Three mathematical representations and an improved ADI method for hyperbolic heat conduction

Abstract

Hyperbolic heat conduction models have been proposed to characterize the breakdown of Fourier’s law, i.e. thermal waves. In this paper, three mathematical representations for hyperbolic heat conduction, namely temperature representation, hybrid representation and heat flux representation, and their corresponding characteristics are first analyzed. The hybrid representation is demonstrated to be preferable for numerical calculations and contains sufficient heat transport information. Specifically designed for solving transient heat conduction problems in the hybrid representation, an improved alternative direction implicit (ADI) method based on staggered grids is then developed. This algorithm focuses on the entire hyperbolic equation set instead of one single hyperbolic equation, and it adopts chasing method rather than iteration, which enables to significantly save computing time and storage space. Characteristics analyses on the definite conditions show that for each side only one boundary condition is necessary for Cattaneo-Vernotte (CV) type hyperbolic heat conduction. The advantages of the hybrid representation are also demonstrated by numerical simulations. Besides, the initial heat flux, which implies the initial phonon momentum in dielectrics, has an important influence on the propagation patterns of thermal waves, changing the way of energy conveying. The mechanism of phonon momentum conservation leads to the vector characteristics of thermal waves, and causes the direction preference in hyperbolic heat conduction.