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arXiv 2105.09415math.NAcs.NA
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具有详细平衡的反应扩散方程组变分算子分裂格式的收敛性分析

Convergence analysis of the variational operator splitting scheme for a reaction-diffusion system with detailed balance

Chun Liu, Cheng Wang, Yiwei Wang, Steven M. Wise

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摘要：

本文对文[C. Liu等人，J. Comput.物理、436，110253，2021]对于具有详细平衡的反应扩散系统。数值方案已被构造基于最近开发的能量变分制剂，其中反应部分被重新制定的反应轨迹，反应和扩散部分耗散相同的自由能。该方案是能量稳定和保正的。本文对算子分裂格式进行了详细的收敛性分析和误差估计。反应轨迹方程中的非线性，以及非线性和奇异对数项的隐式处理，给数值分析带来了挑战。为了克服这些困难，我们利用的对数非线性项，这是隐含在化学反应阶段处理的凸性质。此外，粗略的误差估计和精细的误差估计的组合导致在离散最大范数中的反应阶段中的数值误差的期望界限。此外，离散的最大值原理产生的数值误差函数在扩散阶段的发展范围。作为一个直接的结果，结合不同阶段的数值误差分析和算子分裂的一致性估计，得到了全反应扩散方程组数值格式的收敛性估计.

英文摘要：

We present a detailed convergence analysis for an operator splitting scheme proposed in [C. Liu et al.,J. Comput. Phys., 436, 110253, 2021] for a reaction-diffusion system with detailed balance. The numerical scheme has been constructed based on a recently developed energetic variational formulation, in which the reaction part is reformulated in terms of the reaction trajectory, and both the reaction and diffusion parts dissipate the same free energy. The scheme is energy stable and positivity-preserving. In this paper, the detailed convergence analysis and error estimate are performed for the operator splitting scheme. The nonlinearity in the reaction trajectory equation, as well as the implicit treatment of nonlinear and singular logarithmic terms, impose challenges in numerical analysis. To overcome these difficulties, we make use of the convex nature of the logarithmic nonlinear terms, which are treated implicitly in the chemical reaction stage. In addition, a combination of a rough error estimate and a refined error estimate leads to a desired bound of the numerical error in the reaction stage, in the discrete maximum norm. Furthermore, a discrete maximum principle yields the evolution bound of the numerical error function at the diffusion stage. As a direct consequence, a combination of the numerical error analysis at different stages and the consistency estimate for the operator splitting results in the convergence estimate of the numerical scheme for the full reaction-diffusion system.

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