A Fast Semi-implicit Method for Anisotropic Diffusion

TL;DR

This paper introduces a fast, stable semi-implicit method for anisotropic diffusion that improves computational efficiency and stability over explicit methods, while maintaining accuracy and small oscillations.

Contribution

A novel semi-implicit, directionally-split method for anisotropic diffusion that is second order accurate, stable for large timesteps, and easy to implement in parallel.

Findings

Achieves large speed-ups compared to explicit methods.

Maintains accuracy with small temperature oscillations.

Applicable to both anisotropic and isotropic diffusion on various meshes.

Abstract

Simple finite differencing of the anisotropic diffusion equation, where diffusion is only along a given direction, does not ensure that the numerically calculated heat fluxes are in the correct direction. This can lead to negative temperatures for the anisotropic thermal diffusion equation. In a previous paper we proposed a monotonicity-preserving explicit method which uses limiters (analogous to those used in the solution of hyperbolic equations) to interpolate the temperature gradients at cell faces. However, being explicit, this method was limited by a restrictive Courant-Friedrichs-Lewy (CFL) stability timestep. Here we propose a fast, conservative, directionally-split, semi-implicit method which is second order accurate in space, is stable for large timesteps, and is easy to implement in parallel. Although not strictly monotonicity-preserving, our method gives only small amplitude temperature oscillations at large temperature gradients, and the oscillations are damped in time. With numerical experiments we show that our semi-implicit method can achieve large speed-ups compared to the explicit method, without seriously violating the monotonicity constraint. This method can also be applied to isotropic diffusion, both on regular and distorted meshes.