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On the convergence of waveform relaxation methods for stiff nonlinear ordinary differential equations. (English) Zbl 0836.65087

This paper is concerned with the so called waveform relaxation methods for the numerical solution of initial value problems (IVPs) for systems of ordinary differential equations (ODEs) by using parallel computers. The basic idea in the waveform relaxation methods consists in splitting the original system of ODEs into several subsystems together with an iterative process so that in each iteration these subsystems can be solved independently and the whole iterative process converges to the solution of the IVP under consideration.
Here, the author deals with waveform relaxation methods for nonlinear stiff IVPs, where the solution of subsystems is carried out with algebraically stable Runge-Kutta methods on (possibly) non-uniform grids. By introducing suitable assumptions on the stiff differential system, the author derives a convergence result of the numerical approximations on each iterate to the Runge-Kutta solution on the grid points. Furthermore some estimates of the errors with respect to the exact solution of the iterates are derived in suitable discrete and continuous norms. Finally, some comparisons with related results given by Ch. Lubich and A. Ostermann [BIT 27, 216-234 (1987; Zbl 0623.65125)] on uniform grids are presented.
Reviewer: M.Calvo (Zaragoza)

MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
65Y05 Parallel numerical computation
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L70 Error bounds for numerical methods for ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
34E13 Multiple scale methods for ordinary differential equations

Citations:

Zbl 0623.65125
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References:

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