Positivity-preserving third order DG schemes for Poisson--Nernst--Planck equations

TL;DR

This paper introduces a third order discontinuous Galerkin scheme for Poisson--Nernst--Planck equations that preserves positivity and steady states, with demonstrated accuracy and stability in multiple dimensions.

Contribution

The paper develops a novel third order positivity-preserving DG scheme for PNP equations, including a scaling limiter to ensure positivity of solutions.

Findings

The scheme achieves third order accuracy in numerical tests.

It preserves positivity of cell averages and solutions.

It maintains steady states during simulations.

Abstract

In this paper, we design and analyze third order positivity-preserving discontinuous Galerkin (DG) schemes for solving the time-dependent system of Poisson--Nernst--Planck (PNP) equations, which has found much use in diverse applications. Our DG method with Euler forward time discretization is shown to preserve the positivity of cell averages at all time steps. The positivity of numerical solutions is then restored by a scaling limiter in reference to positive weighted cell averages. The method is also shown to preserve steady states. Numerical examples are presented to demonstrate the third order accuracy and illustrate the positivity-preserving property in both one and two dimensions.