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

106 CHAPTER 5. RUNGE-KUTTA METHODS
While one could use an arbitrary step size for the first step, some knowledge of the
problem is still necessary to prevent a really bad choice. An automatic calculation
of the first value of h is provided by the following algorithm, as presented in [10].
1. Input ɛ abs and ɛ rel , the acceptable absolute and relative tolerances, and calculate
where ‖u‖ =
√
1
∑ ui
n σ
2. Let h 0 = 0.01(d 0 /d 1 ) and calculate
3. Calculate f(t 0 + h 0 , y 1 ) and let
4. Compute a new step size h 1 from
5. Use the following initial step size:
σ = ɛ abs + |y 0 |ɛ rel (5.142)
d 0 = ‖y 0 ‖ (5.143)
d 1 = ‖f(t 0 , y 0 )‖ (5.144)
(for scalar problems this becomes ‖u‖ = |u|/√ σ).
y 1 = y 0 + h 0 f(t 0 , y 0 ) (5.145)
d 2 = ‖f(t 0 + h, y 1 ) − f(t 0 , y 0 )‖
h 0
(5.146)
h p+1
1 max(d 1 , d 2 ) = 0.01 (5.147)
h = min(100h 0 , h 1 ) (5.148)
5.7 Implicit Runge-Kutta Methods
Implicit Runge-Kutta (IRK) methods are more complicated - the diagonal and
supra-diagonal elements of the Butcher array may be nonzero. IRK’s fall into several
classes, including:
• Gauss Methods which are based on Gaussian quadrature 2 , and have an
order of accuracy of 2s, where s is the number of stages. Gauss methods have
the highest attainable order for a given number of stages. Gauss methods are
A-Stable but do not have stiff decay.
• Radau Methods where one end of the interval is included in the iteration
formula; the order of accuracy is 2s − 1. Radau methods have stiff decay.
• Lobatto methods, in which the function is sampled at both ends of the
interval; the order of accuracy is 2s − 2. Lobatto methods are A-Stable but
do not have stiff decay.
2 see [13], for example, for a discussion of Gaussian quadrature.
Math 582B, Spring 2007
California State University Northridge
c○2007, B.E.Shapiro
Last revised: May 23, 2007