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non linear correction

ESAIM: Mathematical Modelling and Numerical Analysis (1)

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Non linear schemes for the heat equation in 1D ∗
Bruno Després
Journal:

ESAIM: Mathematical Modelling and Numerical Analysis / Volume 48 / Issue 1 / January 2014

Published online by Cambridge University Press:

18 December 2013, pp. 107-134

Print publication:

January 2014
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Inspired by the growing use of non linear discretization techniques for the lineardiffusion equation in industrial codes, we construct and analyze various explicit nonlinear finite volume schemes for the heat equation in dimension one. These schemes areinspired by the Le Potier’s trick [C. R. Acad. Sci. Paris, Ser. I348 (2010) 691–695]. They preserve the maximum principle and admita finite volume formulation. We provide a original functional setting for the analysis ofconvergence of such methods. In particular we show that the fourth discrete derivative isbounded in quadratic norm. Finally we construct, analyze and test a new explicit nonlinear maximum preserving scheme with third order convergence: it is optimal on numericaltests.