Optimized Schwarz methods for heterogeneous heat transfer problems

TL;DR

This work tackles rapid, robust solution of the heterogeneous heat equation using optimized Schwarz waveform relaxation with nonoverlapping subdomains. By transforming to the Laplace domain, the authors derive a nonlocal optimal transmission operator and then replace it with three practical local approximations (Versions I–III), each framed as a min–max problem over time frequencies to obtain explicit optimal parameters. Numerical experiments in 1D show that the locally scaled transmissions (especially Version III) significantly outperform the simple scaling (Version I) when diffusion coefficients are highly disparate, and maintain stability across time steps and mesh sizes. The results offer a scalable, robust approach for large-scale heterogeneous heat-transfer simulations, with demonstrated relevance to thermal protection system modeling and potential extension to higher dimensions.

Abstract

We present here nonoverlapping optimized Schwarz methods applied to heat transfer problems with heterogeneous diffusion coefficients. After a Laplace transform in time, we derive the error equation and obtain the convergence factor. The optimal transmission operators are nonlocal, and thus inconvenient to use in practice. We introduce three versions of local approximations for the transmission parameter, and provide a detailed analysis at the continuous level in each case to identify the best local transmission conditions. Numerical experiments are presented to illustrate the performance of each local transmission condition. As shown in our analysis, local transmission conditions, which are scaled appropriately with respect to the heterogeneous diffusion coefficients, are more efficient and robust especially when the discontinuity of the diffusion coefficient is large.

Figures (10)

• Figure 1: Illustration of thermal protection systems.
• Figure 2: 2D illustration of the decomposition.
• Figure 3: Illustration of the convergence factor $\rho$ as a function of $\widetilde{\omega}$ with different values of the parameter $p$. Left: $p \in I_c$. Right: $p \in I_r$.
• Figure 4: Illustration of the convergence factor with respect to $\widetilde{\omega}$, when the optimized $p$ and $q$ are obtained.
• Figure 5: Illustration of the left and rights part in \ref{['eq:V3solution']} for $p\in I_p$.

Theorems & Definitions (22)

• Theorem 3.1: Sufficient condition
• proof
• Lemma 3.1: Restrict parameter $p$
• proof
• Lemma 3.2: Local maxima of $\widetilde{\omega}$
• proof
• Theorem 3.2: Optimized transmission parameter: $\mu \leq 2+\sqrt{3}$
• proof
• Theorem 3.3: Optimized transmission parameter: $\mu > 2+\sqrt{3}$
• proof