Thermodynamically consistent initialization of the Maxwell--Cattaneo---Vernotte heat conduction model: Analytical solutions and engineering applications

Abstract

This paper investigates the numerical initialization of the one-dimensional Maxwell--Cattaneo--Vernotte (MCV) heat conduction model, addressing the critical mathematical challenge of defining the initial time derivative for non-equilibrium states. In modern high-frequency thermal engineering applications, traditional time-integration schemes and commercial finite-element solvers frequently introduce severe numerical artifacts when handling non-Fourier models. This study systematically evaluates three initialization strategies: a zero derivative, a spatially uniform non-zero derivative, and an exact space-dependent derivative. Using an explicit staggered finite-difference scheme, the transient responses to an exponentially distributed initial temperature field under adiabatic boundary conditions are compared against an analytical solution using the Galerkin method. The results demonstrate that assuming a zero or spatially uniform initial derivative introduces significant unphysical oscillatory deviations, leading to heat-flux prediction errors as the relaxation time increases. Conversely, mapping the exact space-dependent derivative onto the staggered grid preserves the local thermodynamic structure of the initial state, yielding robust transient responses that match the analytical benchmark without meaningful computational overhead. These findings establish a thermodynamically consistent initialization technique that can be extended to non-local models as well.