A fast semi-implicit method for anisotropic diffusion

  • Authors:

  • Prateek Sharma;Gregory W. Hammett

  • Affiliations:

  • Theoretical Astrophysics Center and Astronomy Department, University of California, Berkeley, CA 94720, USA;Princeton Plasma Physics Laboratory, Princeton, NJ 08543, USA

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2011

Quantified Score

Hi-index 31.45

Visualization

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.