A Multirate Neumann--Neumann Waveform Relaxation Method for Heterogeneous Coupled Heat Equations

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.

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

“…At each iteration k, imposing continuity of the solution across the interface, one first finds the local solution u k+1 1 (x, t) on Ω 1 by solving the Dirichlet problem (2). Then, imposing continuity of the heat fluxes across the interface, one finds the local solution u k+1 2 (x, t) on Ω 2 by solving the Neumann problem (3).…”

“…where Θ ∈ (0, 1] is the relaxation parameter. Following the work on [3], a multirate DNWR algorithm can be written down for the discrete versions of (2), (3) and (4) using implicit Euler or SDIRK2 in time. Furthermore, the optimal relaxation parameter Θ opt can be found dependent on the parameters α 1 , α 2 , λ 1 , λ 2 , ∆x and ∆t.…”

“…

We introduce a time adaptive multirate method based on the Dirichlet-Neumann waveform relaxation (DNWR) algorithm for the simulation of two coupled linear heat equations with strong jumps in the material coefficients across the interface. Numerical results are included to illustrate the advantages of the time adaptive approach over the multirate approach and the robustness of the multirate DNWR method with respect to its sibling, the multirate Neumann-Neumann waveform relaxation (NNWR) method introduced in a previous work [3].

Model problemThe unsteady transmission problem reads as follows, where we consider a domain Ω ⊂ R d which is cut into two subdomains Ω = Ω 1 ∪ Ω 2 with transmission conditions at the interface Γ = ∂Ω 1 ∩ ∂Ω 2 :where t ∈ [T 0 , T f ] and n m is the outward normal to Ω m for m = 1, 2. The constants λ 1 and λ 2 are the thermal conductivities on Ω 1 and Ω 2 respectively.

…”

“…We introduce a time adaptive multirate method based on the Dirichlet-Neumann waveform relaxation (DNWR) algorithm for the simulation of two coupled linear heat equations with strong jumps in the material coefficients across the interface. Numerical results are included to illustrate the advantages of the time adaptive approach over the multirate approach and the robustness of the multirate DNWR method with respect to its sibling, the multirate Neumann-Neumann waveform relaxation (NNWR) method introduced in a previous work [3].…”

Quasi‐Newton waveform iteration for partitioned surface‐coupled multiphysics applications

“…[8][9][10] Recently, Neumann-Neumann waveform relaxation was suggested 11 and applied to conjugate heat transfer problems. 12,13 Optimal relaxation parameters are determined for the fully discrete version of this specific problem, leading to superlinear convergence. This was later extended to the time adaptive case.…”