Math 151B Homework #3 – due Friday 1/30, in class
1. stability
Consider the intial value problem
y0=−y+1
t,1≤t≤100, y(1) = 1
a. Find the region of absolute stability for explicit Euler’s method.
Answer:
Explicit Euler’s method uses the discrete difference equation
yi+1 =yi+hf(ti, yi)
=yi+h(−yi+ 1/ti)
= (1 −h)yi+h/ti
If we introduce an error of δiinto the computation of yi, we get
yi+1 +δi+1 = (1 −h)(yi+δi) + h/ti
Subract the original equation from this and get
δi+1 = (1 −h)δi
As long as h≤2, Euler’s method is absolutely stable.
b. What do you expect to happen when h= 0.5?
Answer:
If h= 0.5, then (1 −h) is positive and less than 1. So the solution should be stable and
free from overshoot.
c. What do you expect to happen when h= 1.5?
Answer:
If h= 1.5, then (1 −h) is negative and greater than −1. So the solution will be absolutely
stable, but with overshoot problems.
d. What do you expect to happen when h= 2.1?
Answer:
If h > 2 then the solution will be unstable, and numerical errors will grow exponentially.
e. Confirm the above numerically/graphically.
Answer:
