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# 39d for an s stage runge kutta method applied to an m

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Unformatted text preview: direct 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 re-evaluated 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 Runge-Kutta (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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