A second-order accurate in time, positivity-preserving, and unconditionally energy stable operator splitting numerical scheme is proposed and analyzed for the system of reaction-diffusion equations with detailed balance. The scheme is designed based on an 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. At the reaction stage, the reaction trajectory equation is approximated by a second-order Crank-Nicolson type method. The unique solvability, positivity-preserving, and energy-stability are established based on a convexity analysis. In the diffusion stage, an exact integrator is applied if the diffusion coefficients are constant, and a Crank-Nicolson type scheme is applied if the diffusion process becomes nonlinear. In either case, both the positivity-preserving property and energy stability could be theoretically established. Moreover, a combination of the numerical algorithms at both stages by the Strang splitting approach leads to a second-order accurate, structure-preserving scheme for the original reaction-diffusion system. Numerical experiments are presented, which demonstrate the accuracy of the proposed scheme.

翻译：在反应-扩散方程式中,反应部分按反应轨迹重新拟订,反应部分和扩散部分均分解同样的自由能量。在反应阶段,反应轨迹方程式的近似为第二级的Crank-Nicolson-Nicolson类型方法。独特的溶解性、反应-保存和能源稳定操作方程式是根据静态分析确定的。在扩散阶段,如果扩散系数不变,则采用精确的混合器,如果扩散过程非线性,则采用Crank-Nicolson类型办法。无论哪种情况,均可以理论上确定假设-保留属性和能源稳定性。此外,分解法在两个阶段采用的数字算法组合导致对最初反应-分裂制的精确度、结构保持和能源稳定性进行第二级的精确度分析。在扩散阶段,如果扩散系数保持不变,则采用精确的组合法,如果最初的反反应-分裂制式办法的精确度得到试验,则采用Crank-Nicolson型办法。