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

2025/05/29 by Hailiang Liu, Liu, Hailiang, Zhongming Wang +3
Engineering · Mathematics · #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Gas Dynamics and Kinetic Theory #Numerical Analysis (math.NA) #Numerical methods for differential equations

paper · pdf · doi:10.48550/arxiv.2505.24058

openalex publication_date 2025/05/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• The Onsager principle and structure preserving numerical schemes
• Positivity-preserving and energy-dissipating discontinuous Galerkin methods for nonlinear nonlocal Fokker-Planck equations
• A positivity-preserving, energy stable and convergent numerical scheme for the Poisson-Nernst-Planck system
• Efficient, positive, and energy stable schemes for multi-D Poisson-Nernst-Planck systems
• Unconditional positivity-preserving and energy stable schemes for a reduced Poisson-Nernst-Planck system
• A Wasserstein gradient flow approach to Poisson-Nernst-Planck equations
• The Direct Discontinuous Galerkin (DDG) Methods for Diffusion Problems
• On positivity-preserving high order discontinuous Galerkin schemes for compressible Navier–Stokes equations
• A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem

Related

• A class of positive-preserving,energy stable and high order numerical schemes for the Poission-Nernst-Planck system
• Unconditional positivity-preserving and energy stable schemes for a reduced Poisson-Nernst-Planck system
• An optimization-based positivity-preserving limiter in semi-implicit discontinuous Galerkin schemes solving Fokker-Planck equations
• Second-order, Positive, and Unconditional Energy Dissipative Scheme for Modified Poisson-Nernst-Planck Equations
• Positivity-preserving and energy-dissipating discontinuous Galerkin methods for nonlinear nonlocal Fokker-Planck equations