In this paper, we propose and analyze a positivity-preserving, energy stable numerical scheme for certain type 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, and both the reaction and the diffusion parts dissipate the same free energy. This subtle fact opens a path of an 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. To our best knowledge, it is the first time to report an energy-dissipation-law-based operator splitting scheme to a nonlinear PDE with variational structures. Several numerical examples are presented to demonstrate the robustness of the proposed operator splitting scheme.
翻译:在本文中, 我们提议并分析一个符合质量行动法的详细平衡条件的假设- 保存、 能源稳定的数字方案。 数字方案是根据最近开发的动态变异配方构建的, 反应部分以反应轨迹重写, 反应部分和传播部分以相同的自由能量消散。 这个微妙的事实为这些系统的操作员分解计划开辟了一条路径。 在反应阶段, 我们通过处理重新拟订时的所有对数术语, 解决反应轨迹的方程式。 由于它们的性质, 隐含地处理重新拟订时的所有对数。 假设- 保留属性和独特的溶解性可以理论上证明, 其依据是围绕限制值的对数函数的奇异行为。 此外, 反应阶段的这种方案能源稳定性可以通过仔细的调和分析来证明。 类似的技术用于在扩散阶段为标准的半不精确解算法解调和能源稳定地等方程式。 作为结果, 这两个阶段的假设- 预言- 预言- 预言- 预言- 预言- 预言- 数据- 预言- 预言- 预言- 预言- 预言- 预言- 预言- 预言- 预算- 理论- 理论- 预言- 预制- 预算- 理论- 理论- 理论- 理论- 理论- 理论- 理论- 理论- 理论- 理论- 理论- 理论- 理论- 理论- 和- 理论- 理论- 理论- 理论- 理论- 理论- 理论- 理论- 理论- 理论- 理论- 理论- 理论- 预制- 理论- 理论- 理论- 理论- 理论- 理论- 理论- 理论- 理论- 理论- 理论- 理论- 理论- 理论- 和- 理论- 制- 理论- 理论- 理论- 理论- 理论- 理论- 理论- 理论- 理论- 理论- 理论- 理论- 理论- 理论- 理论- 理论- 理论- 和理论- 理论- 理论- 理论- 理论- 理论- 理论- 理论- 理论- 理论- 理论-