Chapter 26

1. Chapter 26

2. Stiffness and Multistep MethodsChapter 26 • Two areas are covered: • Stiff ODEs will be described - ODEs that have both fast and slow components to their solution. • Implicit solution technique and multistep methods will be described.

3. Stiffness • A stiff system is the one involving rapidly changing components together with slowly changing ones. • Both individual and systems of ODEs can be stiff: • If y(0)=0, the analytical solution is developed as:

4. Figure 26.1

5. Insight into the step size required for stability of such a solution can be gained by examining the homogeneous part of the ODE: • The solution starts at y(0)=y0 and asymptotically approaches zero. is the solution.

6. If Euler’s method is used to solve the problem numerically: The stability of this formula depends on the step size h:

7. Thus, for transient part of the equation, the step size must be <2/1000=0.002 to maintain stability. • While this criterion maintains stability, an even smaller step size would be required to obtain an accurate solution. • Rather than using explicit approaches,implicit methods offer an alternative remedy. • An implicit form of Euler’s method can be developed by evaluating the derivative at a future time.

8. Backward or implicitEuler’s method • The approach is called unconditionally stable. Regardless of the step size:

9. Figure 26.2

10. Figure 26.3

11. Multistep Methods The Non-Self-Starting Heun Method/ • Huen method uses Euler’s method as a predictor and trapezoidal rule as a corrector. • Predictor is the weak link in the method because it has the greatest error, O(h2). • One way to improve Heun’s method is to develop a predictor that has a local error of O(h3).

12. Figure 26.4

13. Step-Size Control/ • Constant Step Size. • A value for h must be chosen prior to computation. • It must be small enough to yield a sufficiently small truncation error. • It should also be as large as possible to minimize run time cost and round-off error. • Variable Step Size. • If the corrector error is greater than some specified error, the step size is decreased. • A step size is chosen so that the convergence criterion of the corrector is satisfied in two iterations. • A more efficient strategy is to increase and decrease by doubling and halving the step size.

14. Figure 26.6

15. Integration Formulas/ Newton-Cotes Formulas. Open Formulas. Closed Formulas. fn(x) is an nth order interpolating polynomial.

16. Adams Formulas (Adams-Bashforth). Open Formulas. • The Adams formulas can be derived in a variety of ways. One way is to write a forward Taylor series expansion around xi. A second order open Adams formula: Closed Formulas. • A backward Taylor series around xi+1 can be written: Listed in Table 26.2

17. Figures 26.7

18. Figure 26.8

19. Figure 26.9

20. Higher-Order multistep Methods/ Milne’s Method. • Uses the three point Newton-Cotes open formula as a predictor and three point Newton-Cotes closed formula as a corrector. Fourth-Order Adams Method. • Based on the Adams integration formulas. Uses the fourth-order Adams-Bashforth formula as the predictor and fourth-order Adams-Moulton formula as the corrector.