# First order hyperbolic approach for Anisotropic Diffusion equation

## TL;DR

This paper introduces a high-order finite difference solver based on the first-order hyperbolic system method for anisotropic diffusion problems, achieving uniform accuracy regardless of anisotropy degree.

## Contribution

The paper develops a fifth-order hyperbolic method that simplifies constructing accurate schemes independent of anisotropy levels, with extensions to magnetized electron simulations.

## Key findings

- Achieves high accuracy with fifth-order schemes
- Demonstrates uniform accuracy regardless of anisotropy degree
- Successfully applied to magnetized electron test case

## Abstract

In this paper, we present a high order finite difference solver for anisotropic diffusion problems based on the first-order hyperbolic system method. In particular, we demonstrate that the construction of a uniformly accurate fifth-order scheme that is independent of the degree of anisotropy is made straightforward by the hyperbolic method with an optimal length scale. We demonstrate that the gradients are computed simultaneously to the same order of accuracy as that of the solution variable by using weight compact finite difference schemes. Furthermore, the approach is extended to improve further the simulation of the magnetized electrons test case previously discussed in Refs.[J. Comput. Phys., 284 (2015) 59-69 and 374 (2018) 1120-1151]. Numerical results indicate that these schemes are capable of delivering high accuracy and the proposed approach is expected to allow the hyperbolic method to be successfully applied to a wide variety of linear and nonlinear problems with anisotropic diffusion.