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In this paper we mathematically characterize through a Lie formalism thelocal errors induced by operator splitting when solving nonlinearreaction-diffusion equations, especially in the non-asymptotic regime. Thenon-asymptotic regime is often attained in practice when the splitting timestep is much larger than some of the scales associated with either source termsor the diffusion operator when large gradients are present. In a series ofprevious works a reduction of the asymptotic orders for a range of largesplitting time steps related to very short time scales in the nonlinear sourceterm has been studied, as well as that associated with large gradients but forlinearized equations. This study provides a key theoretical step forward sinceit characterizes the numerical behavior of splitting errors within a moregeneral nonlinear framework, for which new error estimates can be derived bycoupling Lie formalism and regularizing effects of the heat equation. Thevalidity of these theoretical results is then assessed in the framework of twonumerical applications, a KPP-type reaction wave where the influence ofstiffness on local error estimates can be thoroughly investigated; and a muchmore complex problem, related to premixed flame dynamics in the low Mach numberregime with complex chemistry and detailed transport, for which the presenttheoretical study shows to provide relevant insights.

siam journal on numerical analysis

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