International Journal of Numerical Methods and Applications

The International Journal of Numerical Methods and Applications publishes research articles on numerical methods and their applications in various fields, including differential equations, fluid dynamics, and bioinformatics. It also welcomes survey articles on new methods in numerical analysis.

Submit Article

FINITE VOLUME APPROXIMATION OF A CLASS OF TWO-DIMENSIONAL PARABOLIC EQUATIONS WITH DISCONTINUOUS AND HIGHLY OSCILLATING COEFFICIENTS

Authors

  • Drainne Jualix Bambi Pemba
  • Bienvenu Ondami

Keywords:

porous media, homogenization, parabolic equation, finite volume method, method of lines, ordinary differential equation (ODE), implicit numerical scheme

DOI:

https://doi.org/10.17654/0975045223003

Abstract

In this paper, we are interested in the numerical approximation of a class of two-dimensional parabolic equations having discontinuous, periodic, and highly oscillating coefficients. We use the method of lines with a finite volume approach to discretize this equation. This discretization leads to an ordinary differential equation (ODE) that we discretize by the Euler implicit scheme. Numerical experiments comparing the obtained solution and the so-called homogenized solution show that the method accuracy depends on the homogenization parameter.

Received: September 28, 2022 
Revised: November 9, 2022 
Accepted: November 19, 2022

References

B. Amaziane, Global Behavior of Compressible Three-phase Flow in Heterogeneous Porous Media, Transport in Porous Media 10 (1993), pp. 45-56.

B. Amaziane and B. Ondami, Numerical approximations of a second order elliptic equation with highly oscillating coefficients, Notes on Numerical Fluid Mechanics 70 (1999), pp. 1-11.

G. Allaire, Introduction to homogenization theory, CEA-EDF-INRIA school on homogenization, 2010.

http://www.cmap.polytechnique.fr/allaire/homog/lect1.pdf

D. J. Bambi Pemba, Sur l’Approximation Numérique par la Méthode des Volumes Finis de Quelques Problèmes d’Ecoulements en Milieux Poreux, Thèse de Doctorat, Université Marien Ngouabi, Congo-Brazzaville, Defense in 2023.

D. J. Bambi Pemba and B. Ondami, Numerical approximation by the method of line with finite-volume approach of a solute transport equation in periodic heterogeneous porous media, Aust. J. Math. Anal. Appl. 17(2) (2020), 18.

Z. Chen and T. Y. Hou, A mixed multiscale finite element method for elliptic problems with oscillating coefficients, Mathematics of Computation 72(242) (2002), 541-576.

Z. Chen, W. Deng and H. Ye, A new upscaling Method for the Solute Transport Equations, Discrete and Continuous Dynamical Systems - Discrete Contin. Dyn. Syst., 13 (2005).

W. Deng and J. G. Jianmin Huang, Upscaling methods for a class of convection-diffusion equation with highly oscillating coefficients, Journal of Computational Physics 227 (2008), 7621-7642.

D. Estep, M. Pernice, D. Pham, S. Tavener and H. Wang, A posteriori error analysis of a cell-centered finite volume method for semilinear elliptic problems, Journal of Computational and Applied Mathematics 233 (2009), 459-472.

R. E. T. Gallouët and R. Herbin, The Finite Volume Methods, Handbook of Numerical Analysis, Techniques of Scientific Computing, Part III, North-Holland, Amsterdam, 2000.

U. Hornung, Homogenization and Porous Media, Texts in Interdisciplinary Applied Mathematics 6, Springer-Verlag, New York, 1997.

V. V. Jikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differentials Operators and Integral Functionals, Springer-Verlag, Berlin, 1994.

Yu. M. Meshkova and T. A. Suslina, Homogenization of initial boundary-value problem for parabolic systems: operator error estimates, Algebra i Analiz 29(6) (2017), 99-158; English transl.: St. Petersburg Math. J. 29(6) (2018), 935-978. https://arxiv.org/pdf/1801.05035.pdf

Yu. M. Meshkova and T. A. Suslina, Homogenization of initial boundary value problems for parabolic systems with periodic coefficients, Applicable Analysis 95(8) (2016), 1736-1775.

P. Knabner and L. Angermann, Numerical Methods for Elliptic and Parabolic Partial Differential Equations, Texts in Applied Mathematics, Springer-Verlag, New York, 2003.

B. Ondami, Finite volume discretization of a class of parabolic equations with discontinuous and oscillating coefficients, Applied Mathematical Sciences 12(3) (2018), 121-135.

B. Ondami, Approximation of a second-order elliptic equation with discontinuous and highly oscillating coefficients by finite volume methods, Journal of Mathematics Research 8(6) (2016), 34-44.

B. Ondami, The effect of numerical integration on the finite element approximation of a second order elliptic equation with highly oscillating coefficients, Journal of Interpolation and Approximation in Scientific Computing 2 (2015), 128-136.

B. Ondami, Sur Quelques Problèmes d’Homogénéisation des Ecoulements en Milieux Poreux, Thèse de Doctorat, Université de Pau et des Pays de l’Adour, France, 2001.

T. A. Suslina, Homogenization of a periodic parabolic Cauchy problem, Nonlinear Equation and Spectral Theory (2007), 201-233.

Downloads

Published

2023-01-07

Issue

Vol. 23 No. 1 (2023)

Section

Articles

License

Copyright (c) 2023 International Journal of Numerical Methods and Applications

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

Licensing Terms:

Attribution: Credit Pushpa Publishing House as the original publisher, including title and author(s) if applicable.

Non-Commercial Use: For non-commercial purposes only. No commercial activities without explicit permission.

No Derivatives: Modifying or creating derivative works not allowed without written permission.

Contact Pushpa Publishing House for more info or permissions.

How to Cite

FINITE VOLUME APPROXIMATION OF A CLASS OF TWO-DIMENSIONAL PARABOLIC EQUATIONS WITH DISCONTINUOUS AND HIGHLY OSCILLATING COEFFICIENTS. (2023). International Journal of Numerical Methods and Applications, 23(1), 51-65. https://doi.org/10.17654/0975045223003

Similar Articles

11-20 of 50

You may also start an advanced similarity search for this article.