On convergence of waveform relaxation for nonlinear systems of ordinary differential equations

TL;DR

This paper analyzes the convergence of a waveform relaxation method for nonlinear ODE systems, using exponential Krylov subspace techniques, and compares its efficiency with traditional methods through numerical experiments.

Contribution

It provides a theoretical and practical assessment of the convergence and efficiency of a waveform relaxation method combined with EBK for nonlinear differential equations.

Findings

The method converges under certain conditions.

It performs efficiently compared to conventional integrators.

Numerical tests confirm its effectiveness on complex nonlinear problems.

Abstract

To integrate large systems of nonlinear differential equations in time, we consider a variant of nonlinear waveform relaxation (also known as dynamic iteration or Picard-Lindel\"of iteration), where at each iteration a linear inhomogeneous system of differential equations has to be solved. This is done by the exponential block Krylov subspace (EBK) method. Thus, we have an inner-outer iterative method, where iterative approximations are determined over a certain time interval, with no time stepping involved. This approach has recently been shown to be efficient as a time-parallel integrator within the PARAEXP framework. In this paper, convergence behavior of this method is assessed theoretically and practically. We examine efficiency of the method by testing it on nonlinear Burgers, three-dimensional Liouville-Bratu-Gelfand, and three-dimensional nonlinear heat conduction equations and comparing its performance with that of conventional time-stepping integrators.