Stability of ODEs Numerical Methods for PDEs Spring 2007

1. Stability of ODEs Numerical Methods for PDEs Spring 2007 Jim E. Jones • References: • Numerical Analysis, Burden & Faires • Scientific Computing: An Introductory Survey, Heath

2. Stability of the ODE The Continuous Problem

3. 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.

4. 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.

5. Well posed IVP definition The IVP is a well-posed problem if: • A unique solution y(t) exists, and • There 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

6. Difference between solutions may still be large The IVP The perturbed IVP One can show that the difference in solutions is bounded

7. Difference between solutions may still be large The IVP The perturbed IVP One can show that the difference in solutions is bounded The k, and thus the difference between solutions, may be large if L is and/or b >> a

8. Example The IVP The perturbed IVP • Is the IVP well-posed? • What’s the difference between solutions, z(t)-y(t)?

9. Example The IVP The perturbed IVP • 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.

10. Stability definition The IVP is stable if: • A unique solution y(t) exists, and • For every e>0 there exists a d > 0 such that whenever 0 < d0 < d, the IVP has a unique solution z(t) satisfying

11. Stability definition The IVP is stable if: • A unique solution y(t) exists, and • For 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.

12. Absolute Stability definition The IVP is absolutely stable if: • A unique solution y(t) exists, and • For 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.

13. Example The 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?

14. Example • If l is real, what can we say about the stability, absolute stability? • l>0 unstable • l<0 absolutely stable • If l is complex, what can we say about the stability, absolute stability? • Re(l)>0 unstable • Re(l)<0 absolutely stable • Re(l)=0 oscillating solution, stable

15. 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,

16. Stability of Systems of Ordinary Differential Equation If we assume A is diagonalizable, so that its eigenvectors are independent, then the IVP has solution components corresponding to each eigenvalue li

17. Stability of Systems of Ordinary Differential Equation If we assume A is diagonalizable, so that its eigenvectors are independent, then the IVP has solution components corresponding to each eigenvalue li Re(li) > 0 components grow exponentially Re(li) < 0 components decay exponentially Re(li)=0 oscillatory components

18. Stability of Systems of Ordinary Differential Equation If we assume A is diagonalizable, so that its eigenvectors are independent, then the IVP has solution components corresponding to each eigenvalue li Unstable if Re(li) > 0 for any eigenvalue Asymptotically stable if Re(li) < 0 for all eigenvalues Stable if Re(li) 0 for all eigenvalues

19. Stability of the numerical method The Discrete Problem

20. 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) t t=b t=a h

21. Local truncation error definition The 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.

22. Local truncation error definition The 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.

23. Definition of consistency and convergence A difference method is consistent with the differential equation if A difference method is convergent (or accurate) with respect to the differential equation if

24. Definition of consistency and convergence A 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 error This is the total error : the global error See: http://www.cse.uiuc.edu/iem/ode/eulrmthd/

25. Numerical Stability definition 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 e there exists a K such that whenever d < e.

26. Stability of Euler’s method Forward Euler Backward Euler Apply each method to the IVP

27. Solutions generated by Euler’s method Forward Euler Backward Euler 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

28. Stability analysis of forward Euler’s method Forward Euler For 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

29. Stability analysis of backward Euler’s method Backward Euler If 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.

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

31. Ordinary Differential Equation: Example 2.a 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

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

33. 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.

34. Numerical Stability definition 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.