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

Chapter 5
Runge-Kutta Methods
5.1 Taylor Series Methods
We begin by expanding y n as a Taylor series about t n−1 ,
y n = y n−1 + hy ′ n−1 + h2
2 y′′ n−1 + · · · + hp
p! yp n−1 + · · · (5.1)
Using the chain rule for derivatives, and denoting f(t n−1 , y n−1 ) by f, we have
y ′ = f (5.2)
y ′′ = df
dt = ∂f
∂t + ∂f dy
∂y dt
(5.3)
= f t + f y f = F (5.4)
y ′′′ = d dt (f t + f y f) = ∂ ∂t (f t + f y f) + ∂ ∂y (f t + f y f) dy
dt
(5.5)
= f tt + f yt f + f y f t + (f yt + f yy f + f y f y )f (5.6)
= f tt + 2f yt f + f y f t + f yy f 2 + f 2 y f (5.7)
= f y F + G (5.8)
where
and
Hence
F = f t + f y f
G = f tt + 2f yt f + f yy f 2
y n = y n−1 + hf + h2
2 (f t + f y f)+ (5.9)
h 3 (
ftt + 2f yt f + f y f t + f yy f 2 + fy 2 f ) + · · ·
6
(5.10)
Thus we can define a sequence of methods, depending where we truncate the series.
A Taylor method that includes terms through h p therefore has a local truncation
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