Question

Consider two different initial value problems A and B on the interval [0, 2], both with initial data given at t = 0. Suppose the exact solutions to these problems are φA(t) = e^−10t and φB(t) = e^−100t . Suppose you apply Euler’s method to approximate these solutions and that your goal is to keep local truncation error for (tn, yn) under > 0 when tn is near 0. To achieve this would you expect to need a smaller step size for problem A or for problem B? Explain briefly

Answer 1

We are given

10f A(t)e
and
-T00 B(t) e

Clearly,

10h 10t A(th) e10+h)e

Therefore,

ΦΑ(-h)-ΦΑ) 10η 1 Ξe φ4)

while, similarly,

B(th)-B(t) 100h 1 = e Bt)

Clearly, for a given value of ,
-10h 100h e
. Therefore, the relative error in
4(t)
is greater than that in
B(t
and to compensate for this, it is required that problem A have smaller step size than problem B.