Math 269A: Assignment 3
Assigned Monday Oct 22, due Monday Oct 29
Theory
[1] The classic 4th order Runge-Kutta method is
11
211
312
413
1123
2
2
(22
6
n
n
n
n
nn
Yy
h
Yy f
h
Yy f
Yy hf
h
yy f f ff
−
−
−
−
−
=
=+
=+
=+
=+ +++
4
)
11
23 1/2 1
4
(, )
2
kkk
n
nn
n
ffTY
Tt
h
TTt t
Tt
−
−−
=
=
=
==+
=
Rewrite this in the form 111
(, ,
nn nn
)
y
yhtyh
−−−
=+Ψ ; i.e., find an expression for the
function Ψ.
Computation
[2](a) Implement the RK4 method to solve the problem
2
32
2( 1)
(0) 1
dy t t
dt y
y
1
+
+
=−
=−
from HW1. Plot the numerical and exact solutions on the interval [0,2] for a reasonable
choice of time steps.
(b) Compare to the exact solution found in HW1 and use the result to estimate the rate of
convergence for RK4 on this problem. Specifically, find the error between the exact
solution and numerical solution at t = 2.0 using a reasonable set of timesteps dt.
