Stability - (5.10)
Example Consider a 2-step difference method: w i!1 " w i!1 ! h 2f t i , w i
y # " !2y ! 1, y!0" " 1 with
h " 0. 1, w 1 " 1 e !2h ! 1
2
2
. Use it to solve the IVP:
Note that the solution of the differential equation is: y " 12 e !2x ! 12 . So, w 1 " y!t 1 " " y!h". The graphs of
y!t" and #w i $ below show that this difference method is unstable because
y!x" " 1
lim w i " " but lim
x$"
i$"
2
6
5
-. solution y=e
4
-w
i+1
-2x
/2+0.5
=w i-1 +hf(t i ,w i )
3
2
1
0
-1
-2
-3
-4
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
One-step Difference-eqation Methods:
1. Consistent One-step Difference-equation Method:
Definition (5.18): A one-step difference-equation method with local truncation error ! i !h" at the ith step is
said to be consistent with the differential equation it approximates if
lim max |! i !h"| " 0.
h$0
1#i#N
Example One-step methods: Euler, Taylor Method of Order n, Runge-Kutta Methods of Order 2 and 4
are consistent.
The truncation errors of these methods are: ! i !h" " O!h k ", k $ 1.
2. Convergent One-step Difference-equation Method:
Definition (5.19): A one-step difference-equation method is said to be convergent with respect to the
differential equation it approximates if
lim max |w i ! y!t i "| " 0.
h$0
1#i#N
Example Euler Method is a convergent one-step difference-equation method.
We know that
|w i ! y!t i "| % Mh !e L!t i !a" ! 1"
2L
lim max |w i ! y!t i "| % lim Mh !e L!t i !a" ! 1" " 0
h$0 1#i#N
h$0
2L
1
3. Conditions on Stable, Consistent and Convergent Difference-equation Methods:
Theorem (5.20) Suppose that the IVP y # " f!t, y", a # t # b, y!a" " " is approximated by a one-step
difference method in the form:
w0 " "
w i!1 " w i ! h# t i , w i , h
Suppose also that there exists a h 0 & 0 and # t, w, h is continuous and satisfies a Lipschitz
condition in the variable w with Lipschitz constant on the set
D " !t, w, h" : a # t # b, ! " % w % ", 0 # h # h 0 .
Then
a. the method is stable;
b. the difference method is convergent if and only if it is consistent, which is equivalent to
# t, y, 0 " f!t, y", for all a # t # b;
c. if a function ! exists , the local truncation error ! i !h" satisfies |! i !h"| # !!h" whenever 0 # h # h 0 ,
then
!!h" L!t i !a"
e
|y!t i " ! w i | #
L
Example Show the Modified Euler Method is stable, cosistent and convergent if f!t, w" satisfies a Lipshitz
condition with a Lipschitz constant L.
Modified Euler Method:
w0 " "
w i!1 " w i ! h 1 f t i , w i ! 1 f t i ! h, w i ! hf t i , w i
2
2
#!t, w, h" " 1 f t, w ! 1 f t ! h, w ! hf t, w
2
2
a. Stability:
Check the Lipschitz conditionof #: Let !t, w 1 , h" and !t, w 2 , h" in D. Since f satisfies a Lipschitz
condition with the Lipschitz constant L & 0,
|f!t, w 1 " ! f!t, w 2 "| # L|w 1 ! w 2 | for !t, w 1 " and t, w 2 in D
Then
1 f t, w 1 ! 1 f t ! h, w 1 ! hf t, w 1
! 1 f t, w 2 ! 1 f t ! h, w 2 ! hf t, w
2
2
2
2
1
1
f t ! h, w 1 ! hf t, w 1
! f t ! h, w 2 ! hf t, w 2
" |f!t, w 1 " ! f!t, w 2 "| !
2
2
# 1 L|w 1 ! w 2 | ! 1 L w 1 ! hf t, w 1 ! w 2 ! hf t, w 2
2
2
1
" L|w 1 ! w 2 | ! 1 L w 1 ! w 2 ! h f t, w 1 ! f t, w 2
2
2
1
1
# L|w 1 ! w 2 | ! L |w 1 ! w 2 | ! h f t, w 1 ! f t, w 2
2
2
1
# L|w 1 ! w 2 | ! 1 L|w 1 ! w 2 | ! 1 hL 2 |w 1 ! w 2 |
2
2
2
1
" L ! hL 2 |w 1 ! w 2 |
2
For h 0 & 0, there exists a constant L 0 " L ! 12 h 0 L 2 such that
|#!t, w 1 , h" ! #!t, w 2 , h"| "
2
|#!t, w 1 , h" ! #!t, w 2 , h"| # L 0 |w 1 ! w 2 | for any !t, w 1 , h" and !t, w 2 , h" in D.
Therefore, # satisfies a lipschitz condition with a Lipshitz constant: L 0 " L ! 12 h 0 L 2 and the method is
stable.
b. We know that the local truncation error for the Modified Euler Method is
! i !h" " O!h 2 ".
Or
#!t, w, 0" " 1 f!t, w, 0" ! 1 f!t, w, 0" " f!t. w"
2
2
So,
lim max |! i !h"| # lim max |O!h 2 "| " 0
h$0 1#i#N
h$0 1#i#N
and the method is consistant and so is convergent.
1. Multi-step Method:
Consider a multistep method:
w 0 " " 0 , w 1 " " 1 , . . . , w m!1 " " m!1
w i!1 " a m!1 w i ! a m!2 w i!1 !. . . !a 0 w i!!m!1" ! hF!t i , h, w i!1 , . . . , w i!!m!1" "
The local truncation error is:
! i!1 !h" " 1 !y!t i!1 " ! a m!1 y!t i " ! a m!2 y!t i!1 " !. . . !a 0 y!t i!!m!1" "" ! F!t i , h, y!t i!1 ", . . . , y!t i!!m!1" ""
h
F!t i , h, w i , . . . , w i!!m!1" " satisfies a Lipschitz condition with repect to #w k $, in the sense that there exists
L&0
m
|F!t i , h, u i!1 , . . . , u i!!m!1" " ! F!t i , h, v i!1 , . . . , v i!!m!1" "| # L %|u i!!j!1" ! v i!!j!1" |
j"0
for sequences #u k $ and #v k $.
2. Convergent, and Consistent Multi-step Methods:
A multistep method is convergent if
lim max |w i ! y!t i "| " 0;
h$0
0#i#N
A multistep method is consistent if
lim |! i !h"| " 0 for all i " m, m ! 1, . . . , N and
h$0
lim |" i ! y!t i "| " 0 for all i " 1, 2, . . . , m ! 1.
h$0
3. Characteristic Polynomial of a Multistep method:
Let
P!$" " $ m ! a m!1 $ m!2 !. . . !a 1 $ ! a 0 .
P!$" is called the characteristic polynomial of the above multistep method. The stability of a multistep
method with respect to roundoff error is dictated by the magnitudes of the zeros of the characteristic
polynomial. This can be seen through a simple first-order IVP:
y # " 0, y!a" " " & 0.
The exact solution of this problem is: y " ". A multistep method generates w #i s as
w i!1 " a m!1 w i ! a m!2 w i!1 !. . . !a 0 w i!!m!1" .
Let $ be one of solutions of P!$" " 0. Then w n " $ n for each n is a solution to the above equation since
$ i!1 ! a m!1 $ i !. . . !a 0 $ i!!m!1" " $ i!!m!1" !$ m ! a m!1 $ m!1 !. . . !a 0 " " 0
3
If $ #i s are distinct, then the general solution of the equation P!$" " 0 is of the form
w n " c 1 $ 1 !. . . !c m $ m
linear combination of $ #i s.
4. Root Condition:
Let $ 1 , . . . , $ m denote the roots of the characteristic polynomial equation
P!$" " $ m ! a m!1 $ m!2 !. . . !a 1 $ ! a 0 " 0.
If |$ i | # 1 for each i " 1, 2, . . . , m, and all roots with absolute value 1 are simple roots, then the difference
method is said to satisfy the root condition.
5. Strongly and weakly stable and unstable:
a. Methods that satisfy the root condition and have $ " 1 as the only root of the characteristic equation of
magnitude one are called strongly stable.
b. Methods that satisfy the root condition and have more than one distinct root with magnitude one are
called weakly stable.
c. Methods that do not satisfy the root condition are called unstable.
6. Condition for Stable, Consistent and Convergent Multistep Methods:
Theorem (5.24) A multistep method is stable if and only it satisfies the root condition. Moreover, if the
difference method is consistent witht the differential equation, then the method is stable if and
only if it is convergent.
Example Consider Milne’s method:
w i!1 " w i!3 ! h 8 f t i , w i ! 4 f t i!1 , w i!1 ! 8 f t i!2 , w i!2
3
3
3
Determine if F satisfies a Lipschitz condition if f satisfies a Lipschitz condition. Determine if this
method is convergent, consistent and stable. Consider solving the IVP:
y # " !6y ! 6, y!0" " 2, 0 # t # 1.
Compare Milne’s method with Adams-Bashforth 4-step method
w i!1 " w i ! h 55 f t i , w i ! 59 f t i!1 , w i!1 ! 37 f t i!2 , w i!2 ! 9 f t i!3 , w i!3
24
24
24
24
a. Let F!t, h, w i , . . . , w i!2 " "
D"
8
3
f ti, wi !
4
3
f t i!1 , w i!1 !
8
3
f t i!2 , w i!2 . Consider
!t, h, w i , . . . , w i!2 "; a # t # b, ! " % w i % ", 0 # h # h 0
Let !t, h, u i , . . . , u i!2 " and !t, h, v i , . . . , v i!2 " be in D. Since f satisfies a Lipschitz condition, there is a
constant L & 0 such that for t i , u and t i , v in D " t, w ; a # t # b, ! " % w % "
f ti, u ! f ti, v
4
# L|u ! v|.
|F!t, h, u i , . . . , u i!2 " ! F!t, h, v i , . . . , v i!2 "|
" 8 f t i , u i ! 4 f t i!1 , u i!1 ! 8 f t i!2 , u i!2 ! 8 f t i , v i ! 4 f t i!1 , v i!1 ! 8 f t i!2 , v i!2
3
3
3
3
3
3
8
4
8
f ti, ui ! f ti, vi !
f t i!1 , u i!1 ! f t i!1 , v i!1 !
f t i!2 , u i!2 ! f t i!2 , v i!2
#
3
3
3
# 8 L|u i ! v i | ! 4 L|u i!1 ! v i!1 | ! 8 L|u i!2 ! v i!2 | # 8 L|u i ! v i | ! 8 L|u i!1 ! v i!1 | ! 8 L|u i!2 ! v i!2 |
3
3
3
3
3
3
8
" L!|u i ! v i | ! |u i!1 ! v i!1 | ! |u i!2 ! v i!2 |"
3
So, F!t, h, w i , . . . , w i!2 " satisfies a Lipschitz condition with a Lipschitz constant L 0 " 83 L.
b. Check if the method is consistent. The local trunction error of the method as we derived in Lecture
Notes 5.6:
! i!1 !h" " 14 y !5" !t 'i "h 4 , where t 'i is between t i!3 and t i!1 .
45
lim |! i !h"| " lim 14 y !5" !t 'i "h 4 " 0
h$0
h$0 45
'
!5"
assuming y !t i " is bounded. So the method is consistent if a convergent one-step method is used to
compute w 1 , . . . , w m!1 .
c. Check if the method is stable:
P!$" " $ 4 ! 1 " !$ ! 1"!$ ! 1"!$ ! i"!$ ! i" " 0, $ " '1, ' i, |$| " 1
The method satisfies the root condition so it is stable but it is weakly stable.
d. By Theorem 5.24, it is convergent.
e. Check the stability of Adams-Bashforth 4-step method:
P!$" " $ 4 ! $ 3 " $ 3 !$ ! 1" " 0, $ " 0, 0, 0, 1
This method satisfies the root condition and is strongly stable.
20
IVP: dy/dt=-6y+6, t in [0,1.4], y(0)=2
... true solution
15
-. by Adams-Bashforth 4-step method
10
- - by Milne 4-step method
5
0
-5
-10
0
0.2
0.4
0.6
0.8
Example Consider the difference method: w i!1 " w i!1 ! h 2f t i , w i
stable.
5
1
1.2
1.4
. Show that the method is not
P!$" " $ 2 ! 1 " 0, $ " '1
So, the method satisfies the root condition but it is weakly stable. Check the local truncation error:
! i!1 !h" " 1 !y!t i!1 " ! y!t i "" ! 2f t i , y!t i "
h
y ## !t 'i " 2
h ! y!t i " ! 2y # !t i "
" 1 y!t i " ! y # !t i "h !
2
h
y ## !t 'i " 2
h
" !y # !t i " !
2
lim max |! i!1 !h"| & 0
h$0
1#i#N
The method is not consistent. We cannot conclude from Theorem 5.24 that the method is convergent.
6