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.
Zusammenfassung
Wir schlagen ein Iterationsverfahren für die Lösung der nicht-linearen Gleichungen bei der Implementierung von impliziten Runge-Kutta-Verfahren vor, das besonders geeignet für die Parallelverarbeitung ist und immer angewandt werden kann, wenn die KoeffizientenmatrixA nur reelle (und positive) Eigenwerte hat. Für solche Verfahren kann man zwar die Effizienz eines modifizierten Newton-Verfahrens oft auch mit Hilfe einer Ähnlichkeitstransformation vonA verbessern, aber sogar dann hat unser vorgeschlagenes Verfahren Vorteile für die Parallelverarbeitung, wie numerische Beispiele zeigen. Das neue Vorgehen konvergiert für lineare Differentialgleichungssysteme in endlich vielen Schritten wegen der Nilpotenz einer strengen unteren DreiecksmatrixS −1 AS — Λ und reduziert sich auf das modifizierte Newton-Verfahren wennS −1 eine Diagonalmatrix ist. Unser Konvergenzresultat ist auf nichtlineare steife Systeme anwendbar.
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Cooper, G.J., Vignesvaran, R. On the use of parallel processors for implicit Runge-Kutta methods. Computing 51, 135–150 (1993). https://doi.org/10.1007/BF02243848
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DOI: https://doi.org/10.1007/BF02243848