Numerical analysis of the Maxwell-Cattaneo-Vernotte nonlinear model

TL;DR

This paper explores the nonlinear properties of the Maxwell-Cattaneo-Vernotte heat equation, focusing on the impact of temperature-dependent transport coefficients and introducing a stable numerical scheme for solving such equations.

Contribution

It presents an analysis of the nonlinear behavior of the Maxwell-Cattaneo-Vernotte model with temperature-dependent coefficients and proposes a reliable implicit numerical method.

Findings

Temperature dependence significantly affects heat wave propagation.

The proposed scheme is stable and accurately captures nonlinear effects.

Results demonstrate the impact on various initial-boundary conditions.

Abstract

In the literature, one can find numerous modifications of Fourier's law from which the first one is called Maxwell-Cattaneo-Vernotte heat equation. Although this model has been known for decades and successfully used to model low-temperature damped heat wave propagation, its nonlinear properties are rarely investigated. In this paper, we aim to present the functional relationship between the transport coefficients and the consequences of their temperature dependence. Furthermore, we introduce a particular implicit numerical scheme in order to solve such nonlinear heat equations reliably. We investigate the scheme's stability, dissipation, and dispersion attributes as well. We demonstrate the effect of temperature-dependent thermal conductivity on two different initial-boundary value problems, including time-dependent boundaries and heterogeneous initial conditions.