On the convergence of waveform relaxation methods for stiff nonlinear ordinary differential equations
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 \textit{Ch. Lubich} and \textit{A. Ostermann} [BIT 27, 216-234 (1987; Zbl 0623.65125)] on uniform grids are presented.
scientific article; zbMATH DE number 815820
scientific article