' £ ° ¥ ¦<¥ 1"

Z
1
U (x, t) dx
4xi Ii
Z
1
R(x, t) dx
Ri (t) ≈
4xi Ii
U (xi , t) ≈
£ ¥^¤¦
gY&oV W¨^pR a
KWI'II.&g VY I^ Î|&oc V?' ^g& ¥ ©?Vaa5Io D5
7|DD 5 5 ,'V5i?I"'V0PI'
£ ¥° ¦
d
U i (t) = Ri for i = 1, . . . , N.
dt
Z~ , , #%&# 4 £ ¥ ° ¦ ?a5^9Ao&D&g
pPIY&ZV^K^V'KV R
V
(
£ ¥ ¦
& £ ¥ ¦
Ri = 0 for i = 1, . . . , N
% ! &
( ( *
V (U, x) ≡ V = constant.
| £¤W¦ o'V5i?gWI,&D7i5]&^K 4 .
& 4|K
V|KK '& V'I^(I` [D'KV1VD7( V P'I & 3 4 ViPIYYVD
£ ¥:²I¦
Ri = O(4xp )
°
P1FVK^^V ^V VR £ ¥ ¦ YV5' p 7V1VDgVÎV` g&] o_I&L5Io ^7
'5o5¡MIG'V,RI'IK o5i ¥
£ ° ¦ Kg
V|71K'K1 U &g &D7O V_ II1'IV¡ b pW¡[g|^'
1 ôIoga
5Io D5 R |0IVoV 1 giI V CRI Ω ¥ ¥ R ∈ M(Ω) ¥ *[IV5&
o1L& p'51VD&JIop7^V1VÎfo& 'JY9'p 7
L 15 Io Ii17V'I1&o
R
'V5'
P1 YKJDfVIb''IK17PI'?&0 ,1K1Y|&V^V0'0 V`^ I&^ £ ¥^¤¦ 11$
V
oVCc
1o ^7 1Jo1R o ?P1L o^JK I& ¥ RbIR'V K¹O
VK â5I
o&aIoO VIo^po&bFW1ID1&gpbFgK^51Y gRY^'
g Ko, V«K'
^¡iK
'5'pJI|I V'Lp IL VFV' 5'& Îp&" 515&V'55Ja5^9AoIÎo
|oVK
1I^Vic o5iYi`''Io o&'5 ¥ ©?V^g1KI#^5 D`
^1 5^¡M'5DR'Z p'V
aIO1ZV1*
Ig7 RR'K1V'o
'?&$i17V'a^y¯ ¤ °g°I°W±
£ ¦ ¸gI^ Y9'd'V5iOYV^ &' 7|DD 5 5 ^ 'Ki ¥ 1"R'5Yb(YK^
g7 `VVI5 0V7'p ^|^'
'
'K³R'K1oÌ`K¶¯ ¤ ° ° ±" Z5'K10PV^K¡D 5 5
'V| ¥
?^?a5^ VRYZP 1
1o7 R& ÎMDW`1'VcK¹O
VK¡O^]V^'
'
'K³KV £ ¥ ° ¦
I D P 1 V^'
11'KVK ^F^ oi I^'K1 ¥ Ab V§o¨Poo<'K1 f DFoIV^Ko&
'V071K'K1 U Y^K515ÎO V L^poVK ^| ¥ 2 1b7'R'K1o&'¡i7V f (U ) D
5I ^VIg`V0'p V & ^V
V1IV^&`5IoK Î"g U = 0 &Y V_ V ¥ A
'Vc V p^?Vg'IV¡I o5 f (U ) &oZV5g
U 5Ii,t ^'1iI V 1&o
t
Vio`1a Vc V poI' ?x ¡ ¥
U (x, t)
2 ICo&D&g
_DW`gKCDC K&oK_ O' R0&D7[^'
1 ô9'K^ o|V&' b
Yo^]pW¡O,IÎ5Z5K V`¡p o,V 1V^5ªK' 5K#1Y¡p VcV^'
'
'K³R Î ¥
*|IV5&&D 5I ^VK ^$^_'Va§o¨ f (U ) I# VYV&' b YK^KI_ 07^V1VÎ
g& '|K® V]i17o ¥ ©?oM'5'
&®Z 'R'{
DI'7D5IK^¡¸D 'V
'V51 ¡M&Ga O71KV ÎoR?I#
1o 5'W& ^fI(U
Z^)Wx` ¥ KoIVD&'9'KaK[ o0 Io'5'5'
I 0^'?bK.oo57 ¥
# " # ~ +(-.0 0 , + -0 0 p ~R~ , +(0* y1'0 WIKgR i'Vp^1'51'IfI· V & £ ¥^¤¦ #aO7V^K
'51VD&Y&o] KoIVD&?oI7?Y9'[' <?' 1"5 IV0i17o dx
£ ¥ ¦
R=R +R
.
reg
sing
R
^1'9
°²
oIK1IIg7^¡Ia0 V^9Y'Vc
KW15&1b&"'V,' ^g&.^
£ ¥ &¦
i
i
Ri = Rreg + Rsing .
_1 Vi0 gR`'V_ KV1VD&YoI7Y&# V_ 7DoI"&'c
1o
1''& Z&` V
5^
K 741
5VIoZV
1ig7
£ ¥ 1¦
i
Rsing
i
^
i−1/2+
7
£ ¥ ¦
Rsing = Rsing
i
i−1/2+
&o
£ ¥^¤ ¦
Ri = Rreg + Rsing
i+1/2−
+ Rsing
,
i+1/2−
+ Rsing
d
i
i−1/2+
i+1/2−
U i (t) = Rreg + Rsing + Rsing .
dt
2 Ib oC'7bI < Î aF1K1F&MRI5'K@VR _ '&b'VF'51VD&bIop'V
7 ^VIoÎa5Iig1V5JI V0' ^Vo&IM V & £ ¥K¤ 1¦¥ *[©?V51 ¥K¤ b
IÎL15V'II7V¹p5K#
1oK ÎoGYVD,IoI'I 5$'oR# ob'V5i £ ¥^¤ 1¦ ^GbKK
o&D&o5 PI,&¸V^K^V'KVS7'& V ¥ * oo7<'K1 ¥ .ai^'
g 0Zo_5
&#'V|Y&o[1oK^Ko ^V7'& aP1YYVDZ'V ,
1oK ÎoYI ,'R D7o ¥
#"$#%&#
~ R~ + -0 0 ~]R~ , + 0
VV1 C o&
£ ¥^¤I¤¦
Ũ (x) ≈ U (x)
£ ¥^¤ ° ¦
b̃(x) ≈ b(x)
&'|oK
5YD O7i&'·'5Io7 'o
Îo0Ib'VM
KbWI5&1 Ū , b̄ RI5C'VM
KD
i i
¥

{·o7d^ 'V
1

(1)
(p) ·o1R o ¸g1K'[YK V^
Ii
xi . . . x i
Ii
£
^
¥
¤
I
²
¦
¥
¥
1gIR'V
5^R * oo7<'K1
aYK^FV5IK1{'5Io7 'o
Îo
Y9'Z'V0PI^^RYKV|V'I5 ¡º
Z~ , , # " # ! &
( £ ¥ ¦ ! . * V (U,
b) * 2 (Ūi , b̄i)
4 (U, b) 4 *.4 . (Ũ , b̃) °
0a5^9AoIÎo
PI0'VpV^^KV'^V 7'R'
(U, b)
(1)
(p)
xi . . . x i ∈ I i
£ ¥K¤ ¦
(j)
(j)
V
(j)
Ṽ (xi ) := V (Ũ (xi ), b̃(xi )) ≡ V
&o( VOo1R o |g1K
for j = 1 . . . p.
*Io&Î1¡' £ ° ¥ ²1¦ V ^

£ ¥^¤ ¦
R̃ := R(Ũ , b̃) = −f (Ũ)x + s(Ũ , b̃)
0 oc&VV'W¨Kp& C 7DoI ¥ 15
£ ¥^¤²I¦
K(R̃; Ii ) :=
p
X
xi+1/2
(j)
ωj R̃(xi )
j=1
1
≈
4xi
Z
R̃(x)dx
xi−1/2
M1gIR'V ,I$ V|&VV'W¨Kp& ,' ^g& R̃ RIY VK'5'K1`&L
K I &o
i
^

£ ¥^¤ ¦
i
Rreg := K(R̃; Ii ).
,Y^K"7 g¡Zo1'& V'aYK [ o0PI^KRY^V|V 1g7¡º
[~ (,, # # £ ¥^¤ ¦ |bKKAo&D&o5yP1_'V]V^K^V'Ko
7'& V £ ¥^¤ &¦
i
Rreg = 0
. (Ũ , b̃) & .4 £ ¥^¤ ¦ #" #" #
~ , + -0 0 ~R~ , +(0 ?VR¶'V F o? ^VIVD&$oI7$&o V?' ^g& ¥ 15foGP5o$VIi&K 741

¥ 2 IZKgV9' KpIF iIK ε Ya^1' g
¶IVgV&'¡®DW¡I (x
xi+1/2
i+1/2 −
¥
9

1
¡
5

g< Îg
ε, xi+1/2 + ε)
&g b̂ (y) VYV y = x − x ¥ ©?oCIooV&'¡OR&^VY&'
Ûε (y)
ε
i+1/2
£ ¥^¤ ¦
±
Ûε (±ε) = Ũ (xi+1/2 ±) = Ũi+1/2
£ ¥^¤ ¦
b̂ (±ε) = b̃(x
±) = b̃±
,
ε
i+1/2
i+1/2
°
YV5' Ũ &o b̃ I i'VOV^55YD7O iI y'
1o7 'o<'K1oCP 1 £ ¥^¤I¤W¦ £ ¥^¤ ° ¦<¥
`Rda,
gV0 Vc ^VIVD&?oI7?&G V,' ^Vo&.1M'V I £ ^¥ ¤ ¦ D
£ ¥ ° 1¦
i+1/2−
Rsing
£ ¥ ° ¤¦
i+1/2+
Rsing
1
:= lim
ε→0 ε
:= lim
ε→0
1
ε
Z0
−ε
Zε
R̂ε (y)dy
R̂ε (y)dy,
YV5'
£ ¥ °I° ¦
R̂ε (y) := R(Ûε (y), b̂ε (y)).
Z~ , , # # # £ ¥ ° 1¦V£ ¥ ° ¤¦ !
$?a5^9AoIÎo
pPI?'V,V^^KV'^V 7'& V (
£ ¥° ¦
Ṽ
= Ṽ
=V
0
i+1/2−
£ ¥° ¦
#"$# #
i+1/2−
Rsing
i+1/2+
i+1/2+
= Rsing
= 0.
~ ~ ~ 0- _~ -.- / 0-.0 !, ~gR~ c4IbaFgWI`^ 'Vo
Mco& ÎM&"a5^9AoIÎo
^V,P1bIOVoKD^V&AV^ iI
Vi7i9*D 5 5 ,oo9'1I^Vi_ o5i ¥ µ I_oKV^VM V b|5&(KiiDR'5^¡
7'&oKD7oM V0P1K^RYKo_'V51 [º
~gR~ #%&# . 4 £ ¥ ° ²1¦
d
i
i−1/2+
i+1/2−
U (t) = R + R
+R
,
dt
i
reg
sing
sing
R
£ ¥ ° ¤¦ & R
R
$
4 (i − 1) . 4 *) * V i
4& 4 . (Ũ , b̃) £ ¥K¤1¤¦ £ ¥K¤ ° ¦ ! £ ¥^¤ ¦ $ & *.&! & 3 4 4*4 + , .# " $
4 £ ¥ ° ²1¦ & 4 . *
* ( V
, . £ ¥° ¦
d
U (t) ≡ 0.
& i
reg
£ K¥ ¤ ¦ i+1/2−
sing
dt
i
£ ¥ ° ¦ $
i−1/2+
sing
°
©?o^FoVD VCIoF15V'I#D'
o'7Î(&ba5^KoIÎo
o5i ¥ * 'ViV
¨
Vo
ÎRaaYK^V
1o' g<# 515& Vi#YVDc4&^K''VJP&ib1
CI
^^V[^[©?VI'5 ¥K¤ IiIV V« oC'55Y7
IgM1'a'V|C& oVo
5c& ¥ ¯ ¤ ±fIo'ViVÎAI5C o5iF& µ 1' g $I 50I ¥ &o Vi&'VI
¯ ° o° ¤ ± ¥
# # ~ 0-., 0, ,0 ~ )$+(,.-,/ ,.+ W~ + ,
* c VDG
Î,aJY^K VR('oR$71iJ'
G o5i4&^1^ Y'VLP&i5aI
1 ^^V¸K¸ oV'5Îo0
Î ¥ * Vo
Îo ¥ ¥K¤ ¥ ¥ gb|Y^K$'V5'
P1
15'9P¡_ VbKKg&D&o5KVCV'I5 ^$P1$ V
Ig' o
Î& Vo1R o 1&&o
'V ^VIVD&F^W¡151`^ 'Vo
]K
gV9'K1o ¥ ° ¥ &o ¥ ¥ Fo
c VDgI
5d1VI`©?VI'5 ¥^¤ KiV^^M o&M o RI'IK'V|O&' a5^9AoIÎo

155IKo_' 5oVK Î ¥^¤I¥
# #.% # R~ +
3 , ,. ~ ~ -.- ,) ~ , o
]GRg 0 ,
R~ +
3, ,. ~ ~ -.- ,) ~1 , o
]gI^®YK ¸ &*5IK^ $ *..4 4 4*. o&|V' 5'I,'V[V^7
g&'I m &g_'V`?R'5'KI5 η = h + b ¥ Ya?PI^^R go'7`5J& ¥ ¯ ¤ ± ¥ ©?V5¡
51K['V,
1o' g< ÎZV'5'?¡O
1o' g< ^V VcV^'o&'I m V,?R'5
^515 η &o V& I b ¥ ©?o5 K ¯ ¤ £ ° ¥ 1¦ ±J V¡®
gV] V]'5Io7 'o<'
o5ÎI
£ ¥ ° &¦
h̃(x) := η̃(x) − b̃(x).
©?o5'
PI'I19.'VFV^'o&'IÌo| o?R J^515gI `5Io7'IL gI^pY9' V¡
Y^^G'5p&^
1o7'&o ^Vp'V_'5Io7 'o<'K1 ¥ * o& D
oÎ'V_D 1cRC
£ m = 0 η = const¦ ^$o 7 1_ V'IV1VIV#'VY'
1o7 'o<'K1 ¥ 15Lofi5'K1
^y
oI' KVM o& h I,5oV¸^ £ ¥ ° &¦ D0^& 501(' Vg5R'¸¡ og i
c& ¥
^®1'V5c'(1o&& pg7K ^9¡yIY V[a& _VK11 ¥ ©?o^^|1VM^{ o®
?W¡O'oRY'Vca5^9AoIÎo
^Vi ^& Î £ ¥ ° &¦ D`V' 5'I £ 5M¯ ¤ £ ° ¥ ¦ £ ° ¥^¤ ¦ ± ¦<¥
©?o5'
PI'I'V|¡'17'& Dc'5Io7 'o<'K1 DFa5^KoIÎo
]PIC'Vi^ I&,
155IKo_' 5oVK Î ¥ ° ¥
' )$+(,.-.,./ ,+ R~ +
3, , ~ ~ -- ,.) ~ , g
*{¯ ° ¤
¥ ¥ °W±# V|&'VIV5^ (O'
1o7 'o<'K1YV^(V 7 1ÌK#Io

°
Kio7Îg&#V^K^V'ÔPIF V|7oIK^R ?R 1gR Î ¥ V^â_'
P'O gR
o&5$P1G'V`5'&^D5&a?b1VDc^ 1b'0IÎb V 15¡,DCîC7VK ¥ CK15i
5^
WI5&1 (U ) &o[|gI7'IªPoo<'K1 b(x) oa0o7C^&.'
P o
0R&^V V
&J'ViV^KîV'KoSR&'ÎVK ¥ ©?V i&'|5oV(îVKD
K ^¡(¡]'Vp VK'5i1i
o&
Z
£ ¥ ° ¦
1
U (V , x)dx = U .
4xi
i
i
1
,o0g&o |PI,[|1i5c&o·D 5o F'V^0'5DR'K1¸&'
PV^K¡º
^C'V
KI' _I V (U, x) ¥ ¥ U (V (U, x), x) = U ¥ `5DR Î £ ¥ ° 1¦ V1U (V,
Vx) 1`'V
VVD1o £ 7` oFoI'II&VgfV'
^VM¯:° ¤
¥ ¥ °W± ¦ ^5Ig1oK^Ko ^Vª oip gR
Vp
1 ' Io^V[
Ig7 1R&'D&V^ U (V , b(x)) oW1c oO I|i
KLW15&1
IF VpVi D5I$oR' ¥ A,D0V R15 ^ ¯ i° ¤
¥ ¥ °W±$ gRG9b VpV&' U (x)
Ui
&o
I K·K&$V^K^V'Ko £
PI0&^f
KD ¦ V('V
5P5'b(x)
5o5b1oK^Ko ^V 7'R' V 5IiVV(U
'_(x),
D x)£ ¥ ≡° 1V¦ 5Iô
D?YK I iVa 'VaK&
i
V^K^V'KV R' V ¥
©?o|'5Io7 'o
Î D0
IiV^
'(¡^^|K ^V[ o|'5Io7 'o
Î
YK
7
c 'VM'
P o
pRIKV V £ 5(¯ ° ¤ £ ¥^¤ ¦ ± ¦<¥ ©?V[&'IVi5Ṽ,'(x)
oR_1V

Ig' o
Î,D"a5^KoIÎo
CPI i
V^K^V'KV 7'& GDGoR ''IK11 PI'aI'º
9# V,V&'i&'CIÎoIK^¡pK[V^^KV'^V £ ¥ ¥£ ¥ ¦ VIDVaPI`|1K1o&V^^KV'^V
R ¦ L'V5 V = V P1iIK`55^^bIo® V]V^^KV'^V|^KiK ¯ ° ¤ £ ¥^¤ 1¦ ±
i
¥
5P1'5Vb o& Ṽ (x)
≡V
~ -- /(0-0
~ W~ + , + ~ 0 W 0 R~p~ 0- #
*®¯ ± µ I7 'o 0&^D&o 11g³|Io f&'C7'& 0¡ 5IiV'Ko V 1F^ £ ¥ ° ¦<¥
©?V^?1K1a'V,KR I5Y15
o'& CV^K^V'KV
1o' g< Î i
Ii
Ũ ∗ (x) := U (V i , b(x))
Y V^ DiIV^¡¶oI {VI{ VRIKV|Y9'VK 'V ith 5 5^ ¥ 15Oo I 5 i &o
V5o5 Ũ ∗ V¨ ¥ ©G¸go 'VVÎAI|5I' < Î µ I 7 '¸5pI ¥ 1iV

Ig' o
ÎZg1K¡V1|D&
¯ ∗ ), j = i − k, . . . , i + k)
Qi (x) = p(x|(Ij , U j − Ũ
j
YV^[^1'5'g1^& a VcK»5'5o5?&G Vc55^"WI'II U &oM Vc55^"WI'II
j
&Y oM^R 1'V5_'5Io7 'o
Î Ũ ∗ ¥ Ì M o& U 1VK¡¶
5Iô
D_Y9' Ũ¯ ∗ 9
j
j
j=i
¥ 2 Kg&^K¡ µ 1' |
I ¥ '5Io7 'o

U
¡
Ũi (x) := Ũi∗ (x) + Qi (x).
AODpV'RI®^ ¯ ± o&p VDi
1o' g< Î DiVK1AI5pI
VR'IJIo bKK
g&D&o5Z9 V = V PI`&¡ i ¥
i
# # " # ~ -- /(0-0
~ )$+ 0 G 0 + R~ ,. ~ ~ -.- ,) ~1 , o
*y'VD_ <'K1¶bO7'I7,P'I oM7i&'#bKKg&D&o5¸
Ig' o
Îo Ũ b̃
Yo^[b,5Io7 'o<'MKZ V,o Îga
Î[&o[5'ÎCbKKg&D&o5p^ 5'Î
o1R o K(R̃, I ) £
¥`£ ¥K¤W²I¦¦
¥ *[&ôI. VDb
Îa'7 '^
JIV 5Îf
'Vc o&^KR a& aioR'K1o ¥
2 I`5Io5^ 5V b0o 0 o0PI^KRY^V|V&R Î"º$ VV1 0 gR
a, b : Ω → R
I 0'I9AR&^VOPVo<'K1o`5oVM1[1VY7gR D&VIp&^ ¥ 2 1`_o¨]
5^ I V^

i
ã, b̃ : Ii → R
Z i& ®
Ig' o
Îo0R15,'VM^ ÎcIY V[
K ¥ ©?V5®bMV5VI O'V
V9»5'5o5c&oZiI[1g'& 1'b¡
£ ¥° ¦
+
+
Dã := ã−
ā := (ã−
i+1/2 − ãi−1/2 ,
i+1/2 + ãi−1/2 )/2
2 IYDR'5Yo IVa0Io 5'IF oc^'
'
'0V g<Y oK0&fK» o
^V
£ ¥ ¦
D(ãb̃) = ā Db̃ + Dã b̄.
+ 0 " 0+ R~ Ag ~ -0 g~ 0 W~ 0gI^MYK ]_YDK¡po [oIV'& V'`YVDMD?bKKAo&D&o5iP1J'V0^ ICR
'7 £ m ≡ 0 h + b ≡ η̄ ¦<¥ 2 IÌ¡O i& Ũ b̃
xi+1/2
Z
xi−1/2
£ ¥ ¤¦
xi+1/2
R̃(x)dx =
Z
xi−1/2
(−f2 (Ũ )x − g h̃b̃x )(x)dx
xi+1/2
= − Df2 (Ũ ) − g
Z
xi−1/2
(h̃b̃x )(x)dx.
¤
©?V5'
P1 b|V5'[5oViZo1R o _PIC V|^1'51'IGIL Vp IV
_ ¥
&o IVV'W¨KpR',gI
(1)
(2)
xi = xi−1/2 , xi = xi+1/2
Io h̃ ¡MK^VI?PVo
Îo ¥ ©?VDYIÎ
b̃
YK^Gg7_ V_bOoV
£ ¥ °¦
xi+1/2
Z
(h̃b̃x )(x)dx =
xi−1/2
h̃i−1/2 + h̃i+1/2
(b̃i+1/2 − b̃i−1/2 ) = h̄ Db̃
2
*o77'Ko £ ¥ ¤W¦ &o £ ¥ ° ¦ K' £ ¥^¤²1¦ a0I&^M'VcoIV'& V'
£ ¥I ¦
K(R̃, Ii ) :=
1 −Df2 (Ũ ) − g h̄Db̃ .
4xi
7îVKc&D
VDR'K1Z VRà gR? VDYDYa5^9Ao&D&g
OPI?'V,^ I0R`'7º$ A
£ ¥ ¦
V
£ ¥ ²1¦
m̃i−1/2 = m̃i+1/2 = 0,
f2 (Ũi±1/2 ) = g2 (h̃i±1/2 )2
o
b̃i−1/2 + h̃i−1/2 = b̃i+1/2 + h̃i+1/2 = η̄,
g −Df2 (Ũ ) = − D h̃2 = −g h̄Dh̃ = −g h̄D(η̄ − b̃) = g h̄Db̃.
2
LKoII^V £ ¥ ²I¦ ^ £ ¥ 1 ¦ b0î|DR K¡ZI&^M'oR
£ ¥ ¦
K(R̃, Ii ) = 0
PI? o,^ I,RY'7 ¥
+(0 " 0+ R~ g2
,. ¸ 0 ~Z~ )$+ ,-.,./ ,.0
*·¯:° ¤ ±b,'
ooM Vco1'& V' £ ¥ I ¦ iKg
ô,iRKVi1oK^Ko D ¥ ©?VI¡
Io 5'RR ÎZâ'oR
Df2 (Ũ ) = D(m̃ũ + g h̃2 /2)
= m̄Dũ + ūDm̃ + g h̄Dh̃
£ ¥ I¦
`&'Ko'oR
£ ¥ ¦
= m̄Dũ + ūDm̃ + h̄D(Ẽ − g b̃ − ũ2 /2)
= ūDm̃ + h̄DẼ − g h̄Db̃ + (m̄ − h̄ū)Dũ.
m̄ − h̄ū = Dh̃Dũ/4
a0I&^M'oR
1
Df2 (Ũ ) = ūDm̃ + h̄DẼ − g h̄Db̃ + Dh̃(Dũ)2 .
4
°
©?o5'
PI'IP1`||R^V|a& 5YVK^^V ^V]VYV Dm̃ = DẼ = 0
£ ¥ ¦
1
−Df2 (Ũ ) − g h̄Db̃ + Dh̃(Dũ)2 = 0.
4
ì
1o7V5o5I'VcoIR'V'
£ ¥ ¦
1
1
K(R̃, I ) :=
−Df (Ũ ) − g h̄Db̃ + Dh̃(Dũ)2
i
2
4xi
4
DCbKKAo&D&o5[P1F15V'I"V^^KV'DpK ¤ $¥ 2 I0 |I §oR`g Vi
Vo^
I
'
Î 1 Dh̃(Dũ)2 ^M7¶ iIK` o&O9MpV&M&»<p'VI5pIF'V
o1R o F V^4 ¥ *¸¯ ° ¤ ±b0 VRbpVR@'^KiK? VDb 5'ªYVO'V o|g Dh̃
Io Dũ &'0ViK1VIaIG'V,IaI"'V,I'ô7^³5 ¥
# # # , + -0 -.0 g~R 0 ~ ~ -- / + 0 , ~
* Vo
Î ¥ ° ¥ °aaL^1' g
_YIV5&RP&ib1
F&bKKAo&D&o50 KV1VD&
DW¡I' ¥ A$b1VD,g?
1I5oK1G9V'VaV^K^V'KV RIKVGI b
Ig&"'V 1VIoI
¥ *
'VM KoIVD&0DW¡I5 ¥ , VDc&ygOVIVp1VK¡(9 Ṽ
i+1/2− = Ṽi+1/2+ = V
'VD`5170a0

£ ¥ ¤¦
Ûε (y) = U (V , b̂ε (y)),
Yo5' b̂ (·) ^Z&¡ i& '5Io7 'o
Î@&0 o(&7'I 1g1I&V¡ ¥ 2 1
£ ¥ °I° ¦ aε0KiiV^& 5^¡M1'IKZ gR
£ ¥ °¦
R̂ε (y) = R(Ûε (y), b̂ε (y)) = 0.
*]'V_15V'I5171 V _^VO''IK17PI'?&M
1o7 'o<'K1[I$ Vc' Do&"K
'VM KoIVD&,DW¡I ¥ RbIapY^Kb|î^p VM5Io7 'o
Î £ ¥ ¤W¦.£ ¥ ° ¦ I
oI?1'7^V^0K V9&V^0o& '?I"'V,K'5'WI [−ε, ε] ¥
2 Ib VD?a` o M V0 o5ibV'I1 p^¸¯ ¤ ² V° ¤ ±YVDZI C&^ ^&
1`P1K^R`º`©?V|5I ^VIgV^55YD7K^V&F II1'IV¡ZDC
oo ¡]'V_P1V
RIKV y = −ε, −ε/2, ε/2, ε "
1 ' Io^V[ ( VZg1K x
¥ ` o cg1KÌo`V^
Y^ ,i+1/2
^KV& −b̂ε,'x i+1/2
1?'V−
ε/2, xi+1/2 + ε/2, xi+1/2 + ε
ε
RIKV
£ ¥ ¦
b̂ε (y) :=
(
b̃±
i+1/2
for
b̂i+1/2
for
y = ±ε
y = ± ε/2,
I
YV5'c'V_^ 5'iDR'_R&^V|VI`'V^ 741
b̂
7 ^K$VV Zgi5 5
|^V ¥ µ 7&^VK¡1f9i7oIVD¶Z( VK'IVK[
11
ÿi+1/2

IVôR'K1ocIY VZWIKo
&f VYoI^1 ¥ ©?VDf^ 5'iDR'YR&^VY& V` II1'IV¡,a1f
gV
b̃±
i+1/2
7^K1 ^¡|K»5'5 ^¡iKZIpIL¯ ¤ ° ¤ ² ±AIopa`Y^K.D 5o'f oâ^p VFPI^^RYKV
7Vg7< Î ¥
7I
M VK'5'K1'YIf VK5¢0Io]'K11FV5ÎI'KVZ
5^D5 ¥ ¥ K'V
K RI^ [−ε, −ε/2] &g [ε/2, ε] ob 155['VcV^^KV'^V RIKV5Io7'I1o&g

oo Û Io R̂ ^ £ ¥ ¤¦ .£ ¥ ° ¦¥ *y¯:° ¤ ±a0oWI05IK^M Vc
[−ε, −ε/2] ∪
WPε1fYV^|ε 'VY 7DoIR&VD7oR V* ¥ ¡
1o7 'o<'K1
[ε/2, ε]
V,R&^V Û (±ε/2) I
ε
£ ¥ ¦
ε
Ûε (− ) = U (Ṽ (xi+1/2− ), b̂i+1/2 ) =: Ûi+1/2−
2
£ ¥ ²1¦
ε
Ûε ( ) = U (Ṽ (xi+1/2+ ), b̂i+1/2 ) =: Ûi+1/2+ .
2
`&'| o&C'ViPVo
Î
g7(^ £ ¥ ¦ £ ¥ ²I¦ goVc7 'IV1K¡]1('V
U (V, b)
o& D
VD&V^K^V'KV VoV5F5Io D5R'K1 ¥ 2 I5¨V&iV^,909»5YP1Y o_D I
RY `5Io ^V5'ZKy¯ ¤ ±G&gO'V,iR^Vi?R a 'R'ZKy¯:° ¤ ± ¥
*( V p&^VKoOK'5'WI [− ε , ε ] YVD b|5IK^ 4 4*. `^ ¯:° ¤ ±A
Va 1g1I&V¡C^f5Io7'I b̂ (y)2 ≡2b̂ ¥ ©?o5'
PI'I&1P1# V?
¨V1<f71K'K1
ε
i+1/2
YV5'
R(x) = −∂x f (U (x)),
Vp&oV W¨^pR 7DoI# V1V^¸'o5_'[[
1o 5'W& Î_§o¨¸K»5'5o5 ¥ 2 I
VD5a|V
oV|&(IVV'W¨KpR',§o¨ fˆ (y) 1`P1K^R` ¥ * 'V|55 y = 0 o
§oÿYK^JM& IVV'W¨KpR' YKp&V ε IÎ fˆ(Û
$IoyRc'V
i+1/2− , Ûi+1/2+ )
¥ *¶
a5 fˆ pW¡(
5oI^' y = ±ε/2 "9c 10'VOR&^V f (Û
ε
¥ ©?V5]) a0

&¡M
1 ôIobPVo
Îo ¥ ¥ V^55YD70^KV& i+1/2±
£ ¥ ¦
R̂ε (y) := −∂y fˆε (y).
2 'Io5'Ioa_&]17^K¡Z5R&ô& ,'V_ KV1VD&òI7Y&$'Vc' ^g&^ £ ¥ ° 1¦ £ ¥ ° ¤W¦ &oZI&^
£ ¥ I¦
i+1/2−
Rsing
= −fˆ(Ûi+1/2− , Ûi+1/2+ ) + f (Ûi+1/2− )
£ ¥ ¦
i+1/2+
= −f (Û
) + fˆ(Û
, Û
).
R
sing
i+1/2+
i+1/2−
i+1/2+
b~ 0 #%&# $ . £ ¥ &¦ o£ ¥ ¦ ( * !
$ Ya5^9AoIÎo
% , . # ô
?'VY' ^g&R&VD7o$^'VYV^^KV'^V DW¡I5RKL^f ¹O
^5
'O7oR gR
£ ¥ ¦
Û
= Û
,
i+1/2−
i+1/2+
ô
0'V5
£ ¥:² ¦
ˆ Ûi+1/2− , Ûi+1/2+ ) = f (Ûi+1/2+ )
f (Ûi+1/2− ) = f(
Io.P'I £ ¥ I¦ £ ¥ ¦
£ ¥:²¤¦
i+1/2−
i+1/2+
Rsing
= Rsing
= 0.
Z VV1 c gRFaI _K¸K5I"1oK^Ko ^V ^ V|7o _&
gV9'K1 ¥ g ¥ ¥
PI` Ii0V^^KV'^V7'R' V ¥ ©?V
Ṽi+1/2− = Ṽi+1/2+ = V
£ ¥:² ° ¦
Ûi+1/2− = Ûi+1/2+ = U (V , b̂i+1/2 ),
Yo^]D £ ¥ ¦¥ # #
~ ,
~ ~ ,) ~1 ~ ,.0 ~ / * /
R
b̂i+1/2
b pP5o_og1®&¶^ 5 4I5 x
Ioy oOa R&^V b̃± #Yo^
i+1/2
'5o 71_ V Vi®&Y'VM& 1 &_'VZ55^b^ 5 4I5 ¥ Z 1oi+1/2
K'p gR_'V
'R'^7o
^ iV^& ,R&^V b̂
i+1/2
£ ¥:² ¦
+
−
+
min{b̃−
i+1/2 , b̃i+1/2 } ≤ b̂i+1/2 ≤ max{b̃i+1/2 , b̃i+1/2 }.
*y¯ ¤ £ ° ¥ ¦ ±A oo'705& ¥ V1
£ ¥:² ¦
+
b̂i+1/2 := max{b̃−
i+1/2 , b̃i+1/2 }.
©GI1
'V5ZY9'd& IVV'IV'^& µ 2 1 '7 '^
Î &g V9&V^§oV¨ Poo<'K1
£ ¥:² ¦ IoI'I 5,1 9'KK¡·IY VZa& VK1 ¥ AgIg5 o ¡·R&'K1o
I oI5Kg
ô^VO'V_V' 50&'VI5&o b_5Io D5F9C O V|&oVI'
oID
¥
²
RbI V aDGIcKiI 'I1$517bYV^7V1I. gR# V?&oo&cVID

7 V1VD,71|5 îJ'5o^1
¥ V^KJ VJoID
LI oo' J
#& ¥ DoI7'^5VÎ ^¡
o75PVJ&, VM V1 i&Y]D IiYK V
'9'^&L15^59¡1aO
Ig7D50'VOR'V5
9»5'5F K o& Î IfOa& 5 4&^^(7 5g4I7iIV'IKoIo Î' £ 7M¯:° ¤ ±GPI

&^^ ¦<¥ * 2 ÎV' ¤ aO7VR´ VD_7'& ÎoI ¡(|R^V]a& 0§oR ¥ ©?VO?R'5
§oR`$Kp V5
'9'^&^K¡,P'I´ VY^
¢Lo1'Kg9Lo9L,'5 £ IfI5pD'
I'KV1o ¦
'
1 (¥ 2 ^RYKocRY1 V`?R'5J1555^5R ¥ oY _5Io 5'RR'K1Ip1 $'V
a& 5Yo5Î5 I ?V'K" V0§oR
1i`5 K D5I ¥ A`^` V]7 1V]¡Z
R Îg&'¡[ V g1FI' ¥ 5Vô'Vi7V g V|a& iRI1(7^RY^¡ZYK
7V
'K D5&.15^
K¡ ¥
1.2
1
0.8
0.6
0.4
0.2
0
−0.02
−0.015
−0.01
−0.005
0
$&J&L*98$C-( L$&F
0.005
0.01
x ∈ [−0.024, 0.022]
0.015
0.02
_1'IK V| I|71K'K1]PI V Y Kp&V o 1VK YK ¸^'
1 ôIo
gI7 1 ¥ ©?opK 'KoIo 5'RR ÎyD0 gRc'VO¡ ¡oIiîV'IoK £ 'V
R Îg&'¡¸ V ¦ ^_ 71K1yRc'VM& 1 IY VO II1'IV¡I$7(bM V1V^
V^1
£ ¥ ² ¦ ¡
£ ¥:²I²1¦
+
b̂i+1/2 := min{b̃−
i+1/2 , b̃i+1/2 }.
*M¯ ² ± µ 1' g $I'VoR&o $I.P1 V^& bbKKg&D&o5,'V5iGYK IV5&
VID
Ì b̂
& DP¡^V_1VK¡ £ ¥:² ¦
¥ ©?V1 ^p&goID
Ì b̂
iIKgJ&
i+1/2
i+1/2
I5V'IV^5]YVD 75pY'M
^1 5^¡M'5DR'['p VVIVVo^VV'Ì#'V
!
`KiIV]V'IV^5]V 5i¯ ¤ ± ¥ 3&YV^K0aC'5Iii5oO po £ ¥ ² ¦ 7ô5CK
DY1 9'KK¡OV' 5'KV ¥
# $# ~ ~ ~ 0 , g~ (0 0
3 ~1 ~ + ~ #
*·D 5o ÎoF&L'55Fa5^9AoIÎo
ViK0a1FI 1V['oRC V|
| D5IV§oV¨J^ ¯ ¤ £ ° ¥K¤W²I¦ ±a5'`V&?
1o7 R& Î ¥ Fb1VDpK I` 5Î KP¡ VD
g1KºG *1VfVI'R'K1 £ 7 £ ¥ ° ¦ ¦ R'V?ofI#'V5i?& oVo ?
L& ¥ IV
£ ¥:² ¦ ∆x d U (t) = F (U , U , b , b ) − F (U , U , b , b ) = ∆x R
i
i
l
i
i+1 i i+1
r
i−1
i i−1 i
i i
dt
Y9'Z§g¨0¯ ¤ £ ° ¥^¤ ¦ ±
£ ¥:² I¦
£ ¥:² ¦
ˆ i−1 , U i ) +
Fr (U i−1 , U i , bi−1 , bi ) = f(U
0
g 2
g 2
2 hi − 2 hi−1/2+
ˆ i , U i+1 ) +
Fl (U i , U i+1 , bi , bi+1 ) = f(U
0
g 2
g 2
h
−
2 i
2 hi+1/2−
1
`gY
IioI F o^?'i Vc'V| £ ¥^¤ 1¦ ÎM^[ VD` <'K1º
!
!
£ :¥ ² ¦
d
i
i−1/2+
i+1/2−
U i (t) = Rreg + Rsing + Rsing .
dt
2 'I £ ¥^¤ ¦ £ ¥ &¦ &o £ ¥ 1¦
£ ¥ ¦
i
i
4xi Rreg = −f (Ũi+1/2− ) + f (Ũi−1/2+ ) + 4xi S̄reg
£ ¥ ¤W¦
i+1/2−
ˆ Ûi+1/2− , Ûi+1/2+ ) + f (Ûi+1/2− )
= −f(
4xi Rsing
£ ¥ ° ¦
i−1/2+
= −f (Ûi−1/2+ ) + fˆ(Ûi−1/2− , Ûi−1/2+ ),
4xi Rsing
YV5'C V,'51VÎ?o& Y&"'Vc IV
F ^?1K15[¡
£ ¥ I ¦
i
4xi S̄reg

^V
=−
0
g h̄i Db̃i
!
.
ˆ Ûi+1/2− , Ûi+1/2+ ) − fˆ(Ûi−1/2− , Ûi−1/2+ ),
Dfi := f(
Vc'V5i £ ¥ ² ¦ 'Io
£ ¥ ¦
4xi
d
U i (t) = −Dfi + [f (Ũi−1/2+ ) − f (Ûi−1/2+ )]
dt
i
+ [f (Ûi+1/2− ) − f (Ũi+1/2− )] + 4xi S̄reg
.
2 I^KRY^V,'VI 1Vi5'f^ ¯ ¤ ° ¤ ±If'Vo 71bo&511V&OV c gRJ'V
'V|0
oo¸¡ £ ¥ ¦ &o £ ¥ ² ¦
Iô5^ ¥ Ra515 Vp
o ÎoCV1o
1
5'RR'K1b§g¨|K»5'5o5LIVg&LI^ Fî o?PI' £ ¥ ¦¥ YYKôVR 7oR gR
'VCb_§oÖK» o
aKM V0 oI V'1 I5'JI FV'
D 5^¡'V07^VIoÎb IV

'5'pYKZ o,K5¢F&oZ'K11Y1oK^Ko ^VDW¡I'YV5'C Vc IV
F Vio ¥
2 1 £ ¥ °I° ¦ &o £ ¥ ° ¦ b VR¸ gR"'VJ' Do&IWIVD7V^0 VbV^K^V'Ko
DW¡I ¥ 31 |V'
D 5^¡IKc
Ig7D0&?ZoIA³5 M§o¨·V9»5'5o5p&o·[VIA³5
Io'5F'5'ªYVD]o&D&o5FI[&'V5º
£ ¥ ²I¦
i−1/2+
i−1/2+
4xi R̄equil = −f (Ũi−1/2+ ) + f (Ûi−1/2+ ) + 4xi S̄sing
=0
£ ¥ ¦
i+1/2−
i+1/2−
4x R̄
= −f (Û
) + f (Ũ
) + 4x S̄
= 0.
©?o5'
PI'I
£ ¥ I¦
£ ¥ ¦
i
i+1/2−
equil
i−1/2+
4xi S̄sing
i+1/2−
4xi S̄sing
i+1/2−
i
sing
= f (Ũi−1/2+ ) − f (Ûi−1/2+ )
= f (Ûi+1/2− ) − f (Ũi+1/2− )
IoZ Vc'V| £ ¥ ¦ &[g,'5Y'9 Z^[ V,o& V&P1
£ ¥ ¦
d
i−1/2+
i+1/2−
i
+ S̄reg
+ S̄sing
U i (t) = −Dfi + 4xi S̄sing
dt
Yo^ 5K&'K¡·^7 ^V1V^ Vc
1o7 R& Î|§o¨ K» o
5#'5IoÎc&oy KV1VD&
Io'5F'5'p ¥
5. Numerical examples for the shallow water equations
ioWIi o55'PV^^¡ K1V(VÎAI5Ca5^9Ao&D&g
( ViC¡ 9»5
50IVV'11V ¥ * < Î .VÎV1'a5^9AoIÎo
[oVK pK» o
1oVK
II^ViY&o Vif&'Y ÎVcP1J0
DI'$&¡5'g1KD?oIÎo
?DW`5
YV^¶ô5KoVO oM7g&^KR a& co& Îo,YK VZ^ IOR _VK^^V ^V ¥
©?VI5¡M^ R?&VL'V0bKKAo&D&o5MV 1g7¡pDYi75Î5Ii1 9'K1ZI
V[71V'5p ¥ 2 K¢ {I5_oVK [9»5'5g
I$gV9'M1I^Vi ¬ Vi
&oO V^'ZIJoo9'CKi5 ´ ViCI FKiV^5i5'.VIoMa05VI
V ¡ 2 0² 2 ² £ P1ObKKAo&D&o5®oVK (V9»5'5o5Ko ¦ &g 7
Î5^¡ ¥ * Vo <'K1 ¥ JVÎ IgV9']II^Vi] ViiYVD I
bKKg&D&o5[PI ViR^VOVK^^V ^V If V|7oIK^R a& o& ÎooWI
g5]V' 5 ¥ ©?V,a5^9AoIÎo
OV'I5 ¡O'5^K?Ip75Î.V^K^V'Ko'

Ig' o
Î,&g0VI '^D&&o1R o fI'VJ IV
$'5']YV^^f VbVÎAI5
I55VI
¡[
1|YP'I o_VÎAI ¬ '5Io7 'o
Î&o
¨ &IDR'K1
&f o| IV
c 5' ¥ 2 IV (I5CI55VI
¡[5I g|IV'&^V ¥ V5VI V7
oVK |II^Vi_'V5iI 2 £ ´P1Fa5^KoIÎo
]1gIR'V ¦<¥ &'c gR
bYoW1ab,a5^9AoIÎo
coo9'Ì1KVi`'V5i5 2 ² &g 2 &I&^V
V'IV1[V9»5'5`&Vo IV ¥
*Z o^Y
ÎMaCV RDVi D5I' VK'b i|1o7 R F oC1MV'I
5 ^b&'V Fa5^KoIÎo
O'V5iYV5[&VV^^p'_'V07oIK^R ?R boR
Îo ¥ * &^"Vi5'^&" 7'V'Ki|^'
'
'K³R'K1[D`¡Z V
DI'7D5I. V^]I5
© YVoI
0 '¸|5 V ¯:° ± ¥ 2 I|oVK ]1I^ViILgV9'9»5'5g
¬ Vi V µ 2 1yVg`^Y' 15]I o¥
¨V
YPI?'V_15
o'1
¡i YYV
¥ 2 I0'V
7p&^^50 îM7 5gc&'i' I5¸ (5o V''oR|7o& D&L 'I0IiôR' V^' 1' 'V5i1b'V µ 2 1 V_5OD o¥K¤ V¥ 2 1p o ©
Kî9'5
KiV^5i5'y^y V 'V5iIG oM©

Ig& M £ 5¯ #° ² ±LP1c9
¦

oo9'K1 D' 15(1 K(|CVi5'^&#
¨V&iV^goVK F&'V5'YD7|R ¥
©?V,IW9R ÎZ
Ig& g â I[1 ¥ V¤ ° m/s2 VV ^V| oc
IiVV'R'K1 ¥
$#%&# ~ -.- /(0 -.0
~ ~
?© oaoV 1 JIV obg'7G'7#o 1VKpGDG CI KP¡C V?a5^1oIÎo
cV'I5 ¡
&$IVÌK1I'9'Vp ¥ Ì 0'oR 2 D`5Io&V^0&$5&V V'Koi V,iRKo|?R'5
VK^^V ^V]gIo 2 ² 2 0² I ,7ÎoZPIC5&V V'Ko'V_D 1,R
'7YV^K^V'Ko ¥ 5o5 IapK» 1Y'7YV 1V^5pY&'0V 1g7OV ¥
#.% #.% # 0 g~ 0 R~ ~ )$+ ,-.,/ ,.+ ©?o^$'7L^J7VRY ,15'9P¡c o& 2 ² 2 0² & g
ô5|iIK&^'V
5 ¨VI< µ V'I5 ¡cR15L0V1e§RL&7'I ¥ FV1 bacK» LPVo
ÎoLPI
'V,&7'I II1'IV¡pIÎM¡ £ 0 ≤ x ≤ 10¦ º
£²¥^¤¦
b(x) = 5 e− (x−5) ,
Yo^]DY7i&'o&g
(
9 4 ≤ x ≤ 8,
£²¥ ° ¦
4
b(x) =
&'V5'Y^ ,
0
Yo^]DY^'
1 ôIo ¥ ©?V,Ko9'ÎVRD?'VcR'K1o&'¡O71KV Îº
2
5
h + b = 10,
2
hu = 0.
©?o^F'IV¡Z7'R'c7oIVD[gc5¨VI<'K¡ZV' 5'I ¥ c5IiV', o_71KV Î]V ^
o Ko N = 200 VVKPI'S
5^D ¥ *¸I5F ]5i1o''& _ o&F'Vi
¨VI

t = 0.5
µ AV'I5 ¡iâô5MpIK'IKoOoO |'IVoI»] 'IIaCo70 KV1KCV'5^ K1
VIVV^·V
D7Î &o 1gI'VV^(V'5^ Î '{5 PI' Vy
1|o'& ÎC&o
VR@'V L1 &g L∞ 'I#PIb VFa& JVK11 h £ V&'Iº h Kp'VDbI DJVIa
5Io7'IJPVo
Î ¦ &op V0D oI 1 hu ^Z©#&V^ ¤ IoM°0P1J'Và_& 1
Poo<'K1o £²¥^¤¦ &o £²¥ ° ¦ &o]K»5'5`V'
D K1o ¥ 2 I?'V i
'V.V'V
'IÌ _5IiV'gI ]I Vo| D5IG I^ ÎgR0
K$
1'5 ¥ i5&
5K&'K¡¸7'oR_ V L1 Io L∞ 5' 1'C&'pR,'VpKIf&? 1VoA&» 'IFPI
V9»5'5`V'5^ K1o15'9P¡^V| V,5¨VI< µ AV'I5 ¡ ¥
_gWI_I^ M
1iV 7'R'K1o&'¡Z I^'K1oò ^VM^VK D&$
Ig9'K1o`Yo^
I V&F Vi7 1¡]R |71KV ÎoC&o K57 ^VM ^||51IÎc^ [M7 1¡]R 1
1'IKoKVp Ki^ÎY 7o9?Y9'Z'V,
¨V1< µ o 1g7¡ ¥
0 ~Z~ )$+(,.-,/ ,+ #.% # " # , ¸
`VD _K»5'5f LV'IoKi#P1 2
W ,15'9P¡c VYa5ô&D&g
V'I5 ¡
'RaI'oc'V]|R^Vy'IV¡ R I^ Î ¥ ©?V '1¡ R'ZV'IoKiiI
¤
,B&
"
L1
F4 7
2 0 ²
2 ²
$&
L∞
!-';L ?@!-!8A-!!'
'TL 1I'-
$C
$&#'-- 9O -1 $2'
"# &
1
V'5^ Î
7^V1K
1VV^
oI'VV^
7^V1K
1VV^
oI'VV^
7^V1K
1VV^
oI'VV^
L1
¥K¤ h¬f ¤1¥ ° ¬f ¤²
¤1¥ °R¬f I
¥ ¬f ° ¥ ² ¬f ¤
¥ ¬f I
¥ ¬f ¥ °R¬f ¤²
¥ ¬f ¤
5' 1
¤I¥ 1²hu¬f 1²
° ¥ ¬f ¤
° ¥^¤I¤ ¬f °
¥ &² ¬f 1²
° ¥ ° ¬f ¤
° ¥ ¬f °
° ¥ ¬f 1²
° ¥ ¬f ¤
¥ °&¬f I
L∞
¥:² h¬f ¥^¤I¤ ¬f ¤²
¥^¤ ¬f I
¤I¥ I ¬f 1²
¥ ¬f ¤
¤I¥ ¬f °
° ¥:² ¬f 1²
° ¥ ¬f ¤
V¥ 1² ¬f°
5' 1
¥ 1hu
² ¬f ²
V¥ ² ¬f ¤
V¥ ¬f °
¤I¥ I ¬f
¥ ¬f ¤
¥ °&¬f °
¤I¥ &² ¬f
° ¥^¤ ¬f ¤
¤I¥^¤ °&¬f ¤
DI'7D5I"'70I FPIF &o'
'K D5&$&g( V5 K D5I#§oR`5Io'V5¡ I |YD5^¡
o7i _ ao| D5I'V|fP1a o&^^R ?R'5boR'K1o ¥(2 IJ5¨V&iVÎ1 V¡
oWIg5·o7(1CM cI _^{¯:° ± ¥ 5'I.1VFoV 1 |^F ]p&^'IK( V7
'IV¡MR'0 I^'K1o?
¨VI
^¡ ¥
©?o,gI7 1ªPVg< Î]D?IÎZ¡º
(
9 8 ≤ x ≤ 12,
£e²¥ ¦
0.2 − 0.05(x − 10)2
b(x) =
& o5'Y^ I
0
PIcoIVV5fIa^5o& ° ² m ¥ ©?V'5p'1¡¸7'& " V
'9'^&L10 &o'
'9'^&
§oRdYK ]IYYK oIi7 1¡M V pYK^,K17 ^1R' ¥
¦ ºL©G'Io'
'9'^&§oRdYK oIi V ¥ ©?V,^VK D&5IoK Î]D?IÎZ¡º
£e²¥ ¦
1.532
+ 9.812 × 0.66,
m = 1.53,
E=
2 × 0.662
I1
'V5YYK [ o,g1VoVI ¡O5IoK Î
Vo7 'I]ºL©?V,^'oI 1 hu ¤1¥ ² m2/s ^Yîg7 ¥
•

]

O
º
?
©
V

I h V¥ m D_îg7¶YV5® VZ§oR«D
•
V5 K D5I ¥
°
L1
"#&-10,B&
"
$&
!--'ILO ?@!-!2A(!'-'ILMO-1P'-
$C
$&0'
-09O 1 $ '-<
L∞
4 7
V'5^ K1
7^VI^
1VV^
oI'VV^
7^VI^
1VV^
oI'VV^
7^VI^
1VV^
oI'VV^
2 0 ²
2 ²
L1
5' 1
° ¥ ° h¬f ¥ ¥ ² ¬f ¤² ²¥ ¤1¥ ¬f I ¤I¥
¥ ² ¬f ° ¥ ¤1¥ ¬f ¤ ²¥ ° ¥ ¬f ° ¥
²V¥ ¬f ¥:²
¤1¥ ¤ ¬f ¤² V ¥
° ¥ ¬f ¤ I¤ ¥ L∞
¤ ¬f ¬f ¤
¬f °
¤ ¬f 1²
¬f ¤
¬f °
¬f ¬f ¤ °&¬f ²
hu
5' 1
¤I¥ ¤ h¬f ¥:²I² ¬f ¤W²
¥ °&¬f 1
¤I¥ ¤ ¬f ²
¥ °&¬f ¤
²¥ 1² ¬f °
¥:² ¬f ¥:²I² ¬f ¤W²
V¥ ¬f*°
¬f ²
° ¥ hu
¥ ¬f ¤
²¥ ¬f °
¤I¥:² ¬f
° ¥ ¬f ¤
¥ ¬f ¤
¤I¥^¤ ¬f ° ¥ ¬f ¤W²
V¥^¤ ¬f
©?o^F'IV¡Z7'R'c7oIVD[gc5¨VI<'K¡ZV' 5'I ¥ c5IiV', o_71KV Î]V ^
o Ko
VVKPI'i ]g1K ¥ ©?Vc
1|o 7V 4I50KI5
t = 20
Io· oOgI7'NI =b 200
I poKI7 ·K 2 ÎV'M° ¥ *¶I50' 5i1o''& i'oRhc+'Vb
7 I¡p7'& DbKoV5OiIK&^VpoO 1VoA&»[5' 11bFo F KV1KV
D7Î
Io¶1VV^pV
D7Î·'g7P1 VM5IiVR'K1#Io¶ VR 'V L1 &o L∞
'I_PI| V[?R iV5Î h &o VD g&'I hu £ VI Iº[V9'V5 h V1 hu
^{ VDO517[D|y5Io7'I1iIiI^¡VIiD&aPVo
Î ¦ ^ ©#&oK P1iV9»5'5
o
D7Îo ¥ M5I·5K&'K¡( 5i'oR,'V L1 &o L∞ 5' 1'0I pR,'VO^515f&
'IooA&» 5''ILP1`9»5'5`V'5^ ÎoI5'KP¡^V| o,bKKg&D&o5Oo 1g7¡ ¥
2
L1
$&

L∞
(-'TLM3 ?@!(8A!('-'3LM-18
-$&'
! ($&% 9
o
D7Î
7^V1K
1VV^
L1
° ^¥ ¤ h¬f ¤I¥^¤² ¬f ¤ 5' 1
¥ huf¬
¥ ° ¤ ¬f ¤ L∞
I¤ ¥ ¤ h¬f ²¥:²I² ¬f ¤ 9O 18$2'
1 B
'I
¤I¥^¤ hu¬L ¤I¥ I ¬L ¤W²
1.2
surface level h+b
bottom b
1.1
1
surface level h+b
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
5
10
15
20
25
x
18'-L$&! !
h+b
$&D-1O,B&--"
b
L 1O-
$&'
!- -($& 9
9O 18$ '
1 B
¦ ºJ©G'Io'
'9'^&§oR YK i V ¥ ©?V0^VK D&"5IoK Î[DYIÎZ¡º
£e²¥:²I¦
E=



3
2 (9.812 ×
2
2
0.18) 3 ) + 9.812 × 0.2
0.18
+ 9.812 × 0.33
2 × 0.332
9 x ≤ 11.665504281554291,
& o5'Y^ I
m = 0.18,
I1
'V5YYK [ o,g1VoVI ¡O5IoK Î
Vo7 'I]ºL©?V,^'oI 1 hu o¥K¤ m2/s ^Yîg7 ¥
•

]

$
º
?
©
o

h V¥ I m ^`Ki1 ¥
•
©?V^|7 1¡¶R'Z VIVDyZ5¨1<'K¡yV' 5'I ¥ |aMi5 ^Io¶^ Vg7<
^I ¥ bb]IoK¡®V^'
o'i'V 517[YV5 'V 7o ®^p5¨VI<'K¡¶^5& @&|'V

5^$g1VoVI ¡ ¥ 5g
|b|7VK¢0 Vi
1iV'& ÎoI#VIp&^ [V0'V| V R
V|
5^#g1VoVI ¡ ¥ 2 I VDC5I cYV(7'R'K1o&'¡Z V [
¨^7'bco5 Mg7
V Y b§o¨p'|5IiV''V,&VV'W¨Kp& `KiIVMo 1VK]IoO'5o^1
`'V
Kî9'5|V 5V'M¡¶ 1V
* ^¶^KiK 5PI oMa¸55^^V
¨ ¸ V[7V ¥
^ o'ViK5¢c&o( Î,&VV'W¨Kp& ]RIKVC&J& I RC Vp V [o70
5 ¨VI<V7|'oRY'V Y a§o¨[&]5I V'C VDY V O
¨VI
^¡ ¥ 0a,
IiVV
'Vi71K'K1V ^ t = 20 g7^V N = 400 VVKPI' |7I^' ¥ ©?V|
1|o
V 4I5YKI5 h + b Io VY& 1 b &'?VKI7'i^ 2 ÎV' ¥ *iI5$'c5i1
7 R'C o&a'V,'1¡OR'CDaKg5ZpIK'IKoOoM'|'IooA&»5' 1aFo
^VI^MV'5^ Î &g IoVKZV'
D K1y (5 PI' V]
1|o'& Îf&g 7oR
'V L1 Io L∞ 5' 1'JPI?'V,?R YV5Î h IoZ VcD'o&'I hu ^©GIV^
P1cK» 10V'5^ K1o ¥ p&¸5K&'K¡ 5 gR0 V L1 Io L∞ 5''IFI |R
'V0KII"'IVoI»[ 'IfP1?V9»5'5?V
D7Îg5I KP¡KV_ V0a5^9AoIÎo

o 1g7¡ ¥
0.45
surface level h+b
bottom b
0.4
surface level h+b
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
5
10
15
20
25
x
12'- LF$&!2!
h+b
$&:-12,B&--"
b
L;-1 -$&'-( - $& %
(9
¦ º o5 K D5I.§oR ¥ ©?V0Ko9'Î"
1oK Î[^Y1K15[¡º
£²¥ ¦
E = 22.06605,
'I1
o5YYK [ V,IooV&'¡M5IoK Î
Vo7 'I]º$©?oc^'oI 1 hu ¥ °
•
m = 4.42,
m2 /s
^?îg7 ¥
9O 1:$D'
1B
²
2
L1
$&

L∞
(-' LM; ?@!-! A!!'
'8L;-1 -
$&'-!- -($& 9
o
D7Î
KV1K
VIVV^
L1
5' 1
° ¥ h¬f
¤1¥ f¬ ¤²
L∞
° ¥ huf¬
¤I¥ ° ¬f ¤²
¥ h¬f V¥ ¬f ¤
9O 10$/'-1 @
5' 1
° ¥:² hu¬f V¥ °&¬f ¤
RYo7 'I]º$©?oC?R'5YVK1 h ° m ^Yîg7 ¥
©?V^'IV¡Z7'R'c7oIVD]g_
¨VI
^¡OV' 5'I ¥
IiVV 0 o_71KV Î]V ^
o Ko N = 200 VVKPI'i I^1 ¥ ©?Vc
1iV ]7V 4I5,KI h + b
t = 20
&o¸ oOgI7'I b I poKI7 yK 2 ÎV' ¥ *¶I50'(|1o7 R |'oRc'V
'IV¡i7'& `DaKg5pp&^1&^VOVp 1VoA&»]5''I&bFo F KV1KFV'
D K1
&oy1VVÔV'
D K1(' g7P1 VZ
IiVV'R'K1$&oy VR´'V 1 &o ∞
5' 1'cPIi o[?R'5iVK11 h &g¶ o ^'oI 1 hu ^@©GIVK ² PL1O9»5'L5
V
D7Îg ¥ O5Iy5K&'K¡¸7'oR_ V 1 &o ∞ 5''ICI p&, Vp^515LI
1VoA&» 'ILP1`K» 1ò
D7ÎoIL KP¡KVL o,bKKg&D&o5Oo 1g7¡ ¥
•
V'
D K1
^VI^
IVoK
L1
$&
L∞
!-'ILM8 ?@(! IA!('-'8L;-1 '
,B!- -($& %
(9P
L1
5' 1
¥ h°R¬f ¤1¥ ¬f ¤ L∞
¥hu
° ¬L V¥ ¬L ¤ ¥ Vh¤ ¬f ¥ ¬f ¤ 'I
¬f ¤1¥ ¬f ¤²
¥ °
hu
$#"$# ~ , ~ gW#~Rp 0
0
+ 0

* ·'V^c
¨V&iVîbiY^K$'7, VpVÎV1'C155V'1
¡IbIoc ViFPIc
7i&'(71KV Î ¥ ©?V |&' Ii VRY 5¨VI<, I^ Îo M Vi o&^KR ?R'5
oR'K1¶Y9'®V1§gR_& 1 ^ 'VM^K 5R'V'If7g®I71|ZR Îg&'¡· &
K'K1ofV| V¡ &'OV&|IV5'Ô i5I cPIiI
VI5¡ ¥ MgWIO'V5'
P1
2.5
surface level h+b
bottom b
surface level h+b
2
1.5
1
0.5
0
0
5
10
15
20
25
x
12'-L$&! !
h+b
$&:-1 ,@&-"
b
LM;
1P'-,B!- -($& %
9 o1 5Z io 0 V0P1K^RYKo& IPVo
Î&o[^VK D&
1oK Îo
b(x) = sin2 (πx), h(x, 0) = 5 + ecos(2πx) , (hu)(x, 0) = sin(cos(2πx)), x ∈ [0, 1]
YK 5'KDMIooV&'¡¶
1oK ÎoJ 5·¯ ¤ ± ¥ Kg
Z V]
¨V1<| I^'K1®Do&
VRYd
¨V^^59'K¡ PIZ o^[5171`b¸o o V¢'dIOoVK ·I1Ko| ¬ 'V|Y9' N = 12, 800
KDY'Z5IiV'_O'
P o
c I^ Î"Io] '&` o^
'
P o
·71KV Îd1O'Vy
¨VI
[ I^ Î ^ 5IiV'Ko o·Vi D5I5''I ¥
M5IiV'pV· t = 0.1 YV¸'VO I^ Î·^_'K^J |I £ V ,515Î
DR'5,^· ^|iP1C VD,V'IV^5 ¦
¥ ©#&oK Io 5I'IK¸ o L1 5' 1'PI0'V
55^CW15&1|PI 2 2 ² Io Ìo@PIM V¸g1K[WIKoMPI
2 0² IoVi5'^&#I5ÌJI
VI5¡ ¥ i5&¸
^&'^¡] 5_ o&F'Vi K1V
1'V5?&G155V'1
¡iDYIV^51 ¥ 2 1b'V 'V5iI'V,©
5Io7'I1 M D
15(1 ° ¥ & D
_ o&F'V µ 2 1®o_5CboWIcg7]PIF V_oVK |I1KVi
'V|OV
'170Y9'y VM|7 ^³5O&g¸D,'5I¸K ©#&oK &o ¥ 2 I
'V i
o1'V µ 2 1 V_5?DbVÖ& o¥K¤ V¥ CV&''oRaV¢'1'V5
155V'1
¡F^"Io 5'I`P1 2 ¥ ©?ofV¢ V1' ¬ {
Ig' o
ÎCoI"g5
o7M^Z o1
Io? V, IV
5'ªD?&VV'W¨Kp& O¡ZcP1V7'Z1'V5?I55VR

¨ &IDR'K1 ¥ o
i'VM&oV W¨^pR ^I¸&a VO Io'5'5' ^· VMIK1I'9'V

I' ^V'?^'a i o,RI5&^5' 1 ¥ ©?VDYVo5V1|o|oI`g5]K17 ^1R'MK
¯° ± ¥
!-' $&0"#!($&J!-'I&L3$&!!-$& 0LMT
1 K$&"#A 5 (- 4 L1
¥ & µ 2 1

KD
V¥ °²
²
V¥ ¤
V¥
V¥
°
V¥ °
V¥^¤
2
`
1¤L¥
°¥
°¥
°¥
¤1¥
¥
1
µ 2 1
°²
°
¤
²
V¥ V¥ V¥
V¥
V¥ °
V¥^¤
L1
¤1¥
°¥
°¥
°¥
¤1¥
¥°

'Ih I
¬f °
¤ ¬f
° ¥ ¬f
¥ °
1²
¥ ¤
¬f
¥² ¬f ¥
¬f 2 ²
hu
5

L
¥ ¬f °
¤I¥ ¬f °
° ¥:² ¬f
° ¥^¤ ¬f
V¥ ¬f ° ¥ ² ¬f I5
'Ih I
¬f °
¬f
° ¥ 1
¬f
¥ o¤
1²
¥ ¤
¬f
¥² °R¬f ¬f ¥ 5' 1hu

¥ ² ¬f °
¤I¥ ¬f °
° ¥:² ¬f
° ¥^¤ ¬f
V¥ ¬f ° ¥ 1² ¬f I5
1
L1
°¥ ¤
° ¥I
¥
¥:² ¥I
° ¥° ° ¥I
¥
¥:² ¥ $# # 0 -- J ~ + /(0 , 0 ~ 0 G 0 ~ 0 ~
?© obP1K^RY^V`'7fI #&'ao1 5,'0V5iIo7 R'L'VY5&g&V^KK¡,&o V?V'&
g7|'V|$PIa
1|o'& Îof1| V`5 o o& Îi&,7 I¡iR'`71K'K1
YV^·5IVV&Cgp&'V a5^G¡ ZVI¸bKKAo&D&o5'V5i ¥ 2 IC'V|'&i
I IZ1?K <'K1 ²V¥K¤ b| Ì Y&'CV 1g7OP1`K» 1IK1I'9'Vp ¥
!-'2$&:"#(($&J!-' &L $&(!-$& 0LM;
12K$&"#A205!!- 4 L1
¥ & µ 2 1
`
V¥ V¥ V¥ V¥ V¥ V¥ °²
²
¤
°
µ 2 1
V¥ V¥ V¥
V¥
V¥ °
V¥^¤
°²
²
°
¤
2 0 ²
5''Ih I 5
¤I¥ ¬f °
° ¥^¤ ¬f
°¥ ¥ I ¬f
° ¥ ¤
¥ °
° ¥ ¬f 1²
¥ ¬f ¥ ¤
¥ ¬f ¥
5' 1hu

¬f o¤
² ¬f °
¬f
¬f
¬f ¬f I5
5''Ih I 5 L1 5' 1hu

° ¥ ² ¬f
° ¥^¤ °&¬f °
¤I¥^¤² ¬f
¥ I¤ ¥ V¤ ¬f
¤I¥ ° ¬f 1²
¥°
I¤ ¥ ¬f
¤I¥ °&¬f :¥ ²
V¥ ¬f ¤I¥^¤I¤ ¬f ^¥ ¤
¥ ¬f ¤I¥ ¬f ¥
¤I^¥ ¤ ¬f I5
L1
L1
¤I¥
I¤ ¥
° ¥
°¥
V¥
°¥
°¥ ²
° ¥ ¥
¥ ¤
¥
L1
¥
¥° ¤
¥
^¥ ¤
¥ # #.% # F~ + / 0 , 0 -.0 g~ 0 R~ ~ )$+(,.-.,./ ,+ ©?o0PI^KRY^Vi1gI 9*R'K1o&'¡| 7517C?I?V'I1 M¡ 1 fVi¯ ¤ ± ¥ A`?I
o1 5p'_5i1o''& VC5&g&V^KK¡|&. oFV'I1 p'V|`PI?
IiVV'R'K1o
1 ]'IVDK¡¸R&'¡^V[§oR RIc7i&'¶g.#Io· oOg7'V'oR'K1·I`R
'K1o&'¡M7'& ¥ 0 `KÌ 2 ² 2 C² Io i
'VV ¥
©?o,gI7 1« 1g1I&V¡O5Io ^7'?IG1V0o|"º
£²¥ &¦
b(x) =
(
0.25(cos(10π(x − 1.5)) + 1)
0
9 1.4 ≤ x ≤ 1.6,
I V YD71
1.4
1.2
initial surface level
bottom b
1.2
initial surface level
bottom b
1
surface level h+b
surface level h+b
1
0.8
0.6
0.8
0.6
0.4
0.4
0.2
0.2
0
0
0.5
1
1.5
0
2
0
0.5
1
x
1.5
2
x
1> -.$&;'
L$&(>!
$&-1:,@&-"
LD$ ' " &$ TAB!-,%$C-
h+b
b
'
($& <'7$C-P9;$C
!! L )$D, A'-
1 )3$D'
" $&@A'-
&L $
©?o,KVK D&G
Ig9'K1oY&'CIÎ5[YK
9 1.1 ≤ x ≤ 1.2,
& o5'Y^ I
YV5' D_ VIA³5 ]5 V'o& Î
1o7'& ¥ ©Gb I ,gWIp55® V"º V¥ ° £ VKyVo^ ¦ &g o¥ V¤·£ iIKYoV^ ¦
¥ ©?o5I'
'^&^K¡1fPIp p&^ L'V D
^7 o oIo
` V1V^p V^9 aK',b,?WI5IV 1o&R'KV0^
¢bIoi'K11aRL'VoI7
I<'5'^7 D( 5o ±√gh ¥ 3 &¡@Vi5'^&C|5 VV[oW1(K¹O
VK¡ Y9' 'V
5&D
o^& ÎobK1I^^V|7o[7pIK5 V'oR'K1oa&'V0a& ? V 4I5_¯ ¤ ± ¥ &
75'fI^VK D&
Ig9'K1oLI ` VRYiK 2 ÎV' ²¥ ©?V I^'K1i&L î t V¥ ° s P1
ViV^ZoV^ o¥ °V.I&^V(I¸° 55^$VVKPI'SI'^(Y9'· ^|o K| &g
|D'7ÎÌooV&' ¡i
Ig9'K1oIop
1|g&'iYK M

5^71K'K1^b7VRY
K 2 ÎV' P1c'V 2 0² $K 2 Îo PIc'V 2 ² IoyK 2 Îo PIc'V
¥ ©?o, 7VK'YPIY'V_ iIK"VVD7
V¥ V¤ &'c VRY]K 2 K1V' ¤
&o ¤1¤I¥ J o^L ^|1 VCRYo7 '&| WI K^V?R'5LoV^ gIJIK'1¡co1
VòVi ¥ &p
Î ^¡_ 5?'oRL'V5'`&'YVc VV'K1o$Vi5'^&V 5K^DR Îg ¥
£e²¥ 1¦
(hu)(x, 0) = 0
&o
h(x, 0) =
(
1 − b(x) + 1 − b(x)
0 ~1[~ )$+ ,-.,/ ,.+ $# #"$# F~ + / 0 , ,. ¸
*[ Vo
Î ²V¥K¤1¥ °VbCV' 5 p o C7 I¡O7'& ,71KV Îoa&o[ VRbi gR
IVbo| D5I'V|JDOpIK'IKO V«
¨VI
^¡ ¥ *O o^J Y51711aîg7
²
1.2
nx=3000
nx=200
1.15
nx=3000
nx=200
0.4
surface level h+b
1.1
0.2
hu
1.05
1
0
0.95
-0.2
0.9
0.85
0.8
-0.4
0
0.5
'
LF$&! !
1
1.5
2
0
0.5
x
1
1.5
2
x
4)85" $& AB! ,%$C-&L3$ -' -($& <'-
$C
P98$C-(O9O 1 $#,#A'
t s
-1()-12'-71%$& h+b
LM()
hu
1.2
nx=3000
nx=200
1.15
nx=3000
nx=200
0.4
surface level h+b
1.1
0.2
hu
1.05
1
0
0.95
-0.2
0.9
0.85
0.8
-0.4
0
0.5
'
LF$&! !
1
4C<
1.5
2
0
x
0.5
1
1.5
)5" $&AB! ,%$C-0&LJ$2'-- $& <'7$C- 98$C-!39O 1 $ ,PA'
t s
-1()-12'-71%$& h+b
2
x
hu
LM()
²¤
1.2
nx=3000
nx=200
nx=3000
nx=200
0.4
surface level h+b
surface level h+b
1.1
1
0.2
0
-0.2
0.9
-0.4
0.8
0
0.5
1
J!L ()'- LF$&! (
1.5
2
0
0.5
x
1
1.5
2
x
) 5" &$ A@! -,%$C- &LO$0'--($& <'-
$C
98$C-!P9O -1$0,:A'-
t s
-1 ) - 12'
1%$& h+b
hu
1.001
0.002
nx=3000
nx=200
nx=3000
nx=200
1.0006
surface level h+b
0.001
hu
1.0002
0
0.9998
-0.001
0.9994
0.999
0
'- LF$&!2!
0.5
1
1.5
-0.002
2
x
0
0.5
1
x
1.5
2
4)5" &$ A@! -,%$C->&L $ -' -($& <'-
$C
I98$C
!39O -1 $2'
" $&A'
J!L ()
t s
-1()-1 '
1%$& h+b
hu
²°
1.001
0.002
nx=3000
nx=200
nx=3000
nx=200
1.0006
surface level h+b
0.001
hu
1.0002
0
0.9998
-0.001
0.9994
0.999
0
0.5
1
x
1.5
-0.002
2
0
0.5
1
1.5
2
x
4C< )@5" &$ AB! ,%$C-6&L3$#'-- &$ <'7$C-D98$C-! 9O 1 $#'-" $&JA'-
t s
LM()3'- LF$&!2!
- 1()-1 '
1%$& h+b
hu
0.002
1.001
nx=3000
nx=200
nx=3000
nx=200
1.0006
surface level h+b
0.001
hu
1.0002
0
0.9998
-0.001
0.9994
0.999
0
0.5
LM()3'- LF$&!2!
1
1.5
-0.002
2
0
0.5
1
x
x
1.5
2
)35" &$ %AB!-,%$C
&L $/'--($& <'-
$C
9;$C
!;9O 1:$P'-" $&BA'
t s
-1()-1 '
1%$& h+b
hu
²
c'V5ª| iIK5 V'oR'K1 0.01 1p oFVK1a^p oF^ 5'R&#¯ ²V¥ I² ¥ ° ² ±&o
V Yo
o5,'V 2 ªi
oy&'V ,K_bK ¥ O'5p& gR 2 ²
2 0² Io &'YV&bbKVg&D&o5|PIL V77 I¡|R'JIo| o5¡i&^V4&^
p5I V'C VDY5 V'o& Î[I]
1I' C|7o ¥
©?o5I'
'^&^K¡1Y o^D7 V'o&g
¸ V1V^ 7oKK[^ ®b aW1YV'IoI1R'Ko
'V(^
¢]&o ÎM' g<'K15^¡ ¥ 3 I¡{Vi5'D5&Fi
ooMoW1 K¹p5VK¡
Y9' o|&D
VDR'K1o`ÎI^KoM7g¸ iIK#5 V'oR'K1oF&f o?R'507V 4I5 ¥
©?V· I^'K1dIV'&^V I ° 55^,VVKPI' I'^ Y9' KiV^¸ &o i^'7Î
g1VoVI ¡¶
1oK ÎoL
1io&'yYK { V[' V9_g7^V·° VVKPI'
5^D5LD
7VRY[^ 2 Îo ¤ °cP1Y o0 &o'
'9'^&§oR YK V1p7o V^ 2 K1V ¤ PI
VM &o'
'9'^&J§oR Y9'{ V ·Io¶^ 2 K1V' ¤ PIc'V]7V
'K D5&J§oR ¥
©?Vc'IVoKVi'Ki T D`
F1 1.5 P1Y o,oIo[ V^M§gR, 3 PI?'V_ 5Io
§oR ¥ c o^0'Ki1"'VORYo7 'I WIK^VZa& 5,oV^ po1,&^ I¡o1
VCVVi ¥ F&Z5K&'K¡i 5`'oRa o5'F&'o VV'K1oLVi5'D5&1'
^^^& Îo
&oM V,' I^'K1MP1? V,V'Ig&1& O7pIK5 o o& ÎZ^Y15'¡pI ¥
~ 0 / R ~ 0 ,. # R / - ~ ~1 0 R ~ 3
0 (+ -.0 /(+ $# # (
* ® VD|
¨V&iV^Ma]7^_VDR'O'V[o& V' ^V o 1VK RI5| '
'&oI
ÎCoVi"YVD·KI1K10Z'IV^VK¡ RI ¡^VM§gR RI0[D 5I ^VIgF& 1
II1'IV¡ ¥ ©?VD?
¨V&iV^0a1?o7[K·¯ ± ¥
©?o,gI7 1ª'III&o¡| Ia V0P1 ]º
(
K |x − 750| ≤ 1500/8,
£e²¥ ¦
8
b(x) =
I V YD71
0
PI x ∈ [0, 1500] ¥ ©?V,^VK D&"5IoV9'K1oY&'
(
K x ≤ 750,
£e²¥^¤ ¦
I

20 − b(x)
(hu)(x, 0) = 0
h(x, 0) =
I V YD7 ¥
15 − b(x)
©?o_Vi5'^&' VK'I&^V]¡Z V 2 YK
Vo9P1
KD £ &o
Z5Iio&'D71Y9' o|' VK'Fo KV
oV9P1 S55^^ ¦ &' VRY ^ 2 ÎV'
¤² &o ¤ YK ·bK»5'5c5oVKV]'Ki t ¤W² s &o t s ¥ 2 ÎV' ¤ &o
¤ 2 K1V' ¤ &g ° 2 ÎV'_° ¤ &g °I° iIo7 R'p VZVi5'^&L' VK'
¡ 2 C² 2 ² ' g< ÎK¡ ¥ *[ o^
¨V&iVÎ'Vc?R'5YVK11 h(x)
²
1.2
1.1
1
0.409
0.408
V2
0.407
0.8
0.406
0.405
0.7
0.404
12
13
14
15
V1
0.6
1.025
0.5
1.02
V2
surface level, bottom
0.9
0.4
1.015
0.3
1.01
1
2
3
4
5
6
V1
nx=2000
nx=200
bottom
0.2
0.1
0
0
5
10
15
20
25
x
G </)B5" $&@AB!-,%$C
&L*-1 -
$&'-!- -($& 9
9O 1I$D'
1 B
DpD 5I ^VIg|&i V]I^1|¨ ² ° ¥ ² &o®¨ I ¥ ² $YVK^[ V V 4I5Z^515
D|7i&'¸'V5' ¥ Ka o5i_b1
¸bKJPI,'V^
¨VI|oK1G1K^V
h(x) + b(x)
a5^`' IÎ.JV1A1'
^KDR'I'¡ 71K'K1oio Ko
55^^pYo^ &1 ZYK @'V
5II 1M 7VK'?g7^V

5^D ¥
# # _ , ®~ , 0- ~ 0 - ~
©?oc7oIK^R ?R'5Y7¡7 ^Map7o1
,î5o K1oa' Ia V0P1 ]º
£²¥^¤I¤W¦

ht + (hu)
0


x + (hv)y = 

1

(hu)t + hu2 + gh2
+ (huv)y = −ghbx
2
x


1 2

2

 (hv)t + (huv)x + hv + gh
= −ghby
2
y
²I²
0.5
nx=2000
nx=200
bottom
0.425
0.3
0.42
V2
0.415
0.338
0.336
0.2
0.41
0
0.5
1
1.5
2
2.5
0.334
3
V1
0.332
V2
surface level, bottom
0.4
0.33
0.328
0.326
0.1
0
12
12.5
13
13.5
14
14.5
15
V1
0
5
10
15
20
25
x
G </)5" $&@AB( --,$C
&L -12
-$&'
! ($& %
(9
9O 10$D'
1 B
YV5'`&&^ h Df'VY?R LVK11 (u, v) D$ oÌK
K¡cIg'VY§oVD. b V 7'
VbgI7 1d'II1'IV¡&o g ^ VbIW9R Îg&&
1o7'& ¥ AGD"7 &Î7P1 ?&
iIV5&^K³'V , o5i 2 2 ² 2 0² &o i1'IK[bKK
o&D&o5 'V|`P1'V|^ I_&C'7F1oK^Ko ^V Ky° 517 ¥ ©?o5'&^DCI
V51K< ,V ¥ *ca oR'Î1î5o K1o" V b^#&cIVVoVIo
b&o7 I¡0R'
g7D, VpD 1i&c cVK^^V ^V]GIo·9^,o·|1 pK¹O
VKc'^ KP¡
V ^ 'KV¶1V ¥ AZ^MK@V'Kg
^VK(g KoK]'
¨ o{ V oV^VOI
2 K
Î IV'&^da5^KoIÎo
o5iMPI]71i¸Ic V¸ 'V^¡
b|Kio7Îg&|R^V_?R'5?'1¡pR 5VVa'VCV 5V'D?7Îo9g& ^¡
|1 [
1|oKD5& ®&o¶a[YK^bo&p^'
g |7g
¨ o K1o_^ 'VDg&5 ¥ *
VD]7Vg7< Îdb¸IV^¡ 5Io D5[IVZIV·^|o K1o&CbKKAo&D&o5 'V5i
2 ªV K1Vy^¶V'5K1oc
Îo# 'KD&^K¡¸I5o5&^K³¸ a î5o K1o
²
3
nx=2000
nx=200
bottom
2
1.5
2.01
2.004
1
2.002
V2
2.003
2.005
V2
surface level, bottom
2.5
2.001
2
2
1.999
1.995
0.5
0
0
1
2
3
4
1.998
14
5
V1
0
5
14.5
15
15.5
16
16.5
17
17.5
18
V1
10
15
20
25
x
G </)5" $&@A@! -,%$C- &L 1P'-,B!- -($& %
9 ¡ g7^V[IV0a5^9Ao&D&g
¬ &Î1 K V ^¸&'·VK'
Îo ¥ |'
PC
¯:° ¤ ±YPI| V5'IKDi&îV^5i5'& Î ¥ [ I]¸o| D5I?
¨V&iV^ZYo^
D0paZî5o ÎoIG5 V'o& Î IJOIo|î5o ÎoIGiRKoM'1¡]a& 5
Io 5Iio&',Io o5icYK (p15V'IK^³& Î[& 2 ² Z° I IoYo^D
oIAbKKAo&D&o5OPI? o,|R^V|a& 5 ¥
F IÎ?'VF ¡'5 KO VF <'IVIoÎJIpIK [0, 25] × [0, 25] ¥ ©?VFgI7'I
'II1'IV¡p^?1K15[¡º
£²¥^¤ ° ¦
b(x, y) =
&'^50 o&a'V,&7'I
(
0.2 − 0.05(x − 10)2
0
K 8 ≤ x ≤ 12,
I V YD ¥
^YPVo
Î]& x IoK¡ ¥ 7 I¡ZR , I^ Î]5IZ
²
22
21
h+b
initial h+b
bottom b
20
16
19
surface level h+b
surface level h+b, bottom b
18
nx=4000
nx=400
20
14
12
10
8
18
17
16
6
4
15
2
0
0
250
500
750
1000
1250
14
1500
0
500
1000
x
1500
x
G </) 1#'-L$&!#(
LM2
1#$&" ,($ A -,!" $C -"#
(4 h+b
t
s
J!L ()I-1D "#! $&*'-
'- G
!('@A&--!9O -1 1 -.$& ( $& -1
,B&
" AB-$&A1 1() -1 "#!-($&J'--0'- G
$& G
-:!('(
22
20
h+b
initial h+b
bottom b
20
18
nx=4000
nx=400
19
16
surface level h+b
surface level h+b, bottom b
14
12
10
8
18
17
6
16
4
2
0
0
250
500
750
1000
1250
1500
15
0
500
1000
x
x
1500
G </) 1#'-L$&!#(
LM2
1#$&" ,($ A -,!" $C -"#
R
h+b
t s
J!L ()I-1D "#! $&*'-
'- G
!('@A&--!9O -1 1 -.$& ( $& -1
,B&
" AB-$&A1 1() -1 "#!-($&J'--0'- G
$& G
-:!('(
²
22
21
h+b
initial h+b
bottom b
20
16
19
surface level h+b
surface level h+b, bottom b
18
nx=5000
nx=500
20
14
12
10
8
18
17
16
6
4
15
2
0
0
250
500
750
1000
1250
14
1500
0
500
x
1000
1500
x
4) 1 '- LF$&! ! h + b LM3-1 %$&" ,($ DA-,(" $C;-"# t (4 s LM()
1 "#!($&@'--#'
/4
-#!!' A&
! 9O 1#-1 -.$&%! -0$& 1 ,@& -"
AB-$&A1 1()-12 "#!-($& '-
:'
4
$&04
-:!('(
22
20
h+b
initial h+b
bottom b
20
18
nx=5000
nx=500
19
16
surface level h+b
surface level h+b, bottom b
14
12
10
8
18
17
6
16
4
2
0
0
500
1000
15
1500
0
500
x
1000
x
1500
4) 1 '- LF$&! ! h + b LM3-1 %$&" ,($ DA-,(" $C;-"# t R s LM()
1 "#!($&@'--#'
/4
-#!!' A&
! 9O 1#-1 -.$&%! -0$& 1 ,@& -"
AB-$&A1 1()-12 "#!-($& '-
:'
4
$&04
-:!('(
²
22
21
h+b
initial h+b
bottom b
20
nx=4000
nx=400
20
16
19
surface level h+b
surface level h+b, bottom b
18
14
12
10
8
6
18
17
16
4
15
2
0
0
250
500
750
1000
1250
14
1500
x
0
500
1000
1500
x
4C< ) 1#'- LF$&! (
LM2
1#$&" ,($ A -,!" $C/-"#
(4 h+b
t
s
J!L ()I-1D "#! &$ *'-
'- G
!('@A&--!9O -1 1 -.$& ( $& -1
,B&
" AB-$&A1 1() -1 "#! -($&J'--0'- G
$& G
-:!('(
22
20
nx=4000
nx=400
h+b
initial h+b
bottom b
20
18
19
16
surface level h+b
surface level h+b, bottom b
14
12
10
8
18
17
6
16
4
2
0
0
250
500
750
x
1000
1250
1500
15
0
500
1000
x
1500
4C< ) 1#'- LF$&! (
LM2
1#$&" ,($ A -,!" $C/-"#
R
h+b
t s
J!L ()I-1D "#! &$ *'-
'- G
!('@A&--!9O -1 1 -.$& ( $& -1
,B&
" AB-$&A1 1() -1 "#! -($&J'--0'- G
$& G
-:!('(
22
21
h+b
initial h+b
bottom b
20
16
19
surface level h+b
surface level h+b, bottom b
18
nx=4000
nx=400
20
14
12
10
8
18
17
16
6
4
15
2
0
0
250
500
750
1000
1250
14
1500
0
500
1000
x
1500
x
) 1P'
LF$&!P( h + b LMI-1D$&" , $ :A-,(" $CI
"# t (4 s LM()I-1/"#!($&'-
'
G
!('@A&--!9O -1 1D -.$&*( $& -1
,@&-" AB-$&A1 1 ()-12 "#!-($& '--0'- G
$& G
0!!' 22
20
h+b
initial h+b
bottom b
20
18
nx=4000
nx=400
19
16
surface level h+b
surface level h+b, bottom b
14
12
10
8
18
17
6
16
4
2
0
0
250
500
750
1000
1250
1500
15
0
500
1000
x
x
1500
) 1P'
LF$&!P( h + b LMI-1D$&" , $ :A-,(" $CI
"# t R s LM()I-1/"#!($&'-
'
G
!('@A&--!9O -1 1D -.$&*( $& -1
,@&-" AB-$&A1 1 ()-12 "#!-($& '--0'- G
$& G
0!!' ¤
20
20
15
15
10
10
5
5
5
10
15
20
25
5
10
15
20
25
12! -' &L*
1P ? !!!/,B9;!! 1 1!1
$&0-1 .$& '--($&0'7$C-
h
4(LM3
1 A,!" >5 (-04 4P$CT-"#
L"# '-A%$&!( !
8!'3LM"
t = 0.5 < Q2- ( J!L ()*-!'
'T9O -1 $
LM" "#!'-1J 1 ()*-!'- 'T9O -1 $
100 × 100
200 × 200
L" "#!'-1J

IiVV ZP'I]º
£e²¥^¤ ¦
1 2
u + g(h + b) = 22.06605,
2
hu(x, y, 0) = 4.42,
hv(x, y, 0) = 0.
©?V aoR'05I' 7IocV
D7K¡0'C V?Io
*Kio7Îg&7V
'K D5&V7 I¡c7'&
& £e²¥ ¦ I&g, o?5 # <'K1IV oaoVg7'V'g I^ Îi5I_Y75_^ 2 Îo
¥ Fo"^V9'Î
Ig9'K1cD"1K15,¡,Ya^|o K1o&7pIKg7'V'oR'K1,& gR
'IV¡ R 1.YV5' h DCg7'V'g(V?&¡ V¥ 1² K¸ V|W¨ 6.5 ≤ x ≤ 7.5
¥ 2 K1V c° &gy° ^ VDW¡ oMD'V g&o5i1,KcK '1<0Y9'
12 ≤ y ≤ 13
V0ViIZaK» ?oV9P1 |7oaY9' 100 × 100
5^DaIo 200 × 200

5^D#PIJ5Iio&'D71 ¥ ©?V`K»5'5o5Yg5b5| VYo5Î h &o'VYKo9'Îo'1¡
R £e²¥^¤ ° ¦ ^V' 5 [&FK» ` î t = 0.5 Io t = 1 ¥ |&D7i'V] V
I|ZVi5'^&a iY9' 2 ² _¥ Ì Z o& 2 ² D|V&pa5^9AoIÎo
·PI
|R^VV^^KV'D ¥ ©?oM
1|g&'^ I·&? oOVi5'^&f' V9_I iV' 5 ·K
2 Îo C° ² Io¸° ¥ ©?V|' V9CKoV^R gR 2 &(' IÎc Vp
1iVK5¨
7p&^gPR'V'LI o`§oR 15'¡a5êYK V1? VV'ÎoLPR'V'bYV^ZIVg&
K[ o, 7VK'?1'IKoZY9' 2 ² _¥
°
20
20
15
15
10
10
5
5
5
10
15
20
25
5
10
15
20
25
12!
'2&L -1 ?@(!!/,B9;!! 1P1(1 h $&:-1 -.$&J'--($& '7$C-
F4(OLM8
1PA,!" 5!- 4 4#$C -"#
O LM-"# '-A%$&!! (-2!'OLM"
t=1 < 4 - LM()*-!'- 'T9O -1 $
LM" "#('-1J 1 () -!'- '9I -1>$
100 × 100
200 × 200
LM-" "#('-1J
Z
Z
Y
Y
X
X
0.02
0.02
0.01
0.01
Height h
Height h
0
0
25
-0.01
5
10
X
20
0
5
15
10
25
-0.01
20
0
15
10
Y
10
X
15
5
20
25
5
20
0
Y
15
25
0
12 = 2&L -1 ?@!-!! ,B9;!(>
121!1 h $&:-1 -.$&@'-- $& :'7$C-
F 4(OLMI-1/A-,(" 5 (-4 4>$CI
"#
9O 1 $
L" "#!'
1J L )
t = 0.5
200 × 200
G </ 1 ) !'
-'I,$&'
! -!'
'I,%$&'
!0
4C< I
Z
Z
Y
Y
X
X
0.02
0.02
0.01
0.01
Height h
Height h
0
0
25
-0.01
20
0
5
10
X
20
0
5
15
10
25
-0.01
15
10
Y
10
X
15
5
20
25
Y
15
5
20
0
25
0
12 = 2&L -12 ?@!-!!2,B9;!(:
1 1!1 $&>-12 -.$&B'-- $& :'7$C-
h
4( L2-1#A,!" 65!
4 4:$C "#
9O 1$
L" "#!'-1J
LM()
t = 1
200 × 200
G </ 1 )('- -'I,%$&'-!0
!'
-'O,%$&'
! 4C< 6. Conclusion
* VDcoIgcaO1W1iI¶RI ^5Í`71|p'55 ^¡¸V5IK1gyVÎAI5
bKKg&D&o5p'V5iIô5KoVKVPIo7'M&opV¢ ZI5a Vi ¥ ©?VF
¨V
K^5
71K'K1(Ib'Vp Vi0D,5iIg''& ¸¡([V_5,IaoIK^5V1KoZ5¨5
Ki5|PI| V oIK^R«a& |oR'K1o ¥ ©?V]V' 5'& Î{Io ^'
g K1®I
V[
Ig' o
ÎyV'Kg
^VK_ VIo^yo&V^p'VM'15,' ^|oK|_ o5 &o
515Î V´PV o5$PIb&p&VoKD5& ÎiRLoIo ¥ ©?V`
1o' g< Îg$b ?5K o5
oI {I@a5^9Ao&D&g
JVÎAIi155V'& 1JV1 5K^DR 1 ¡·oo9'(K» o

I5R 1'J1O& bKKAo&D&o5 &o@15
o'& ]1gIR'V [P1i V '51VD&p&o
7^VIoÎag& '?&# oc
5^KAW15&Ip' ^g&D ¥ ©?V0oo9'c9»5'5g
c
1o7 'o<'K1
iW¡(gM IiYoR_ KiV^5&o¸4I7 GYVD·aIo^·VDW¡(&yI5y1 R , 1K
K 5I'I7gI
,î5o Îo ¥ `'VcoIV'& V',&VV'11Z&&^ I¡Mo&oVK
|R^V|a& 5YVK^^V Di&oZDYKZ o^` 5g70iI'0I5o5& ¥
References
¯ ¤ ±
!" $ #%&$('$)*+,.-/0#213'145
6 7$$'
89:<; .* & 3 4 4 & . ! ..4 4& 4 . . 0& & /0& 3 ¥
¥ µ 1|o ¥ ° ² £ ° ¦
° 1² ° ²¥
¯ °R± 6 '
/- * # '
1)'
1 6* 8<'
1# $):;
& # . 4 . & & . .
-( 3 4*# 3 ¥
¥ µ 1|o ¥ ° £ ° ° ¦ ²1² o¥
¯ ± 9'-/&' ' '
1 ): / 4. $ & / / 4
- 4 & 4 ( ' 444
¥ µ 1|o ¥ J¡ ¥¤ Z£ ° 1¦ 1² ² ° ¥
¯ ±6' '
-/- ' $ ! 00 ' 7: 54
) 3)! * . .
3 4 &
3 ¥ `V ¥ o& ¥ £ ° 1¦ ¤ V¤ °W ¤ ¥
¯ ² ±!6' ' $ ! ' 7: 4 .4 4 #" $ *4 4 *.4 . 4 . R 3 & ¥ 3 ¥
3
' ¥ o ¥ 5 ¥"£ 3 V¦C¤ Z£ ° &¦ ° 1²1² ° ¤I¤ ¥
¯ ±!!9#211$'%$&$(' (' 0* ) '
1+* ,.-
#21 :"; & 3 4 .4 &4 . # 0&-& !# & 0/ & 4 / / * ¥ ¥ 2 K ¥ fI ¥f¤£ ° ¦ K<
'IVD ¥
¯ ± 1 324 $1):
& ¯
¯
¯¤
4)( ! . 4 4 . K65 7 . .&
8 .4
© ¥ ¥1 I7'®&g c¥ 5IVô LV9'I5 1"<'V' & |K µ IiVR Îg&

^5o5c&oZ¬LV1Ko55'^Vo1I^Vi V'KoI5 ¤ °I° ¥
± 324 $1) 9' #21 '
1: ; ): =<?> 3 - #4
A@ 4 . 4 ). ! ( , . / . 4 #
& 7BB;BC/ * 5 ¥ µ IiV ¥ L¡ ¥ £¤ ¦ ¤1¤ ¥
± D 32?4 $1 '1E; ): =<D> 3 - 4
C@4*.! 4)( %, . ) . 4 . & 0B;BC/ &1 3 & ¥ µ 1| ¥g² ° £¤ ¦ ¤I¤ ²¥
± , 32?4 $1) ; ): 3 - 4 ). (
.24 # & < /D . + * ¥ µ IiV ¥
¦ ¤ °I° ¥
L¡ ¥¤ ¤i£¤
²
¯ ¤1¤ ±5 32?4 $1 '
1+; ): 4 ). ! (
. 4 4*. (& ¥
¥ µ 1iV ¥J¤ £ ° V¤W¦
¤ ° ¤1¥
¯ ¤ °W± '- ' - ) ' : , . 3$ & * / 4 4 ¥ 3 & ¥ LV' oV ¥ £¤ ²1¦ ² V¥
¯ ¤ ± ' 1 '1 4 1): *.4*# 4$ =< 4 . / 2 * ( .4 4& # & 5 ¥ µ IiV ¥
L¡ ¥.¤ ° £ ° V¤W¦ o° ¤ ° ¥
¯ ¤ ± . -/#24 3 . '
8 : . * * ( 4.#" . ! 3 `5 ¥ £ ° o¤¦ ¤I¤ ° ¥
¯ ¤W² ± . #2-7-/01):D!5 4 2 . 4 2 & . ! ) 3 R' ¥ 3 ¥
3
' ¥ V ¥ 5 ¥G£ 3 V¦C¤ ° £ ° ° ¦ ¤ ¤ ¥
¯ ¤ ± # '
10- '
1 ; ): 4/ ) . & 4 ¥ µ 1iV ¥ L¡ ¥.¤ ° £7¤ ¦ o° °W°I° o¥
¯ ¤ ±! ' % ) : 4. . 4 & . 5$ * * / /! 4 %/ o 1V i.V10K^ IK'.IVV^^ <
$
.4 . 5¬J5I^ µ ¬ A¬ *2 A Y Y Y5Vo
1V7 .£ 2 &g
¦ 3 I' ¤ o¥ WIKD&oK 1 7'º WYY?
I ¥ VV^i ¥ P 0^5'I¨ WoVV^^R ^Ig &W¡ ¥ K Y 1¨ ¥ i £ K 2 o ¦<¥
¯ ¤ ±*+ : > 3 4 *4 $ ! ) . 3)! ! / ! * & # [ ¥ µ 1|o ¥ J¡ ¥.¤ £7¤ 1¦ ! ²V¥
¯ ¤ ± . #2)0 &
'
1
.09$'1): / )/ 4 4 ¥ µ 1iV ¥ L¡ ¥.¤I¤W²M£7¤ ¦ o° ° ¤ ° ¥
¯ ° ± * 0-7-/ * '
1&2 ' &0& 9& '
1
*+3*'/ # - : 54
, 4 . 4*4 4- . 0& & /0& ¥ µ 1|o ¥ L¡ ¥ ° ¤ £ ° ¦ ¥
¯ ° ¤ ± D* 0-/-7
?' !5#210-'
1
; ):=5 & 3 4 < 4 1. #0& & !. & 2! ( & ¥
µ IiV ¥ L ¡ ¥ °1° £ ° I¦ V° ² V¥
!
¯:°I°R± ' 7: .4 ! . / 4 . 4 ;
* ..4 & 3 ¥ V ¥ oI ¥ £ ° ¦ ° ¤I¥
¯:° ±*+ '$ $0- '
1
* 4 - '-7-: ; . 3/! . 4 .# $ . /0& ¥ µ IiV ¥ L¡ ¥J¤² y£¤ ¦
° ² ¥
¯:° ±
0# "'1 ; ): ; 4 . .. & ( 4 5 ¥ µ IiV ¥ L¡ ¥"¤ &²Z£ ° ° ¦ ¤ ¤ ° ¥
¯:° ² ± ; ,
): =<D> 3 . 4 . 4 # &
3 R' ¥ µ Ii ¥ £¤ I¦ ¤ ² ¤ ° ¤1¥
¯:° ± ; ): . ( . !4 " .&5 3 ¥

¥ R D7 ¥ µ 1|o ¥ £7¤ 1¦ ¤ ¤ ¥
¯:° ± ; ): ).
/ 4 3$ & ) / 4
+ - 4 . .4 4 # & 1^ ; 54 4
; . %
'5 ( .4 !. C¥ µ VV' µ ¥
I1Vo I µ ¥ ¥ &o ¬ ¥ ©#IV|1 £ ¬JK 1º ¥ Fg& 5'Io ¦ 1< o
`&'?^ 3 R'V5pR'^51I^Vi ¤ V'^VI ¤ ° ² ° ¥
¯:° ± ; '
1 & : 4/ / # ).
/ 4
+ - 4 4 1( 4 ¥ µ IiVV ¥ L¡ ¥ £¤ 1¦ ¤I¥
¯:° ± '? C901 1): B $! .. )1 *4 & $
4 !. 0& & * .& ( 4 3 & & * .- ¥ µ IiV ¥ L¡ ¥¤ [£¤ ¦ ² ° ¥
¯ ± 02 #% '
1 9 00 ' : 3$ 4 & 4* 4 . $ (* . #&
0& & !# ¥
µ IiV ¥ L¡ ¥.¤ £ ° ° ¦ ² 1 ° ¤I¥
¯ ¤ ±,' #210- '
1
; ):5 , 4 4 & . 4 4 # . 0& & .&5 ¥
µ IiV ¥ L¡ ¥ ° ]£ ° 1²1¦ V° °I° ¥
¯ °R±,' #21- '1 ; ): 5 & !
4 , . 54
4 . 4
.4 & . *4 5 ¥

¥ µ IiVV ¥ ° O£ ° ¦ ¥
¯ 1 ±,' 5
#210-"'
1 ; ):D5 & 3 4 , 4 3$ 4 ). ! ( ! . 4
.4 & *4 ¥ µ IiV ¥ L¡ ¥ ° ¤ £ ° ¦ ² ² V¥
¯ ±,' !5#210-'
1 ; ): ; & 4 & 3 4 , 4 $ 4). ( ! . 4
4 & . *42 µ Ii ¥ µ 1|o ¥ L¡ ¥L¤
£° ¦ ¤ ¤ ¥

Download Report

High-order well-balanced schemes

Paperzz.com

Your Paperzz