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On the theory of parallel Runge-Kutta methods. (English) Zbl 0712.65071

Author’s summary: The purpose of this paper is to create a theoretical framework for parallelization of Runge-Kutta methods. We investigate the inherent potential for parallelism by considering digraphs of Runge-Kutta matrices. By highlighting the important rôle of the underlying sparsity pattern, this approach narrows the field down to certain types of methods. These are further investigated by two techniques: perturbed collocation and elementary differentials. Our analysis leads to singly diagonally implicit fourth-order L-stable methods that can be implemented on two processors (in MIMD architecture) with computational cost of two ‘conventional’ stages. We also debate local error control and present a technique that can be used to this end.
Reviewer: R.Scherer

MSC:

65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
65Y05 Parallel numerical computation
65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
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