Maxwell-Cattaneo Regularization of Heat Equation

Authors: F. Ekoue, A. Fouache d'Halloy, D. Gigon, G Plantamp, E. Zajdman

Abstract:

This work focuses on analysis of classical heat transfer equation regularized with Maxwell-Cattaneo transfer law. Computer simulations are performed in MATLAB environment. Numerical experiments are first developed on classical Fourier equation, then Maxwell-Cattaneo law is considered. Corresponding equation is regularized with a balancing diffusion term to stabilize discretizing scheme with adjusted time and space numerical steps. Several cases including a convective term in model equations are discussed, and results are given. It is shown that limiting conditions on regularizing parameters have to be satisfied in convective case for Maxwell-Cattaneo regularization to give physically acceptable solutions. In all valid cases, uniform convergence to solution of initial heat equation with Fourier law is observed, even in nonlinear case.

Keywords: Maxwell-Cattaneo heat transfers equations, fourierlaw, heat conduction, numerical solution.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1331145

References:

[1] V. P. Vaslov, "The existence of a solution of an ill-posed problem is equivalent to the convergence of a regularization process", Uspekhi Mat. Nauk, vol. 23, issue 3(141), pp. 183-184, 1968.
[2] J. Fourier, "The analytic theory of heat", "Théorie analytique de la chaleur", Paris : Firmin Didot Père et Fils, 1822.
[3] C. I. Christov and P. M. Jordan, "Heat Conduction Paradox Involving Second-Sound Propagation in Moving Media", Physical review letters, vol. 94, no. 15, Apr. 2005.
[4] C. Cattaneo, "On a form of heat equation which eliminates the paradox of instantaneous propagation", C. R. Acad. Sci. Paris, pp. 431 - 433, July 1958.
[5] P. Vernotte, " Les paradoxes de la théorie continue de l'équation de la chaleur",C. R. Acad. Sci. Paris, Vol. 246, pp. 3154-3155, 1958.
[6] Y. Taitel, "On the Parabolic, Hyperbolic, and Discrete formulations of the heat conduction equation", Pergamon Press, Vol. 15, pp. 369-371, 1972.
[7] B. Bertman and D. J. Sandiford, "Second sound in solid helium", Scient. Am., vol. 222, no. 5, pp. 92-101, 1970.
[8] M. Zhang, "A relationship between the periodic and the Dirichlet BVPs of singular differential equations", Proceedings of the Royal Society of Edinburgh, Section A Mathematics, vol. 128, pp. 1099-1114, 1998.