Discrete analysis of Schwarz Waveform Relaxation for a simplified air-sea coupling problem with nonlinear transmission conditions

TL;DR

This paper analyzes a Schwarz waveform relaxation method for a simplified ocean-atmosphere coupling model with nonlinear interface conditions, demonstrating convergence properties and potential improvements through relaxation parameters.

Contribution

It provides a discrete analysis of SWR convergence for nonlinear interface conditions in a simplified coupling problem, including numerical validation.

Findings

Convergence of SWR is similar for linearized and full nonlinear friction cases.

Adding a relaxation parameter can improve convergence speed.

Numerical experiments confirm theoretical convergence behavior.

Abstract

In this study we present a non-overlapping Schwarz waveform relaxation (SWR) method applied to a one dimensional model problem representative of the coupling between the ocean and the atmosphere. This problem includes nonlinear interface conditions analogous to a quadratic friction law. We study the convergence of the corresponding SWR at a semi-discrete level for a linear friction and for a linearized quadratic friction at the interface. Using numerical experiments we show that the convergence properties in the linearized quadratic friction case are very close to the ones obtained with the full nonlinear problem for the range of parameter values of interest. We investigate the possibility to improve the convergence speed by adding a relaxation parameter at the interface.