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.

在本文中，我们提出并分析了涉及质量作用定律并具有详细平衡条件的某类反应扩散系统的保正性，能量稳定的数值方案。数值方案是基于最近开发的高能变分公式构建的，其中根据反应轨迹对反应部分进行了重新公式化。反应和扩散部分都耗散相同的自由能这一事实为设计用于这些系统的能量稳定的，操作员分配方案开辟了道路。在反应阶段，我们通过对所有对数项进行凸形式化处理，从而以隐式形式对它们进行隐式处理，从而解决了反应轨迹的方程式。从理论上可以证明其保持正性和独特的可溶性，基于对数函数在极限值附近的奇异行为。此外，通过仔细的凸度分析可以证明该方案在反应阶段的能量稳定性。类似的技术被用来为标准的半隐式求解器在扩散阶段建立正性和能量稳定性。结果，这两个阶段的组合导致原始反应扩散系统的保正性和能量稳定的数值方案。给出了几个数值示例，以证明所提出的算子拆分方案的鲁棒性。类似的技术被用来为标准的半隐式求解器在扩散阶段建立正性和能量稳定性。结果，这两个阶段的组合导致原始反应扩散系统的保正性和能量稳定的数值方案。给出了几个数值示例，以证明所提出的算子拆分方案的鲁棒性。类似的技术被用来为标准的半隐式求解器在扩散阶段建立正性和能量稳定性。结果，这两个阶段的组合导致原始反应扩散系统的保正性和能量稳定的数值方案。给出了几个数值示例，以证明所提出的算子拆分方案的鲁棒性。