Optimized Schwarz waveform relaxation for heterogeneous Cattaneo-Vernotte non-Fourier heat transfer

Abstract

The non-Fourier heat transfer in heterogeneous media is crucial for material science and biomedical engineering. The optimized Schwarz waveform relaxation (OSWR) method is an efficient approach for solving such problems due to its divide-and-conquer strategy. Despite the wave-type nature of non-Fourier heat transfer, the short phase-lag time leads to more parabolic-like behavior. To address this, in the OSWR method, we employed Robin boundary conditions to transmit information along the interface. Using Fourier analysis, we derived and rigorously optimized the convergence factors of the OSWR algorithm with scaled Robin and Robin-Robin transmission conditions. The resulting optimized transmission parameters were provided in explicit form for direct application in the OSWR algorithm, along with corresponding convergence factor estimates. Interestingly, the results show that a larger heterogeneity contrast actually accelerates the convergence, rather than deteriorating it. Furthermore, the OSWR algorithm with the Robin-Robin condition exhibits mesh-independent convergence asymptotically. However, the presence of the phase-lag time is found to slow down the convergence of the OSWR algorithm. These theoretical findings were validated through numerical experiments.

Keywords

• cattaneo-vernotte equation
• optimized schwarz waveform relaxation
• heterogeneous media
• robin transmission condition
• non-fourier heat transfer