A structure-preserving, operator splitting scheme for reaction-diffusion equations with detailed balance
A structure-preserving, operator splitting scheme for reaction-diffusion equations with detailed balance
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DOI：
10.1016/j.jcp.2021.110253
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发表时间：
2021-07
期刊：
影响因子：
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通讯作者：
Chun Liu;Cheng Wang;Yiwei Wang
中科院分区：
文献类型：
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作者：
Chun Liu;Cheng Wang;Yiwei Wang
在本文中，我们提出并分析了一类涉及质量作用量定律的反应扩散方程组的正性守恒、能量稳定的数值格式。数值格式是基于最近发展的能量变分公式，其中反应部分根据反应轨迹进行了重新表述。反应部分和扩散部分都消耗相同的自由能，这一事实为为这些系统设计能量稳定的算符分裂方案开辟了一条道路。在反应阶段，由于对数项的凸性，我们通过隐含地处理所有重新公式形式中的对数项来求解反应轨迹方程。基于对数函数在极限值附近的奇异行为，从理论上证明了对数函数的保正性和唯一性。此外，通过仔细的凸性分析，可以证明该格式在反应阶段的能量稳定性。类似的技巧被用来建立扩散阶段标准半隐式求解器的正性保持性质和能量稳定性。因此，这两个阶段的结合导致了一个正性守恒和能量稳定的原始反应扩散系统的数值格式。给出了几个数值算例，验证了所提出的算子分裂格式的稳健性。
In this paper, we propose and analyze a positivity-preserving, energy stable numerical scheme for a certain type of reaction-diffusion systems involving the Law of Mass Action with the detailed balance condition. The numerical scheme is constructed based on a recently developed energetic variational formulation, in which the reaction part is reformulated in terms of reaction trajectories. The fact that both the reaction and diffusion parts dissipate the same free energy opens a path of designing an energy stable, operator splitting scheme for these systems. At the reaction stage, we solve equations of reaction trajectories by treating all the logarithmic terms in the reformulated form implicitly due to their convex nature. The positivity-preserving property and unique solvability can be theoretically proved, based on the singular behavior of the logarithmic function around the limiting value. Moreover, the energy stability of this scheme at the reaction stage can be proved by a careful convexity analysis. Similar techniques are used to establish the positivity-preserving property and energy stability for the standard semi-implicit solver at the diffusion stage. As a result, a combination of these two stages leads to a positivity-preserving and energy stable numerical scheme for the original reaction-diffusion system. Several numerical examples are presented to demonstrate the robustness of the proposed operator splitting scheme.