Positive and energy stable schemes for Poisson-Nernst-Planck equations and related models
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In this thesis, we design, analyze, and numerically validate positive and energy-
dissipating schemes for solving Poisson-Nernst-Planck (PNP) equations and Fokker-Planck (FP) equations with interaction potentials. These equations play an important role in modeling the dynamics of charged particles in semiconductors and biological ion channels, as well as in other applications. These model equations are nonlinear/nonlocal gradient flows in density space, and their explicit solutions are rarely available; however, solutions to such problems feature intrinsic properties such as (i) solution positivity, (ii) mass conservation, and (iii) energy dissipation. These physically relevant properties are highly desirable to be preserved at the discrete level with the least time-step restrictions.
We first construct our schemes for a reduced PNP model, then extend to multi-dimensional PNP equations and a class of FP equations with interaction potentials. The common strategies in the construction of the baseline schemes include two ingredients: (i) reformulation of each underlying model so that the resulting system is more suitable for constructing positive schemes, and (ii) integration of semi-implicit time discretization and central spatial discretization. For each model equation, we show that the semi-discrete schemes (continuous in time) preserve all three solution properties (positivity, mass conservation, and energy dissipation). The fully discrete first order schemes preserve solution positivity and mass conservation for arbitrary time steps. Moreover, there exists a discrete energy function which dissipates along time marching with an $O(1)$ bound on time steps.
We show that the second order (in both time and space) schemes preserve solution positivity for suitably small time steps; for larger time steps, we apply a local limiter to restore the solution positivity. We prove that such limiter preserves local mass and does not destroy the approximation accuracy. In addition, the limiter provides a reliable way of restoring solution positivity for other high order conservative finite difference or finite volume schemes.
Both the first and second order schemes are linear and can be efficiently implemented without resorting to any iteration method. The second order schemes are only slight modifications of the first order schemes. Computational costs of a single time step for first and second order schemes are similar, hence our second-order in time schemes are efficient than the first-order in time schemes, given a larger time step could be utilized (to save computational cost). We conduct extensive numerical tests that support our theoretical results and illustrate the accuracy, efficiency, and capacity to preserve the solution properties of our schemes.