Vol. 9, No. 4, 2021

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Discrete and nonlocal solutions for the lattice Cattaneo–Vernotte heat diffusion equation

Estefania Nuñez del Prado, Noël Challamel and Vincent Picandet

Vol. 9 (2021), No. 4, 367–396
DOI: 10.2140/memocs.2021.9.367
Abstract

We consider one-dimensional lattice and nonlocal heat diffusion problems. Some experimental, numerical and analytical works have shown significant deviations from Fourier’s law in the case of problems dealing with temporal and spatial microscales. Hence, we propose a study based on a one-dimensional Cattaneo–Vernotte thermal lattice. Some exact analytical solutions based on the method of separation of variables and the use of trigonometric series are formulated for this spatially discrete diffusion problem. The discrete thermal lattice model is compared to a nonlocal continuous Cattaneo–Vernotte model. The length scale for the nonlocal equation is calibrated from the lattice spacing, by applying a continualization method to the lattice heat equation. Then, an error analysis is performed to study the efficiency of the nonlocal model with respect to the local one. It is concluded that the Cattaneo–Vernotte thermal lattice response may be efficiently approximated by a continuous nonlocal Cattaneo–Vernotte heat equation.

Keywords
lattice, thermodynamics, Cattaneo–Vernotte, nonlocal heat equation, heat transfer
Mathematical Subject Classification
Primary: 06B05, 34K31, 35K05, 37K60, 58J35
Milestones
Received: 18 March 2021
Revised: 1 September 2021
Accepted: 26 October 2021
Published: 14 March 2022

Communicated by Martin Ostoja-Starzewski
Authors
Estefania Nuñez del Prado
Institut Dupuy de Lôme, Centre de Recherche
CNRS UMR 6027
Université de Bretagne Sud
Lorient
France
Noël Challamel
Institut Dupuy de Lõme, Centre de Recherche
CNRS UMR 6027
Université de Bretagne Sud
Lorient
France
Vincent Picandet
Institut Dupuy de Lõme, Centre de Recherche
CNRS UMR 6027
Université de Bretagne Sud
Lorient
France