The Computable Differential Equation Lecture ... - Bruce E. Shapiro
The Computable Differential Equation Lecture ... - Bruce E. Shapiro
The Computable Differential Equation Lecture ... - Bruce E. Shapiro
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110 CHAPTER 5. RUNGE-KUTTA METHODS<br />
If we choose<br />
w j = P ′ (t n + c j h), j = 1, 2, . . . , ν (5.179)<br />
<strong>The</strong>n ⇐=<br />
r(t n + c j h) = w j = P ′ (t n + c j h) (5.180)<br />
Since the two polynomials r(t) and P ′ (t) are both the same order (ν − 1) and they<br />
coincide at ν points (order +1) then they must be identical polynomials. Hence we<br />
conclude that<br />
r(t) = P ′ (t) (5.181)<br />
<strong>The</strong>n by equation 5.171<br />
Integrating,<br />
∫ t<br />
t n<br />
P ′ (s)ds =<br />
P (t) − y n =<br />
P ′ (t) =<br />
ν∑<br />
k=1<br />
ν∑<br />
k=1<br />
=<br />
ν∑<br />
k=1<br />
ν∑<br />
k=1<br />
q k ((t − t n )/h<br />
P ′ (t n + c k h) (5.182)<br />
q k (c k )<br />
q k ((t − t n )/h<br />
f(t n + c k h, P (t n + c k h)) (5.183)<br />
q k (c k )<br />
∫ t<br />
f(t n + c k h, P (t n + c k h))<br />
If we let t = t n + c j h in 5.185 then<br />
P (t n + c j h) = y n + h<br />
hence<br />
Let<br />
q k ((s − t n )/h<br />
ds (5.184)<br />
t n<br />
q k (c k )<br />
∫ (t−tn)/h<br />
q k (s)<br />
f(t n + c k h, P (t n + c k h))<br />
hds ⇐= (5.185)<br />
0 q k (c k )<br />
ν∑<br />
k=1<br />
P (t n + c j h) = y n +<br />
∫ cj<br />
f(t n + c k h, P (t n + c k h))<br />
0<br />
q k (s)<br />
ds ⇐= (5.186)<br />
q k (c k )<br />
ν∑<br />
a j,k f(t n + c k h, P (t n + c k h)) ⇐= (5.187)<br />
k=1<br />
K j = P (t n + c j h) = y n +<br />
ν∑<br />
a j,k f(t n + c k h, K k ) ⇐= (5.188)<br />
k=1<br />
<strong>The</strong>n if we let t = t n+1 in 5.185 then,<br />
P (t n+1 ) = y n + h<br />
= y n + h<br />
ν∑<br />
∫ 1<br />
q k (s)<br />
f(t n + c k h, P (t n + c k h)) ds<br />
0 q k (c k )<br />
⇐= (5.189)<br />
ν∑<br />
b k f(t n + c k h, K k ) ⇐= (5.190)<br />
k=1<br />
k=1<br />
which is a ν-stage implicit Runge-Kutta method.<br />
Math 582B, Spring 2007<br />
California State University Northridge<br />
c○2007, B.E.<strong>Shapiro</strong><br />
Last revised: May 23, 2007