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Volume 39, Issue 3
Convergence of Numerical Schemes for a Conservation Equation with Convection and Degenerate Diffusion

R. Eymard, C. Guichard & Xavier Lhébrard

DOI: 10.4208/jcm.2002-m2018-0287

J. Comp. Math., 39 (2021), pp. 428-452.

Published online: 2021-04

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  • Abstract

The approximation of problems with linear convection and degenerate nonlinear diffusion, which arise in the framework of the transport of energy in porous media with thermodynamic transitions, is done using a $θ$-scheme based on the centred gradient discretisation method. The convergence of the numerical scheme is proved, although the test functions which can be chosen are restricted by the weak regularity hypotheses on the convection field, owing to the application of a discrete Gronwall lemma and a general result for the time translate in the gradient discretisation setting. Some numerical examples, using both the Control Volume Finite Element method and the Vertex Approximate Gradient scheme, show the role of $θ$ for stabilising the scheme.

  • Keywords

Linear convection, Degenerate diffusion, Gradient discretisation method, $θ$-scheme.

  • AMS Subject Headings

65N30, 35K65.

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

robert.eymard@u-pem.fr (R. Eymard)

cindy.guichard@sorbonne-universite.fr (C. Guichard)

xavier.lhebrard@ens-rennes.fr (Xavier Lhébrard)

  • BibTex
  • RIS
  • TXT
@Article{JCM-39-428, author = {Eymard , R.Guichard , C. and Lhébrard , Xavier}, title = {Convergence of Numerical Schemes for a Conservation Equation with Convection and Degenerate Diffusion}, journal = {Journal of Computational Mathematics}, year = {2021}, volume = {39}, number = {3}, pages = {428--452}, abstract = {

The approximation of problems with linear convection and degenerate nonlinear diffusion, which arise in the framework of the transport of energy in porous media with thermodynamic transitions, is done using a $θ$-scheme based on the centred gradient discretisation method. The convergence of the numerical scheme is proved, although the test functions which can be chosen are restricted by the weak regularity hypotheses on the convection field, owing to the application of a discrete Gronwall lemma and a general result for the time translate in the gradient discretisation setting. Some numerical examples, using both the Control Volume Finite Element method and the Vertex Approximate Gradient scheme, show the role of $θ$ for stabilising the scheme.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2002-m2018-0287}, url = {http://global-sci.org/intro/article_detail/jcm/18745.html} }
TY - JOUR T1 - Convergence of Numerical Schemes for a Conservation Equation with Convection and Degenerate Diffusion AU - Eymard , R. AU - Guichard , C. AU - Lhébrard , Xavier JO - Journal of Computational Mathematics VL - 3 SP - 428 EP - 452 PY - 2021 DA - 2021/04 SN - 39 DO - http://doi.org/10.4208/jcm.2002-m2018-0287 UR - https://global-sci.org/intro/article_detail/jcm/18745.html KW - Linear convection, Degenerate diffusion, Gradient discretisation method, $θ$-scheme. AB -

The approximation of problems with linear convection and degenerate nonlinear diffusion, which arise in the framework of the transport of energy in porous media with thermodynamic transitions, is done using a $θ$-scheme based on the centred gradient discretisation method. The convergence of the numerical scheme is proved, although the test functions which can be chosen are restricted by the weak regularity hypotheses on the convection field, owing to the application of a discrete Gronwall lemma and a general result for the time translate in the gradient discretisation setting. Some numerical examples, using both the Control Volume Finite Element method and the Vertex Approximate Gradient scheme, show the role of $θ$ for stabilising the scheme.

Eymard , R.Guichard , C. and Lhébrard , Xavier. (2021). Convergence of Numerical Schemes for a Conservation Equation with Convection and Degenerate Diffusion. Journal of Computational Mathematics. 39 (3). 428-452. doi:10.4208/jcm.2002-m2018-0287
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