  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.
