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Proper Generalized Decomposition using Taylor expansion for non-linear diffusion equations

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  • Deeb, Ahmad
  • Kalaoun, Omar
  • Belarbi, Rafik

Abstract

For a physical problem described by a parameterized mathematical model, different configurations of the problem require computing the solution over a range of parameters in order to study the phenomenon when parameters change. In other words, it is a process of looking for a continuum of solutions of the equation, relative to these parameters, in order to find the ones that fit the experimental data. However, solving a direct problem for each parametric configuration will generate a cascade of direct problems, which will cost a huge amount of time, especially when we deal with non-linear equations. Therefore, the parametric solution is a suitable alternative strategy to compute the solution of the equation. In this paper, we will use the Proper Generalized Decomposition (PGD) method to solve non-linear diffusion equations and produce parametric solutions. To treat the non-linear functions, we will not use the Discrete Empirical Interpolation Methods (DEIM), which has proven their utility, but the non-linear terms will be replaced by their Taylor series expansion up to an order m. This will produce a new model, which we call here the ”developed equation” and therefore the PGD is applied on. Polynomial equations appear for each tensor element computation. While space and time tensor elements’ equations are to be solved using Finite Elements Methods (FEM) and Borel–Padé–Laplace (BPL) integrator respectively, Newton solver is used for tensors relative to the parameters’ equations. Here, rational polynomial functions arise for parametric tensor elements, which are known to extrapolate solutions. Numerical simulations are done for a non-linear diffusion equation with exponential diffusion coefficient as first trial, and with a magnetic diffusion coefficient as a second one.

Suggested Citation

  • Deeb, Ahmad & Kalaoun, Omar & Belarbi, Rafik, 2023. "Proper Generalized Decomposition using Taylor expansion for non-linear diffusion equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 208(C), pages 71-94.
    • Handle: RePEc:eee:matcom:v:208:y:2023:i:c:p:71-94
    DOI: 10.1016/j.matcom.2023.01.008
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    References listed on IDEAS

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    1. Deeb, Ahmad & Hamdouni, Aziz & Razafindralandy, Dina, 2022. "Performance of Borel–Padé–Laplace integrator for the solution of stiff and non-stiff problems," Applied Mathematics and Computation, Elsevier, vol. 426(C).
    2. Theeraek, P. & Phongthanapanich, S. & Dechaumphai, P., 2011. "Solving convection-diffusion-reaction equation by adaptive finite volume element method," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 82(2), pages 220-233.
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    4. Gambino, G. & Lombardo, M.C. & Sammartino, M., 2012. "Turing instability and traveling fronts for a nonlinear reaction–diffusion system with cross-diffusion," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 82(6), pages 1112-1132.
    5. Amin, Fahs & Zakeri, Ali & Wanko, Adrien, 2021. "Time-dependent solution for natural convection in a porous enclosure using the Darcy–Lapwood–Brinkman model," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 182(C), pages 39-65.
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