Unformatted text preview: c) and
2
3
f (tn 1 + c1h)
6 f (tn 1 + c2 h) 7
7:
f =6
6
7
...
4
5
f (tn 1 + csh)
; (3.3.23) ; ; Let
^
Y = T 1Y
; ^ = T 1l
l ^
A = T 1 AT ; ; ^ = T 1f :
f
; (3.3.24) Butcher 9] chose the collocation points ci = i, i = 1 2 : : : s, where i is the i th
root of the s thdegree Laguerre polynomial Ls(t) and is chosen so that the numerical
method has favorable stability properties. butcher also selected T to have elements Tij = Li 1 ( j ):
; Then 2
6
6
^ =6
A6
6
4 ...
36 3
7
7
7:
7
7
5 (3.3.25) ^
Thus, A is lower bidiagonal with the single eigenvalue . The linearized system (3.3.9)
is easily solved in the transformed variables. (A similar transformation also works with
Radau methods 17].) Butcher 9] and Burrage 5] show that it is possible to nd Astable SIRK methods for s 8. These methods are also Lstable with the exception of
the sevenstage method. Problems 1. Verify that (3.3.17d) is correct when f (t y) = ay with a a constant.
2. Consider the method yn = yn 1 + h (1 ; )f (tn 1 yn 1) + f (tn yn)]
; ; ; with 2 0 1]. The method corresponds to the Euler method with = 0, the
trapezoidal rule with = 1=2, and the backward Euler method and when = 1.
2.1. Writ...
View
Full Document
 Spring '14
 JosephE.Flaherty
 Numerical Analysis, yn, Tn, Numerical ordinary differential equations

Click to edit the document details