Discrete heat conduction equation: Dispersion analysis and continuous limits

Abstract

The dispersion relation in ω-k space has been derived and analyzed for discrete heat conduction equation (DE). The DE is inherently nonlocal both in time and space and can be used to describe non-Fourier heat transport occurring on ultrashort time and length scales. An important distinction between the DE and the continuum description is that the frequencies and wave numbers available to the discrete system are limited, whereas in the continuum description there are no such limitations. Analytical expressions for the attenuation distance, group and phase velocities have been obtained and analyzed as functions of frequency. In the continuum limit, depending on the basic invariant of the continualization procedure, the dispersion relation for the DE reduces to either dispersion relation for the classical Fourier parabolic equation (PE) or to dispersion relation for the hyperbolic heat equation (HE). However, at the moderate and high frequencies the spectrum of the DE differs significantly from the spectra of the PE and HE. This implies that the continuum-based approaches, including both the PE and HE, unable to describe high-frequency and short-wavelength processes and the DE is preferable. This work provides a relatively simple analytical tool to interpret the transient wave-like behavior of heat transport on ultrashort space and time scales.