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Unformatted text preview: s. All
values of ci, i = 1 2 : : : s, are on (0 1] with, as designed, cs = 1. The one-stage Radau
method is the backward Euler method. The tableau of the two-stage Radau method is
; 1 3
4 35 1
4 We'll conclude this Section with a discussion of singly implicit Runge-Kutta (SIRK)
methods. These methods are of order s, which is less than the Legendre (2s), Radau
(2s ; 1), and DIRK (s + 1) techniques. They still have excellent A- and L-stability
properties and, perhaps, o er a computational advantage.
A SIRK method is one where the coe cient matrix A has a single s-fold real eigenvalue. These collocation methods were Originally developed by Butcher 9] and have
been subsequently extended 5, 10, 6, 20]. Collocating, as described, leads to the system
(3.3.12-3.3.15). The intermediate solutions Yi, i = 1 2 : : : s, have the vector form speci ed by (3.2.9d) with the elements of A given by (3.3.15a). Multiplying (3.2.9d) by a
nonsingular matrix T 1 , we obtain
; T 1 Y = ynT 1l + hT 1 ATT 1f
; ; ; ; where Y, l, A, and f are, respectively, given by (3.2.6a...
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- Spring '14