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

Abstract

We consider partitioned time integration for heterogeneous coupled heat equations. First and second order multirate, as well as time-adaptive Dirichlet-Neumann waveform relaxation (DNWR) methods are derived. In 1D and for implicit Euler time integration, we analytically determine optimal relaxation parameters for the fully discrete scheme.Similarly to a previously presented Neumann-Neumann waveform relaxation (NNWR) first and second order multirate methods are obtained. We test the robustness of the relaxation parameters on the second order multirate method in 2D. DNWR is shown to be very robust and consistently yielding fast convergence rates, whereas NNWR is slower or even diverges. The waveform approach naturally allows for different timesteps in the subproblems. In a performance comparison for DNWR, the time-adaptive method dominates the multirate method due to automatically finding suitable stepsize ratios. Overall, we obtain a fast, robust, multirate and time adaptive partitioned solver for unsteady conjugate heat transfer.

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

“…the DNWR algorithm is retrieved, as examined in the continuous case in [1] and in the discrete case in [3]. However (3d) involves both φ k a and U k−1+θ a : the θ parameter appears thus here within (close to Robin) condition (ν a φ a (0) − αθU a (h a /2) = .…”