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.