A second order accurate (in time) numerical scheme is proposed and analyzed for the Poisson-Nernst-Planck equation (PNP) system, reformulated as a non-constant mobility $H^{-1}$ gradient flow in the Energetic Variational Approach (EnVarA). The centered finite difference is taken as the spatial discretization. Meanwhile, the highly nonlinear and singular nature of the logarithmic energy potentials has always been the essential difficulty to design a second order accurate scheme in time, while preserving the variational energetic structures. The mobility function is updated with a second order accurate extrapolation formula, for the sake of unique solvability. A modified Crank-Nicolson scheme is used to approximate the logarithmic term, so that its inner product with the discrete temporal derivative exactly gives the corresponding nonlinear energy difference; henceforth the energy stability is ensured for the logarithmic part. In addition, nonlinear artificial regularization terms are added in the numerical scheme, so that the positivity-preserving property could be theoretically proved, with the help of the singularity associated with the logarithmic function. Furthermore, an optimal rate convergence analysis is provided in this paper, in which the higher order asymptotic expansion for the numerical solution, the rough error estimate and refined error estimate techniques have to be included to accomplish such an analysis. This work combines the following theoretical properties for a second order accurate numerical scheme for the PNP system: (i) second order accuracy in both time and space, (ii) unique solvability and positivity, (iii) energy stability, and (iv) optimal rate convergence. A few numerical results are also presented.
翻译:为Poisson-Nernst-Planck等式(PNP)系统提出并分析了第二个顺序准确(及时)数字方案,该系统在 Energistic Variational 方法( EnVarA) 中被改成非固定流动 $H ⁇ -1}美元梯度流。中央有限差异被理解为空间离散。同时,对数能源潜力的高度非线性和单数性质一直是设计第二顺序准确方案的基本困难,同时保留变异能结构。流动函数以第二顺序准确外推公式更新,以达到独特的稳定性估算。经修改的Crank-Nicolson 方案用于接近对数术语,因此其内部产品与离散时间导导出相应的非线性能量差异。此外,对对数方法增加非线性人工调节术语(因此,对正数保值保值属性可以进行理论性更新,而精准性精确性精确性精确度的第二顺序则有助于精确性精确性估算值的精确性估算值。 精度分析中,对精度的精度和精细度的精度进行这种精度分析,对精度的精度和精度是精度的精度分析,对精度的精度是精度的精度的精度分析,对精度的精度和精确性分析,对精度的精度的精度的精度是精度。