Math

Math 5315 / CSE 7365, Fall 2018 Due November 30

Homework 6 – Numerical ODEs

1. Prove that the Runge-Kutta method,

k1 = h f(tn, yn),

k2 = h f

( tn +

1

2 h, yn +

1

2 k1

) ,

k3 = h f

( tn +

3

4 h, yn +

3

4 k2

) ,

yn+1 = yn + 1

9 (2k1 + 3k2 + 4k3) ,

has local truncation error O ( h4 ) .

2. Consider the linear multistep method

yn+1 =

p∑ j=0

ajyn−j + h

p∑ j=0

bjf(tn−j, yn−j) (1)

for the problem y′(t) = f(t, y(t)), y(t0) = y0. (2)

Assuming that the method uses uniform time steps h and that y ∈ C(m+1)([t0, tf]), prove that it has local truncation error O

( hm+1

) if and only if

p∑ j=0

(−j)k aj + k p∑

j=0

(−j)k−1 bj = 1, k = 0, 1, . . . , m.

Hint: replace each f with an appropriate y′, and Taylor expand everyone around tn.