M465 -- Homework 4

We consider the numerical solution of the initial value problem

$$\left\{ \begin{array}{l} y^{\prime }=f\left( t,y\right) \\ y\left( 0\right) =\alpha \neq 0 \end{array}\right.$$

via the leapfrog method:

$$\left\{ \begin{array}{l} w_{0}=\alpha \\ w_{1}=\alpha _{1} ... ...-w_{j-1}}{2h}=f\left( t_{j},w_{j}\right) } \end{array}\right.$$

1) Determine the local truncation error for the leapfrog method (see Definition 5.14, page 298).

For the remainder of this assignment, assume $f(t,y)=\lambda y$.

2) Discuss the accuracy of the leapfrog method by comparing with $e^{\lambda h}$. Assume that $w_{j}=\sigma ^{j}$, where $\sigma \in \Bbb{C}$. Assume also $\lambda \in \Bbb{C}$.

3) Analyze the stability of leapfrog method. Again assume that $w_{j}=\sigma ^{j}$, where $\sigma \in \Bbb{C}$ and assume $\lambda \in \Bbb{% C}$. In particular, discuss the stability of the method when

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a) $\ \lambda$ is real.

b) $\ \lambda$ is purely imaginary.

c) $\ \lambda$ is complex (i.e., with non-zero real and imaginary parts).

4) Write a computer program to implement the leapfrog method. Discuss the results of various runs of your program in the context of your stability analysis.

5) Repeat exercise 2) for the fourth order Runge-Kutta method (page 281).

6) Determine the stability diagram for the fourth order Runge-Kutta method. Do this numerically, not analytically. Again, assume that $w_{j}=\sigma ^{j}$, where $\sigma \in \Bbb{C}$ and assume $\lambda \in \Bbb{C}$.

This assignment is due on Wednesday, 7 April 1999, at 12:40 p.m.