On the convergence of waveform relaxation methods for stiff nonlinear ordinary differential equations
type: Item
instance of: scholarly article
title: On the convergence of waveform relaxation methods for stiff nonlinear ordinary differential equations
zbMATH Open document ID: 0836.65087
DOI: 10.1016/0168-9274(95)00052-V
author: Karel J. in 't Hout
published in: Applied Numerical Mathematics
publication date: 1996-01-07T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65L20
Mathematics Subject Classification ID: 65Y05
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 65L70
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 34E13
zbMATH DE Number: 815820
zbMATH Keywords: algebraic stability
zbMATH Keywords: stiff equations
zbMATH Keywords: error bounds
zbMATH Keywords: parallel computation
zbMATH Keywords: waveform relaxation methods
zbMATH Keywords: systems
zbMATH Keywords: splitting
zbMATH Keywords: iteration
zbMATH Keywords: nonlinear
zbMATH Keywords: Runge-Kutta methods
zbMATH Keywords: non-uniform grids
zbMATH Keywords: convergence
zbMATH Keywords: comparisons
reviewed by: Manuel Calvo
MaRDI profile type: Publication
cites work: Contractivity of Waveform Relaxation Runge–Kutta Iterations and Related Limit Methods for Dissipative Systems in the Maximum Norm
cites work: The use of Runge-Kutta formulae in waveform relaxation methods
cites work: Parallel methods for initial value problems
cites work: Stability Criteria for Implicit Runge–Kutta Methods
cites work: Implementing an ODE code on distributed memory computers
cites work: Q4725642
cites work: Sur la B-stabilité des méthodes de Runge-Kutta
cites work: On the existence of solutions to the algebraic equations in implicit Runge-Kutta methods
cites work: Q3687624
cites work: Massive parallelism across space in ODEs
cites work: Q4000061
cites work: Q3221299
cites work: A Note on Unconditional Maximum Norm Contractivity of Diagonally Split Runge–Kutta Methods
cites work: Multi-grid dynamic iteration for parabolic equations
cites work: Convergence of Dynamic Iteration Methods for Initial Value Problems
cites work: Sets of convergence and stability regions
cites work: Remarks on Picard-Lindelöf iteration
cites work: Remarks on Picard-Lindelöf iteration. II
full work available at URL: 0168-9274(95)00052-v
OpenAlex ID: W1980309222
Recommended article: The use of Runge-Kutta formulae in waveform relaxation methods
Recommended article: On the convergence of the discretized multi-splitting waveform relaxation algorithm
Recommended article: Q4860183
Recommended article: Waveform relaxation methods for implicit differential equations
Recommended article: Optimized Schwarz Waveform Relaxation Methods for Advection Reaction Diffusion Problems
Recommended article: Efficient Parallel Algorithms for Solving Initial-Boundary Value and Time-Periodic Parabolic Partial Differential Equations
Recommended article: Metodi waveform relaxation per la risoluzione numerica di grandi sistemi di equazioni differenziali ordinarie
Recommended article: Q4010711
Recommended article: Q4853481
Recommended article: Q4215158
description: scientific article; zbMATH DE number 815820