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

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 step size, and the mesh resolution. This gives a very efficient method which needs only two iterations. Numerical results confirm the analysis and show that the one-dimensional nonmultirate optimal relaxation parameter is a very good estimator for the multirate one-dimensional case and even for multirate and nonmultirate two-dimensional examples using both implicit Euler and SDIRK2.