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Operator splitting for nonlinear reaction-diffusion systems with an entropic structure : singular perturbation and order reduction

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In this paper, we perform the numerical analysis of operator splitting techniques for nonlinear reaction-diffusion systems with an entropic structure in the presence of fast scales in the reaction term. We consider both linear diagonal and quasi-linear non-diagonal diffusion; the entropic structure implies the well-posedness and stability of the system as well as a Tikhonov normal form for the nonlinear reaction term [23]. It allows to perform a singular perturbation analysis and to obtain a reduced and well-posed system of equations on a partial equilibrium manifold as well as an asymptotic expansion of the solution. We then conduct an error analysis in this particular framework where the time scale associated to the fast part of the reaction term is much shorter that the splitting time step Δt thus leading to the failure of the usual splitting analysis techniques. We define the conditions on diffusion and reaction for the order of the local error associated with the time splitting to be reduced or to be preserved in the presence of fast scales. All the results obtained theoretically on local error estimates are then illustrated on a numerical test case where the global error clearly reproduces the scenarios foreseen at the local level. We finally investigate the discretization of the corresponding problems and its influence on the splitting error in terms of the previously conducted numerical analysis.

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Authors and Affiliations

  1. UMPA, UMR 5669 - CNRS, Unité de Mathématiques Pures et Appliquées, Ecole Normale Supérieure de Lyon, 46 Allée d’Italie, 69364, Lyon Cedex 07, France

    Stéphane Descombes

  2. CNRS - MAPLY, UMR 5585, Laboratoire de Mathématiques Appliquées de Lyon, Université, Claude Bernard, Lyon 1, 69622, Villeurbanne Cedex, France

    Marc Massot

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  1. Stéphane Descombes
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  2. Marc Massot
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Correspondence to Stéphane Descombes.

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Mathematics Subject Classification (2000): 65M12, 35K57, 35B25, 35Q80, 34E15, 80A32, 92E20

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Descombes, S., Massot, M. Operator splitting for nonlinear reaction-diffusion systems with an entropic structure : singular perturbation and order reduction. Numer. Math. 97, 667–698 (2004). https://doi.org/10.1007/s00211-003-0496-3

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  • DOI: https://doi.org/10.1007/s00211-003-0496-3

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