Analysis of operator splitting in the non-asymptotic regime for nonlinear reaction-diffusion equations. Application to the dynamics of premixed flames

Stéphane Descombes (1) , Max Duarte (2, 3) , Thierry Dumont (4) , Frédérique Laurent (5) , Violaine Louvet (4) , Marc Massot (5)

1 JAD - Laboratoire Jean Alexandre Dieudonné
2 LBNL - Lawrence Berkeley National Laboratory [Berkeley]
3 LJAD - Laboratoire Jean Alexandre Dieudonné
4 MMCS - Modélisation mathématique, calcul scientifique
5 EM2C - Laboratoire d'Énergétique Moléculaire et Macroscopique, Combustion

In this paper we mathematically characterize through a Lie formalism the local errors induced by operator splitting when solving nonlinear reaction-diffusion equations, especially in the non-asymptotic regime. The non-asymptotic regime is often attained in practice when the splitting time step is much larger than some of the scales associated with either source terms or the diffusion operator when large gradients are present. In a series of previous works a reduction of the asymptotic orders for a range of large splitting time steps related to very short time scales in the nonlinear source term has been studied, as well as that associated with large gradients but for linearized equations. This study provides a key theoretical step forward since it characterizes the numerical behavior of splitting errors within a more general nonlinear framework, for which new error estimates can be derived by coupling Lie formalism and regularizing effects of the heat equation. The validity of these theoretical results is then assessed in the framework of two numerical applications, a KPP-type reaction wave where the influence of stiffness on local error estimates can be thoroughly investigated; and a much more complex problem, related to premixed flame dynamics in the low Mach number regime with complex chemistry and detailed transport, for which the present theoretical study shows to provide relevant insights.

Mots clés: operator splitting; error bounds; reaction-diffusion; traveling waves; combustion

Domaines: Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]; Milieux fluides et réactifs

Stéphane Descombes, Max Duarte, Thierry Dumont, Frédérique Laurent, Violaine Louvet, et al.. Analysis of operator splitting in the non-asymptotic regime for nonlinear reaction-diffusion equations. Application to the dynamics of premixed flames. SIAM Journal on Numerical Analysis, 2014, 52 (3), pp.1311-1334. ⟨10.1137/130926006⟩. ⟨hal-00837089v4⟩