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UvA-DARE (Digital Academic Repository) Parallel Iteration across the steps of high-order Runge-Kutta Methods for nonstiff initial value problems van der Houwen, P.J.; Sommeijer, B.P.; van der Veen, W.A.

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We construct and analyze parallel iterative solvers for the solution of the linear systems arising in the application of Newton's method to ;;-stage implicit Runge-Kutta (RK) type discretizations of implicit differential equations (ID Es). These 1 incar solvers arc partly iterative and partly direct. Each linear system iteration again requires the solution of linear subsystems. hut now only of IDE dimension, which is ;; times less than the dimension of the linear system in Newton's melhod. Thus. the effective costs on a parallel computer system arc only one LU-decomposition of !DE dimension for each Jacobian update, yielding a considerable reduction of the effective LU-costs. The method parameters can he chosen such that only a few iterations by the linear solver are needed. The algorithmic properties arc illustrated hy solving the transistor problem (index 1) and the car axis problem (index 3) taken from the CW! test set.1997 Elsevier Science B. V.Kn·words: Numerical analysis; Implicit differential equations; DAEs: Runge-Kutta methods; Parallelism IntroductionWe consider initial value problems (lVPs) for systems of implicit differential equations (IDEs or DAEs)

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