A multirate Neumann-Neumann waveform relaxation method for heterogeneous coupled heat equations

TL;DR

This paper introduces a multirate Neumann-Neumann waveform relaxation method for efficiently solving heterogeneous coupled heat equations, enhancing parallelization in time with proven convergence and optimal relaxation parameters.

Contribution

It develops a multirate waveform relaxation algorithm with a linear interpolation at the interface, and provides a convergence analysis and optimal relaxation parameter estimation.

Findings

The method converges in only two iterations.

The 1D nonmultirate optimal relaxation parameter effectively estimates multirate cases.

Numerical results validate the theoretical analysis across 1D and 2D examples.

Abstract

An important challenge when coupling two different time dependent problems is to increase parallelization in time. We suggest a multirate Neumann-Neumann waveform relaxation algorithm to solve two heterogeneous coupled heat equations. In order to fix the mismatch produced by the multirate feature at the space-time interface a linear interpolation is constructed. The heat equations are discretized using a finite element method in space, whereas two alternative time integration methods are used: implicit Euler and SDIRK2. We perform a one-dimensional convergence analysis for the nonmultirate fully discretized heat equations using implicit Euler to find the optimal relaxation parameter in terms of the material coefficients, the stepsize and the mesh resolution. This gives a very efficient method which needs only two iterations. Numerical results confirm the analysis and show that the 1D nonmultirate optimal relaxation parameter is a very good estimator for the multirate 1D case and even for multirate and nonmultirate 2D examples using both implicit Euler and SDIRK2.