Mathematical analysis and numerical simulation of the Guyer-Krumhansl heat equation

TL;DR

This paper investigates the mathematical properties and numerical solutions of the Guyer-Krumhansl heat equation, highlighting its potential as a standard model for heat conduction in heterogeneous materials.

Contribution

It provides a thorough mathematical analysis, including well-posedness and stability, of the Guyer-Krumhansl heat equation, advancing its applicability in engineering contexts.

Findings

Proved the well-posedness of key initial-boundary value problems.

Analyzed the stability of solutions using finite difference methods.

Compared the Guyer-Krumhansl model to the classical Fourier heat equation.

Abstract

The Guyer-Krumhansl heat equation has numerous important practical applications in both low-temperature and room temperature heat conduction problems. In recent years, it turned out that the Guyer-Krumhansl model can effectively describe the thermal behaviour of macroscale heterogeneous materials. Thus, the Guyer-Krumhansl equation is a promising candidate to be the next standard model in engineering. However, to support the Guyer-Krumhansl equation's introduction into the engineering practice, its mathematical properties must be thoroughly investigated and understood. In the present paper, we show the basic structure of this particular heat equation, focusing on the differences in comparison to the Fourier heat equation {obtained when $(\tau_{q}, \mu^{2})\rightarrow (0,0)$}. Additionally, we prove the well-posedness of a particular, practically significant initial and boundary value problem. The stability of the solution is also investigated in the discrete space using a finite difference approach.