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Stability of ODEs Numerical Methods for PDEs Spring 2007 Jim E. Jones References: •Numerical Analysis, Burden & Faires •Scientific Computing: An Introductory Survey, Heath
Stability of ODEs Numerical Methods for PDEs
Spring 2007
Jim E. Jones
References: •Numerical Analysis, Burden & Faires•Scientific Computing: An Introductory Survey, Heath
Stability of the ODE
The Continuous Problem
Ordinary Differential Equation: Initial Value Problem (IVP)
)(
,),,()(
ay
btaytfty
Last lecture we saw that f satisfying a Lipschitz condition was enough to guarantee that the IVP is well-posed.
Lipschitz condition definition
A function f(t,y) satisfies a Lipschitz condition in the variable y on a set D in R2 if a constant L > 0 exists with
whenever (t,y0) and (t,y1) are in D. The constant L is called a Lipschitz constant for f.
|||),(),(| 0101 yyLytfytf
The IVP
is a well-posed problem if:• A unique solution y(t) exists, and• There exists constants 0 >0 and k > 0 such that for any in (0,0),
whenever (t) is continuous with |(t)| < for all t in [a,b], and when |0| < , the IVP
has a unique solution z(t) satisfying
Well posed IVP definition
)(
,),,()(
ay
btaytfty
0)(
,),(),()(
az
btatztftz
],[,|)()(| batktytz
The IVP
The perturbed IVP
One can show that the difference in solutions is bounded
Difference between solutions may still be large
)(
,),,()(
ay
btaytfty
0)(
,),(),()(
az
btatztftz
|)(|max1
|||)()(| ],[
)(
0)( t
L
eetytz bat
atLatL
The IVP
The perturbed IVP
One can show that the difference in solutions is bounded
Difference between solutions may still be large
)(
,),,()(
ay
btaytfty
0)(
,),(),()(
az
btatztftz
|)(|max1
|||)()(| ],[
)(
0)( t
L
eetytz bat
atLatL
The k, and thus the difference between solutions, may be large if L is and/or b >> a
The IVP
The perturbed IVP
Example
1)0(
,100,100)(
y
tyty
000001.1)0(
,100,100)(
z
tztz
•Is the IVP well-posed?•What’s the difference between solutions, z(t)-y(t)?
The IVP
The perturbed IVP
Example
1)0(
,100,100)(
y
tyty
000001.1)0(
,100,100)(
z
tztz
•Is the IVP well-posed?•What’s the difference between solutions, z(t)-y(t)?
To characterize the time growth (or decay) of initial perturbations, we need the concept of stability.
The IVP
is stable if:
• A unique solution y(t) exists, and
• For every >0 there exists a > 0 such that whenever 0 < 0 < , the IVP
has a unique solution z(t) satisfying
Stability definition
)(
,),,()(
ay
taytfty
0)(
,),,()(
az
taztftz
),[,|)()(| attytz
The IVP
is stable if:
• A unique solution y(t) exists, and
• For every >0 there exists a > 0 such that whenever < 0 < , the IVP
has a unique solution z(t) satisfying
Stability definition
)(
,),,()(
ay
taytfty
0)(
,),,()(
az
taztftz
),[,|)()(| attytz
For a stable ODE the difference between the solutions is bounded for all time.
The IVP
is absolutely stable if:
• A unique solution y(t) exists, and
• For every 0 the IVP
has a unique solution z(t) satisfying
Absolute Stability definition
)(
,),,()(
ay
taytfty
0)(
,),,()(
az
taztftz
0|)()(|lim tytzt
For an absolutely stable ODE the difference between the solutions goes to zero as t increases.
The IVP
Example
)(
,,)(
ay
tayty
•If is real, what can we say about the stability, absolute stability?
•If is complex, what can we say about the stability, absolute stability?
Example
•If is real, what can we say about the stability, absolute stability?
>0 unstable<0 absolutely stable
•If is complex, what can we say about the stability, absolute stability? Re()>0 unstable
Re()<0 absolutely stableRe()=0 oscillating solution, stable
Let y=(u,w)t
and in this case the rhs can be described by a matrix,
Systems of Ordinary Differential Equation: Initial Value Problem (IVP)
7)(
5)(
,,)(3)()(
)()(3)(
aw
au
btatwtutw
twtutu
)(
,),,()(
ay
btaytfty
)(
,),(31
13)()(
ay
btatytyAty
Stability of Systems of Ordinary Differential Equation
)(
,),()(
ay
btatyAty
If we assume A is diagonalizable, so that its eigenvectors are independent, then the IVP
has solution components corresponding to each eigenvalue i
Stability of Systems of Ordinary Differential Equation
)(
,),()(
ay
btatyAty
If we assume A is diagonalizable, so that its eigenvectors are independent, then the IVP
has solution components corresponding to each eigenvalue i
Re(i) > 0 components grow exponentiallyRe(i) < 0 components decay exponentiallyRe(i)=0 oscillatory components
If we assume A is diagonalizable, so that its eigenvectors are independent, then the IVP
has solution components corresponding to each eigenvalue i
Stability of Systems of Ordinary Differential Equation
)(
,),()(
ay
btatyAty
Unstable if Re(i) > 0 for any eigenvalueAsymptotically stable if Re(i) < 0 for all eigenvalues Stable if Re(i) 0 for all eigenvalues
Stability of the numerical method
The Discrete Problem
• Numerical Solution: rather than finding an analytic solution for y(t), we look for an approximate discrete solution wi (i=0,1,…,n) with wi approximating y(a+ih)
Ordinary Differential Equation: Initial Value Problem (IVP)
)(
,),,()(
ay
btaytfty
tt=a t=b
h
Local truncation error definition
The difference method
has local truncation error
for each i=0,1,…,n-1
)),(),(( 111
0
iiiiii wtfcwtfchww
w
))](,())(,([)()(
111
1 iiiiii
i tytfctytfch
tyty
The local truncation error is a measure of the degree to which the true IVP solution fails to satisfy the difference equation.
Local truncation error definition
The difference method
has local truncation error
for each i=0,1,…,n-1
)),(),(( 111
0
iiiiii wtfcwtfchww
w
))](,())(,([)()(
111
1 iiiiii
i tytfctytfch
tyty
The local truncation error is error in single step, assuming the previous step is exact, scaled by the mesh size h.
A difference method is consistent with the differential equation if
A difference method is convergent (or accurate) with respect to the differential equation if
Definition of consistency and convergence
.0|})(|{maxlim 00 hinih
.0|})(|{maxlim 00 iinih tyw
A difference method is consistent with the differential equation if
A difference method is convergent (or accurate) with respect to the differential equation if
Definition of consistency and convergence
.0|})(|{maxlim 00 hinih
.0|})(|{maxlim 00 iinih tyw
This is the error in a single step: the local error
This is the total error : the global error
See: http://www.cse.uiuc.edu/iem/ode/eulrmthd/
Apply the numerical method to
to generate discrete solution w and apply the same method to the perturbed
Problem to generate solution u
The method is stable if for every there exists a K such that
whenever < .
Numerical Stability definition
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,),,()(
ay
taytfty
)(
,),,()(
az
taztftz
iKwud iii ,||
Apply each method to the IVP
Backward Euler
Forward Euler
Stability of Euler’s method
),( 111 iiii wthfww ),(1 iiii wthfww
)(
,,)(
ay
tayty
The quantity inside the parens is called the growth factor . For the difference between the solution and the perturbed solution, we have
and the requirement for stability is that || 1
Backward Euler
Forward Euler
Solutions generated by Euler’s method
ii hw )1(
i
i hw
1
1
iid ||
For complex, stability requires that h must lie inside the circle with radius 1 in the complex plane centered at -1.
If we consider real for which the ODE is stable, < 0, the stability requirement is
Forward Euler
Stability analysis of forward Euler’s method
ii hw )1(
/2h
If we consider for which the ODE is stable, Re( < 0, the stability of backward Euler is assured: the growth factor is less than 1 in magnitude. Backward Euler is unconditionally stable.
Backward Euler
Stability analysis of backward Euler’s method
i
i hw
1
1
Examples from last time
• The example (2.a and 2.b) from last time illustrate the difference in stability of forward and backward Euler.
Ordinary Differential Equation: Example 2.a
1)0(
,1000,2)(
y
tyty
01
1
5
4
)21(2
ww
whhwww iiii
01
1111
6
5
)21(2
ww
whwhwww iiiii
Forward Euler with h=0.1
Backward Euler with h=0.1
Both computed solutions go to zero as t increases like the true ODE solution
tey 2
Ordinary Differential Equation: Example 2.b
1)0(
,1000,2)(
y
tyty
01
1
5
6
)21(2
ww
whhwww iiii
01
1111
16
5
)21(2
ww
whwhwww iiiii
Forward Euler with h=1.1
Backward Euler with h=1.1
Backward Euler go to zero as t increases. Forward Euler blows up.
Examples from last time
• The example (2.a and 2.b) from last time illustrate the difference in stability of forward and backward Euler.
• Another example at http://www.cse.uiuc.edu/iem/ode/stiff/– Here the step size for stability (h=0.02) is tighter than
one needs to control truncation error if one is not interested in resolving the fast decaying initial transient component of the solution.
A numerical method may be unstable, using the previous definition, because
the underlying ODE itself is unstable. To focus specifically on the numerical
method, we can alternatively define stability as:
A method is stable if the numerical solution at any arbitrary but fixed time t
remains bounded as h goes to zero.
Numerical Stability definition
