MATH 572 Numerical Methods for Scientific Computing II Winter 2005
hw#1 due: Tuesday, January 25
0. Write a brief description of your scientific interests and/or reasons for taking this course.
If you work in a lab or research group, please give your supervisor’s name and describe
briefly the project you’re working on. (1 paragraph is fine)
1. Consider the problem y0=ay +b, y(0) = y0, where a, b are constants and a6= 0. Find
the exact solution y(t). Let unbe the numerical solution at time t=nh given by Euler’s
method. Find an expression for unand show that limn→∞ un=y(t).
2. Consider the problem y0=−y2, y(0) = 1.
a) Compute y(1) using Euler’s method with stepsize h= 0.1,0.05,0.025,0.0125. Present
the results in a table with the following information.
column 1: h
column 2: un
column 3: un−yn
column 4: (un−yn)/h
b) Find the principal error function E(t) and evaluate E(1). Compare with column 4.
c) Perform Richardson extrapolation on the values in column 2.
3. The numerical solution of the problem y0=f(y), y(0) = y0by Euler’s method has
an asymptotic expansion of the form un=yn+hEn+h2Dn+O(h3) as h→0, where
En=E(tn), Dn=D(tn) for certain functions E(t), D(t).
a) The equation for E(t) was given in class. Find the equation for D(t).
b) Consider the problem y0=y, y0= 1. In class we showed that E(t) = −t
2et. Find D(t).
c) Compute y(1) using Euler’s method with stepsize h= 0.1,0.05,0.025,0.0125. Present
the results in a table with the following data.
column 1: h
column 2: un
column 3: un−yn
column 4: (un−yn)/h
column 5: un−(yn+hEn)
column 6: (un−(yn+hEn))/h2
Evaluate E(1), D(1) and compare with the results in the table.
4. Let y0=f(y), where f(y) is bounded and satisfies a Lipschitz condition. In class it
was shown that an explicit 1-step method of the form un+1 =un+hF (un, h) converges if
F(u, h) satisfies two conditions: (a) |F(u, h)−f(u)| ≤ Ch, (b) |F(u, h)−F(v, h)| ≤
e
L|u−v|,
for 0 ≤h≤h0, where h0, C,
e
Lare positive constants. Find F(u, h) for the midpoint rule
and show that it satisfies these two conditions.
