Non linear schemes for the heat equation in 1D

Bruno Després

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2014)

  • Volume: 48, Issue: 1, page 107-134
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Després, Bruno. "Non linear schemes for the heat equation in 1D." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 48.1 (2014): 107-134. <http://eudml.org/doc/273136>.

@article{Després2014,
abstract = {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.},
author = {Després, Bruno},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {finite volume schemes; heat equation; non linear correction; nonlinear correction; numerical examples; Le Potier correction method; maximum preserving; Courant-Friedrichs-Lewy (CFL) condition; convergence; stability},
language = {eng},
number = {1},
pages = {107-134},
publisher = {EDP-Sciences},
title = {Non linear schemes for the heat equation in 1D},
url = {http://eudml.org/doc/273136},
volume = {48},
year = {2014},
}

TY - JOUR
AU - Després, Bruno
TI - Non linear schemes for the heat equation in 1D
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2014
PB - EDP-Sciences
VL - 48
IS - 1
SP - 107
EP - 134
AB - 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.
LA - eng
KW - finite volume schemes; heat equation; non linear correction; nonlinear correction; numerical examples; Le Potier correction method; maximum preserving; Courant-Friedrichs-Lewy (CFL) condition; convergence; stability
UR - http://eudml.org/doc/273136
ER -

References

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