A multigrid waveform relaxation method for solving the nonlinear silicon problem with relaxing boundary conditions

Abstract

This paper introduces a multigrid FAS Waveform Relaxation method (FAS-MGWR) for solving a heat transfer model in a thin and homogeneous silicon bar with constant density and heat capacity. This method exhibits versatility, making it applicable to a range of problems including electronic device engineering, numerical simulation of nanofluids, among others. The Finite Difference Method with central differences (CDS) for spatial discretization and the Crank-Nicolson method for temporal approximation were utilized. Comparison with literature results and code verification demonstrated that irrespective of the combinations of physical and numerical parameters, the apparent order of discretization error converges to the theoretical asymptotic order. The study underscores the superior performance of the proposed FAS-MGWR method, notable for its parallel architecture, particularly in terms of computational time, compared to existing literature. Notably, the FAS-MGWR method was found to have excellent convergence factor and speed-up in relation to its singlegrid version, underscoring its efficiency and practical advantage in addressing complex thermal problems.

Keywords:

• Convective
• convergence factor
• Dirichlet
• full approximation scheme
• speed-up

Acknowledgments

The authors extend their gratitude to Stefan M. Filipov, István Faragó, and Ana Avdzhieva for generously providing their code, which was instrumental in facilitating comparisons with the work conducted in this study.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This study was funded partly by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), Brazil–Finance Code 001.