Preserving Monotonicity in Anisotropic Diffusion

TL;DR

This paper demonstrates that standard anisotropic diffusion algorithms can violate physical constraints like monotonicity and entropy, and proposes slope limiter-based methods to ensure physically consistent and stable simulations, especially in astrophysical contexts.

Contribution

It identifies the monotonicity violation in existing anisotropic diffusion algorithms and introduces slope limiter techniques to preserve physical constraints in simulations.

Findings

Centered algorithms can produce negative temperatures in large gradients.

Slope limiters guarantee non-negative temperatures and stability.

Limited methods are especially useful for astrophysical plasma simulations.

Abstract

We show that standard algorithms for anisotropic diffusion based on centered differencing (including the recent symmetric algorithm) do not preserve monotonicity. In the context of anisotropic thermal conduction, this can lead to the violation of the entropy constraints of the second law of thermodynamics, causing heat to flow from regions of lower temperature to higher temperature. In regions of large temperature variations, this can cause the temperature to become negative. Test cases to illustrate this for centered asymmetric and symmetric differencing are presented. Algorithms based on slope limiters, analogous to those used in second order schemes for hyperbolic equations, are proposed to fix these problems. While centered algorithms may be good for many cases, the main advantage of limited methods is that they are guaranteed to avoid negative temperature (which can cause numerical instabilities) in the presence of large temperature gradients. In particular, limited methods will be useful to simulate hot, dilute astrophysical plasmas where conduction is anisotropic and the temperature gradients are enormous, e.g., collisionless shocks and disk-corona interface.