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Numerical analysis of a dual‐phase‐lag model involving two temperatures
Author(s) -
Bazarra Noelia,
Fernández José R.,
Magaña Antonio,
Quintanilla Ramón
Publication year - 2019
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.6082
Subject(s) - mathematics , backward euler method , discretization , uniqueness , partial differential equation , mathematical analysis , convergence (economics) , finite element method , numerical analysis , physics , economics , thermodynamics , economic growth
In this paper, we numerically analyse a phase‐lag model with two temperatures which arises in the heat conduction theory. The model is written as a linear partial differential equation of third order in time. The variational formulation, written in terms of the thermal acceleration, leads to a linear variational equation, for which we recall an existence and uniqueness result and an energy decay property. Then, using the finite element method to approximate the spatial variable and the implicit Euler scheme to discretize the time derivatives, fully discrete approximations are introduced. A discrete stability property is proved, and a priori error estimates are obtained, from which the linear convergence of the approximation is derived. Finally, some one‐dimensional numerical simulations are described to demonstrate the accuracy of the approximation and the behaviour of the solution.