Some convergence results of waveform relaxation for a class of second-order quasilinear parabolic equations

Abstract

Waveform relaxation method at the PDE level can simplify the original differential system by several decoupling approaches, which is very useful for the iterative solution of nonlinear differential equations. In this paper, the convergence of waveform relaxation method at the PDE level for quasilinear equations is analysed for the first time. We mainly consider a class of general quasilinear parabolic equations both in divergence form and in non-divergence form. Some superlinear and linear convergence results under several convergence conditions for these quasilinear equations are given, and numerical experiments illustrated these superlinear or linear convergence rates, which claim that waveform relaxation method is available as an iterative framework for nonlinear problems.

Keywords:

• Waveform relaxation
• quasilinear parabolic equations
• convergence analysis
• divergence form
• non-divergence form

2010 AMS Subject Classifications:

• 65M12
• 65M15
• 65B99

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the Natural Science Foundation of China (NSFC) [grant numbers 11871393, 61663043] and International Science and Technology Cooperation Program of Shaanxi Key Research & Development Plan [grant number 2019KWZ-08].