非渐近 regime 中非线性反应扩散方程中算子分裂的分析。预混火焰动力学的应用
Analysis of operator splitting in the non-asymptotic regime for nonlinear reaction-diffusion equations. Application to the dynamics of premixed flames
AI总结：
本文通过李形式法分析非线性反应扩散方程中算子分裂的局部误差，特别是在非渐近 regime 中。通过两个应用验证理论结果：KPP型反应波和预混火焰动力学。
摘要：
本文通过李形式法数学地刻画了在求解非线性反应扩散方程时，算子分裂引起的局部误差。非渐近 regime 通常出现在分裂时间步远大于源项或扩散算子相关尺度时，尤其是在大梯度存在的情况下。在一系列先前工作中，研究了与非常短时间尺度相关的非渐近阶数的减少，以及与大梯度相关的线性化方程。本研究提供了关键的理论进展，因为它在一个更一般的非线性框架中表征了分裂误差的数值行为，通过结合李形式法和热方程的正则化效应可以推导出新的误差估计。这些理论结果的有效性通过两个数值应用框架验证：KPP型反应波，其中可彻底研究刚性对局部误差估计的影响；以及更复杂的问题，与低马赫数预混火焰动力学相关，其中本理论研究显示出提供相关见解。
英文摘要：
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.
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