A multirate Neumann-Neumann waveform relaxation method for heterogeneous coupled heat equations
This work addresses the need for parallelization in time for coupled time-dependent problems, offering an efficient iterative solver for heterogeneous heat equations.
The authors propose a multirate Neumann-Neumann waveform relaxation method for coupling heterogeneous heat equations with different time steps, achieving convergence in only two iterations by using an optimal relaxation parameter derived from 1D analysis. Numerical results confirm the method's efficiency in 1D and 2D for both implicit Euler and SDIRK2 schemes.
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.