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Unformatted text preview: ue is known as a singly diagonally implicit
RungeKutta (SDIRK) method. Thus, the coe cient matrix of an SDIRK method has
26 the form 2
6
A=6
6
4 3 a
7
a21 a
(3.3.11)
7
... ... . . . 7 :
5
as1 as2
a
Thus, with the approximation (3.3.10), the system Jacobian in (3.3.10c) is
2
3
I ; haJ
0
6 ;ha21 J I ; haJ
07
6
(I ; hA J) = 6 ..
7
...
... 7 :
...
4.
5
;has1 J ;has2J
I ; haJ
The Newton system (3.3.10) is lower block triangular and can be solved by forward
(
substitution. Thus, the rst block of (3.3.10c) is solved for Y1 ) . Knowing Y1 the
(
second equation is solved for Y2 ), etc. The Jacobian J is the same for all stages thus,
the diagonal blocks need only be factored once by Gaussian elimination and forward and
backward substitution may be used for each solution.
The implicit midpoint rule (3.3.2) is a onestage, secondorder DIRK method. We'll
examine a twostage DIRK method momentarily, but rst we note that the maximum
order of an sstage DIRK method is s + 1 2].
Example 3.3.5. A twostage DIRK formula has the ta...
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This document was uploaded on 03/16/2014 for the course CSCI 6820 at Rensselaer Polytechnic Institute.
 Spring '14
 JosephE.Flaherty

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