A positivity-preserving hybrid DDG method for Poisson--Nernst--Planck systems

TL;DR

This paper introduces a high-order hybrid DDG method for Poisson--Nernst--Planck systems that guarantees positivity of solutions and energy dissipation, improving stability and physical fidelity over previous schemes.

Contribution

It develops a novel positivity-preserving high-order DDG scheme with a hybrid limiter and theoretical guarantees, extending applicability to higher dimensions.

Findings

Ensures positivity of cell averages via modified flux

Achieves point-wise positivity with a hybrid limiter

Demonstrates high-order accuracy and energy dissipation in numerical tests

Abstract

In earlier work [H. Liu and Z. Wang, J. Comput. Phys., 328(2017)], an arbitrary high-order conservative and energy-dissipative direct discontinuous Galerkin (DDG) scheme was developed. Although this scheme enforced solution positivity using cell averages as reference values, it lacked a theoretical guarantee for the positivity of those cell averages. In this study, we develop a novel arbitrary high-order DDG method with rigorously proven positivity-preserving properties. Specifically, the positivity of the cell averages is ensured through a modified numerical flux in combination with forward Euler time discretization. To achieve point-wise positivity of ion concentrations, we introduce a hybrid algorithm that integrates a positivity-preserving limiter. The proposed method is further extended to higher-dimensional problems with rectangular meshes. Numerical results confirm the scheme's high-order accuracy, guaranteed positivity preservation, and consistent discrete energy dissipation.