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product of two matrices as
2
3
a11 J a12J
a1sJ
6 a21 J a22 J
a2sJ 7
6
7
A J = 6 ..
(3.3.10b)
... . . . ... 7 :
4.
5
as1J as2J
assJ
Then, (3.3.9b) can be written concisely as
(I ; hA J) Y( ) = ;F( )
where A was given by (3.2.6c) and
2 ()
Y1
(
6 Y2 )
6
()
Y = 6 ..
4. Ys( ) 3
7
7
7
5 2
6
()
F =6
6
4 (3.3.10c)
3 F(1 )
F(2 ) 7 :
7
... 7
5
()
Fs (3.3.10d) The approximation of the Jacobian does not change the accuracy of the computed solution, only the convergence rate of the iteration. As long as convergence remains good,
the same Jacobian can be used for several time step and only be reevaluated when
convergence of the Newton iteration slows.
Even with this simpli cation, with m ranging into the thousands, the solution of
(3.3.10) is clearly expensive and other ways of reducing the computational cost are necessary. Diagonally implicit RungeKutta (DIRK) methods o er one possibility. A DIRK
method is one where aij = 0, i < j and at least one aii 6= 0, i j = 1 2 : : : s. If, in
addition, a11 = a22 = : : : = ass = a, the techniq...
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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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