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

CHAPTER 5. RUNGE-KUTTA METHODS 101
and hence the most general form of the iteration formula is
K 1 = f(t n−1 + c 1 h, y n−1 + ha 11 K 1 + ha 12 K 2 ) (5.108)
= f(t n−1 , y n−1 ) = f (5.109)
K 2 = f(t n−1 + c 2 h, y n−1 + ha 21 K 1 + ha 22 K 2 ) (5.110)
= f(t n−1 + ah, y n−1 + haK 1 ) ⇐= (5.111)
= f(t n−1 + ah, y n−1 + haf) ⇐= (5.112)
y n = y n−1 + hb 1 K 1 + hb 2 K 2 (5.113)
= y n−1 + hb 1 f + hb 2 f(t n−1 + ah, y n−1 + haf) (5.114)
where we have use the shorthand notation f = f(t n−1 , y n−1 ). To determine the
coefficients to minimize error, we expand the last term in a Taylor series about
(t n−1 , y n−1 )
f(t n−1 + ah,y n−1 + haf) = f + ahf t + haff y + (5.115)
a 2 h 2
2 (f tt + 2ff ty + f 2 f yy + f y (f t + ff y )) + O(h 3 ) (5.116)
To simplify the notation we let (after Lambert) F = f t + ff y and G = f tt + 2ff ty +
f 2 f yy
(
y n = y n−1 + hb 1 f + hb 2 f + ahf t + haff y + a2 h 2 )
2 (G + f yF ) (5.117)
Hence we need
= y n−1 + h(b 1 + b 2 )f + h 2 b 2 aF + O(h 3 ) (5.118)
b 1 + b 2 = 1 (5.119)
b 2 a = 1 2
(5.120)
to match a Taylor expansion. This gives us a family of two stage explicit methods
that are second order:
0 0
a a 0
1 − 1 1
2a 2a
(5.121)
When a = 1, this gives the trapezoidal method, and when a = 1/2, it gives the
explicit midpoint method.
Two popular explicit three-stage third-order methods include Heun’s thirdorder
method
0 0
1/3 1/3 0
(5.122)
2/3 0 2/3 0
1/4 0 3/4
c○2007, B.E.Shapiro
Last revised: May 23, 2007
Math 582B, Spring 2007
California State University Northridge