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

104 CHAPTER 5. RUNGE-KUTTA METHODS
Figure 5.5: The function RungeKuttaOrderConditions in Mathematica.
5.6 Step Size Control for ERK Methods
Computing the local truncation error for a Runge-Kutta method is quite unwieldy
and requires the analytic calculation of numerous partial derivatives, making it
impractical for an efficient implementation. Instead, we are forced to use an approximation
of the error. Suppose we have computed two different solutions: y, of
order p, and z, of order p + 1. Then
y n = y(t n ) + ch p+1 + O(h p+2 ) (5.132)
z n = y(t n ) + O(h p+2 ) (5.133)
for some constant c that depends on the method but is independent of h. Subtracting,
ch p+1 = |y n − z n | (5.134)
The idea is that we have some tolerance ɛ that we do not want to exceed, and when
|y n − z n | approaches ɛ, we choose a new step size that satisfies
|y n − z n | old
h p+1
old
≈ c ≈ |y n − z n | new
h p+1
new
< νɛ
h p+1
new
(5.135)
Math 582B, Spring 2007
California State University Northridge
c○2007, B.E.Shapiro
Last revised: May 23, 2007