Non linear schemes for the heat equation in 1D
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 1, p. 107-134
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Inspired by the growing use of non linear discretization techniques for the linear diffusion equation in industrial codes, we construct and analyze various explicit non linear finite volume schemes for the heat equation in dimension one. These schemes are inspired by the Le Potier's trick [C. R. Acad. Sci. Paris, Ser. I 348 (2010) 691-695]. They preserve the maximum principle and admit a finite volume formulation. We provide a original functional setting for the analysis of convergence of such methods. In particular we show that the fourth discrete derivative is bounded in quadratic norm. Finally we construct, analyze and test a new explicit non linear maximum preserving scheme with third order convergence: it is optimal on numerical tests.

DOI : https://doi.org/10.1051/m2an/2013096
Classification:  65J05,  65M08,  65M12
Keywords: finite volume schemes, heat equation, non linear correction
@article{M2AN_2014__48_1_107_0,
     author = {Despr\'es, Bruno},
     title = {Non linear schemes for the heat equation in 1D},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {48},
     number = {1},
     year = {2014},
     pages = {107-134},
     doi = {10.1051/m2an/2013096},
     zbl = {1292.65098},
     mrnumber = {3177839},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2014__48_1_107_0}
}
Després, Bruno. Non linear schemes for the heat equation in 1D. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 1, pp. 107-134. doi : 10.1051/m2an/2013096. http://www.numdam.org/item/M2AN_2014__48_1_107_0/

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