Stability of ODEs Numerical Methods for PDEs Spring 2007Jim E. Jones

References: Numerical Analysis, Burden & FairesScientific Computing: An Introductory Survey, Heath

Stability of the ODEThe Continuous Problem

Ordinary Differential Equation: Initial Value Problem (IVP)Last lecture we saw that f satisfying a Lipschitz condition was enough to guarantee that the IVP is well-posed.

Lipschitz condition definitionA 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.

Well posed IVP definitionThe IVP

is a well-posed problem if:A unique solution y(t) exists, andThere exists constants e0 >0 and k > 0 such that for any e in (0,e0), whenever d(t) is continuous with |d(t)| < e for all t in [a,b], and when |d0| < e, the IVP

has a unique solution z(t) satisfying

Difference between solutions may still be largeThe IVP

The perturbed IVP

One can show that the difference in solutions is bounded

Difference between solutions may still be largeThe IVP

The perturbed IVP

One can show that the difference in solutions is boundedThe k, and thus the difference between solutions, may be large if L is and/or b >> a

ExampleThe IVP

The perturbed IVP

Is the IVP well-posed?Whats the difference between solutions, z(t)-y(t)?

ExampleThe IVP

The perturbed IVP

Is the IVP well-posed?Whats the difference between solutions, z(t)-y(t)?

To characterize the time growth (or decay) of initial perturbations, we need the concept of stability.

Stability definitionThe IVP

is stable if:A unique solution y(t) exists, andFor every e >0 there exists a d > 0 such that whenever 0 < d0 < d, the IVP

has a unique solution z(t) satisfying

Stability definitionThe IVP

is stable if:A unique solution y(t) exists, andFor every e >0 there exists a d > 0 such that whenever 0 < d0 < d, the IVP

has a unique solution z(t) satisfying

For a stable ODE the difference between the solutions is bounded for all time.

Absolute Stability definitionThe IVP

is absolutely stable if:A unique solution y(t) exists, andFor every d0 the IVP

has a unique solution z(t) satisfying

For an absolutely stable ODE the difference between the solutions goes to zero as t increases.

ExampleThe IVP

If l is real, what can we say about the stability, absolute stability?

If l is complex, what can we say about the stability, absolute stability?

ExampleIf l is real, what can we say about the stability, absolute stability?

l>0 unstablel0 unstableRe(l)

Systems of Ordinary Differential Equation: Initial Value Problem (IVP)Let y=(u,w)t

and in this case the rhs can be described by a matrix,

Stability of Systems of Ordinary Differential EquationIf we assume A is diagonalizable, so that its eigenvectors are independent, then the IVP

has solution components corresponding to each eigenvalue li

Stability of Systems of Ordinary Differential EquationIf we assume A is diagonalizable, so that its eigenvectors are independent, then the IVP

has solution components corresponding to each eigenvalue liRe(li) > 0 components grow exponentiallyRe(li) < 0 components decay exponentiallyRe(li)=0 oscillatory components

Stability of Systems of Ordinary Differential EquationIf we assume A is diagonalizable, so that its eigenvectors are independent, then the IVP

has solution components corresponding to each eigenvalue liUnstable if Re(li) > 0 for any eigenvalueAsymptotically stable if Re(li) < 0 for all eigenvalues Stable if Re(li) 0 for all eigenvalues

Stability of the numerical methodThe Discrete Problem

Ordinary Differential Equation: Initial Value Problem (IVP)

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)tt=at=b h

Local truncation error definitionThe difference method

has local truncation error

for each i=0,1,,n-1

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 definitionThe difference method

has local truncation error

for each i=0,1,,n-1

The local truncation error is error in single step, assuming the previous step is exact, scaled by the mesh size h.

Definition of consistency and convergenceA 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 convergenceA difference method is consistent with the differential equation if

A difference method is convergent (or accurate) with respect to the differential equation if

This is the error in a single step: the local errorThis is the total error : the global errorSee: http://www.cse.uiuc.edu/iem/ode/eulrmthd/

Numerical Stability definitionApply the numerical method to

to generate discrete solution w and apply the same method to the perturbedProblem to generate solution u

The method is stable if for every e there exists a K such that

whenever d < e.

Stability of Eulers method

Apply each method to the IVP Backward Euler

Forward Euler

Solutions generated by Eulers method

The quantity inside the parens is called the growth factor r. For the difference between the solution and the perturbed solution, we have

and the requirement for stability is that |r| 1 Backward Euler

Forward Euler

Stability analysis of forward Eulers methodFor complex l, stability requires that hl must lie inside the circle with radius 1 in the complex plane centered at -1.

If we consider real l for which the ODE is stable, l < 0, the stability requirement is Forward Euler

Stability analysis of backward Eulers methodIf we consider l for which the ODE is stable, Re(l) < 0, the stability of backward Euler is assured: the growth factor is less than 1 in magnitude. Backward Euler is unconditionally stable. Backward Euler

Examples from last timeThe example (2.a and 2.b) from last time illustrate the difference in stability of forward and backward Euler.

Ordinary Differential Equation: Example 2.aForward Euler with h=0.1Backward Euler with h=0.1Both computed solutions go to zero as t increases like the true ODE solution

Ordinary Differential Equation: Example 2.bForward Euler with h=1.1Backward Euler with h=1.1Backward Euler go to zero as t increases. Forward Euler blows up.

Examples from last timeThe 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.

Numerical Stability definitionA numerical method may be unstable, using the previous definition, becausethe underlying ODE itself is unstable. To focus specifically on the numericalmethod, 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.