The Computable Differential Equation Lecture ... - Bruce E. Shapiro

CHAPTER 4. IMPROVING ON EULER’S METHOD 87
The midpoint method is
y n = y n−1 + h n f
(
t n−1/2 , 1 )
2 [y n + y n−1 ]
(4.123)
The midpoint method is second-order and A-stable.
The modified Euler Method is
y n = y n−1 + h n
2 [f(t n−1, y n−1 ) + f(t n , y n−1 + hf(t n−1 , y n−1 ))] (4.124)
Heun’s Method is
y n = y n−1 + h n
4
[
f(t n−1 , y n−1 ) + 3f
(
t n−1 + 2 3 h, y n−1 + 2 )]
3 hf(t n−1, y n−1 )
(4.125)
Both Heun’s method and the modified Euler method are second order and are
examples of two-step Runge-Kutta methods, which we shall discuss later. It is
clearer to implement these in two “stages,” eg., for the modified Euler method,
while for Heun’s method,
ỹ n = y n−1 + hf(t n−1 , y n−1 ) (4.126)
y n = y n−1 + h n
2 [f(t n−1, y n−1 ) + f(t n , ỹ n )] (4.127)
ỹ n = y n−1 + 2 3 hf(t n−1, y n−1 ) (4.128)
y n = y n−1 + h [
(
n
f(t n−1 , y n−1 ) + 3f t n−1 + 2 )]
4
3 h, ỹ n (4.129)
c○2007, B.E.Shapiro
Last revised: May 23, 2007
Math 582B, Spring 2007
California State University Northridge