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

CHAPTER 5. RUNGE-KUTTA METHODS 111
The maximum order of any collocation method is equal to the number of stages.
Gauss-Legendre numerical methods are ν-stage, order 2ν methods that are based
on collocation, and hence they give the highest possible order for any ν-stage
method. 3
Example 5.3. Calculate the Butcher array for the ν = 1 using P 1 (t) = t − 1/2
Solution. The only root is at c 1 = 1/2, hence
Hence the Butcher Array is
q(t) = t − 1/2 (5.191)
q 1 (t) = 1 (5.192)
a 11 = 1 1
b 1 = 1 1
∫ 1/2
0
∫ 1
0
1/2 1/2
1
ds = 1 2
(5.193)
ds = 1 (5.194)
(5.195)
Example 5.4. Calculate the Butcher Array for the two-stage Gauss-Legendre method
with ν = 2 where P 2 (t) = t 2 − t + 1/6
Solution. The roots of 6t 2 − 6t + 1 = 0 are at
Hence
q(t) =
(
c 1 , c 2 = 6 ± √ 36 − 24
12
t − 1 2 − √
3
6
) (
t − 1 2 + √
3
6
= 1 2 ± √
3
6
)
(5.196)
(5.197)
= t 2 − t + 1 2
(5.198)
q 1 (t) = t − 1 √
3
2 + 6
q 1 (c 1 ) = 1 √
3
2 + 6 − 1 √ √
3 3
2 + 6 = 3
q 1 (t)
q 1 (c 1 ) = √ √
3
3t −
2 + 1 2
∫
√ (
1/2+ 3/6
√ )
√3s 3
a 11 =
−
0
2 + 1 2
(√
3
=
2 s2 + 1 − √ )∣
3 ∣∣∣∣
1/2+ √ 3/6
s
2
=
√
3
2
( √
1 3
2 + 6
) 2
+
(
0
1 − √ 3
2
3 For a proof of this statement see [13] section 3.4.
(5.199)
(5.200)
(5.201)
ds (5.202)
) ( √ )
1 3
2 + = 1 6 4
(5.203)
(5.204)
c○2007, B.E.Shapiro
Last revised: May 23, 2007
Math 582B, Spring 2007
California State University Northridge