Abstract.The implementation of implicit Runge-Kutta methods requires the solution of large systems of non-linear equations. Normally these equations are solved by a modified Newton process, which can be very expensive for problems of high dimension. The recently proposed triangularly implicit iteration methods for ODE-IVP solvers [5] substitute the Runge-Kutta matrix A in the Newton process for a triangular matrix T that approximates A, hereby making the method suitable for parallel implementation. The matrix T is constructed according to a simple procedure, such that the stiff error components in the numerical solution are strongly damped. In this paper we prove for a large class of Runge-Kutta methods that this procedure can be carried out and that the diagonal entries of T are positive. This means that the linear systems that are to be solved have a non-singular matrix.AA!S subject classification: Primary: 65L06, Secondary: 15A23.
The use of implicit methods for numerically solving stiff systems of differential equa,. tions requires the solution of systems of nonlinear equations. Normally these are solved by a Newtontype process, in which we have to solve systems of linea.r equations. The Jacobian of the derivative function determines the structure of the matrices of these linear systems. Since it often occurs that the components of the derivative function only depend on a small number of variables, the system can be considerably sparse. Hence, it can be worth the effort to use a sparse matrix solver instead of a dense LU-decomposition. This paper reports on experiences with the direct sparse matrix solvers MA28 by Duff (1), Yl2M by Zlatev et al. [2) a.nd one special-purpose matrix solver, all embedded in the parallel ODE solver PSODE by Sommeijer [3). The authors a.re grateful to B. P. Sommeijer for his careful reading of the manuscript and for suggesting severe.I improvements. The research reported in this pa.per was supported by STW (Dutch Foundation for Technical Sciences). 43 44 J. J. B. DE SWARl' AND J. G. BLOM numerical solution of the linear systeDlS is given in Section 4. Finally, in Section 5, numerical experiments give insight into the beha.vior of these matrix solvers.
Link to publication General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: http://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. If the nonlinear systems arising in implicit Runge-Kutta methods like the Radau IIA methods are iterated by (modified) Newton, then we have to solve linear systems whose matrix of coefficients is of the form I − A ⊗ hJ with A the Runge-Kutta matrix and J an approximation to the Jacobian of the righthand side function of the system of differential equations. For larger systems of differential equations, the solution of these linear systems by a direct linear solver is very costly, mainly because of the LU-decompositions. We try to reduce these costs by solving the linear systems by a second (inner) iteration process. This inner iteration process is such that each inner iteration again requires the solution of a linear system. However, the matrix of coefficients in these new linear systems is of the form I − B ⊗ hJ where B is similar to a diagonal matrix with positive diagonal entries. Hence, after performing a similarity transformation, the linear systems are decoupled into s subsystems, so that the costs of the LU-decomposition are reduced to the costs of s LUdecompositions of dimension d. Since these LU-decompositions can be computed in parallel, the effective LU-costs on a parallel computer system are reduced by a factor s 3 . It will be shown that matrices B can be constructed such that the inner iterations converge whenever A and J have their eigenvalues in the positive and nonpositive halfplane, respectively. The theoretical results will be illustrated by a few numerical examples. A parallel implementation on the four-processor Cray-C98/4256 shows a speed-up ranging from at least 2.4 until at least 3.1 with respect to RADAU5 applied in one-processor mode.
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