A hyperbolic finite difference scheme for anisotropic diffusion equations: preserving the discrete maximum principle

TL;DR

This paper analyzes and demonstrates a hyperbolic finite difference scheme that preserves the discrete maximum principle for anisotropic diffusion equations, ensuring stability and accuracy in plasma simulations.

Contribution

It provides a mathematical analysis and parameter selection method to guarantee the discrete maximum principle in hyperbolic system approaches for anisotropic diffusion.

Findings

The scheme can preserve the DMP with proper parameter choice.

Optimal conditions for parameters are derived mathematically.

Numerical experiments confirm DMP preservation with linear discretization.

Abstract

A hyperbolic system approach is proposed for robust computation of anisotropic diffusion equations that appear in quasineutral plasmas. Though the approach exhibits merits of high extensibility and accurate flux computation, the monotonicity of the scheme for anisotropic diffusion cases has not been understood. In this study, the discrete maximum principle (DMP) of the hyperbolic system approach is analyzed and tested in various anisotropic diffusion cases. A mathematical analysis is conducted to obtain an optimal condition of an arbitrary parameter to guarantee the DMP, and numerical experiments reveal an adoptive selection of the parameter for DMP-preserving results. It is confirmed that, with an appropriate preconditioning matrix and parameter choice, the hyperbolic system approach preserves the DMP even with a linear discretization.