Implicit numerical schemes for generalized heat conduction equations

TL;DR

This paper develops and compares implicit finite difference schemes for the generalized heat conduction equation, addressing non-Fourier heat transfer in complex materials, validated against analytical and finite element solutions.

Contribution

It introduces a shifted field approach for implicit schemes tailored to the Guyer--Krumhansl equation, enhancing numerical treatment of non-Fourier heat conduction.

Findings

Implicit schemes accurately match analytical solutions.

Comparison shows advantages over finite element methods.

Effective for modeling non-Fourier heat transfer phenomena.

Abstract

There are various situations where the classical Fourier's law for heat conduction is not applicable, such as heat conduction in heterogeneous materials or for modeling low-temperature phenomena. In such cases, heat flux is not directly proportional to temperature gradient, hence, the role -- and both the analytical and numerical treatment -- of boundary conditions becomes nontrivial. Here, we address this question for finite difference numerics via a shifted field approach. Based on this ground,implicit schemes are presented and compared to each other for the Guyer--Krumhansl generalized heat conduction equation, which successfully describes numerous beyond-Fourier experimental findings. The results are validated by an analytical solution, and are contrasted to finite element method outcomes obtained by COMSOL.