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

CHAPTER 5. RUNGE-KUTTA METHODS 109
Hence absolute stability requires
∣ y n ∣∣∣ ∣ =
y n−1
∣ 1 + h ( )∣ 6λ − 4hλ
2
4 h 2 λ 2 − 4hλ + 6 + 18λ ∣∣∣
h 2 λ 2 (5.168)
− 4hλ + 6
=
∣ 1 + 6z − z2
z 2 − 4z + 6∣ (5.169)
=
6 + 2z
∣z 2 − 4z + 6∣ ≤ 1 (5.170)
where z = hλ. The region of absolute stability is shown in figure 5.6
5.8 IRK Methods based on Collocation
In the method of collocation, given an initial value problem that we have solved
numerically on the interval [t 0 , t n ] we seek a ν th order polynomial P (t) such that
P (t n ) = y n (5.171)
P ′ (t n )(t n + c j h) = f(t n + c j h, P (t n + c j h)), j = 1, 2, . . . , ν (5.172)
where the numbers c 1 , . . . , c n are called collocation parameters. Thus P satisfies
the initial value problem defined by the numerical solution at t n . A collocation
method finds such a polynomial P and sets y n+1 = P (t n+1 ).
Theorem 5.1. The Implicit Runge-Kutta Method
c
A
b T (5.173)
and the collocation method of equation 5.171 are identical if
where
a ji = 1
q i (c i )
b j = 1
q j (c j )
q j (t) =
∫ cj
0
∫ 1
ν∏
0
i=1,i≠j
q i (s)ds (5.174)
q j (s)ds (5.175)
(t − c i ) (5.176)
Proof. Define the ν th order polynomial
r(t) =
ν∑
k=1
q k ((t − t n )/h)
w k (5.177)
q k (c k )
then
r(t n + c j h) =
ν∑
k=1
q k (c j )
q k (c k ) w k =
ν∑
δ jk w k = w j (5.178)
k=1
c○2007, B.E.Shapiro
Last revised: May 23, 2007
Math 582B, Spring 2007
California State University Northridge