On the use of parallel processors for implicit Runge-Kutta methods |
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Authors: | G. J. Cooper R. Vignesvaran |
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Affiliation: | 1. School of Mathematical and Physical Sciences, University of Sussex, BN1 9QH, Brighton, England 2. Department of Mathematics, Eastern University, Chenkalady, Sri Lanka
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Abstract: | An iteration scheme, for solving the non-linear equations arising in the implementation of implicit Runge-Kutta methods, is proposed. This scheme is particularly suitable for parallel computation and can be applied to any method which has a coefficient matrixA with all eigenvalues real (and positive). For such methods, the efficiency of a modified Newton scheme may often be improved by the use of a similarity transformation ofA but, even when this is the case, the proposed scheme can have advantages for parallel computation. Numerical results illustrate this. The new scheme converges in a finite number of iterations when applied to linear systems of differential equations, achieving this by using the nilpotency of a strictly lower triangular matrixS ?1 AS — Λ, with Λ a diagonal matrix. The scheme reduces to the modified Newton scheme whenS ?1 AS is diagonal.A convergence result is obtained which is applicable to nonlinear stiff systems. |
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