Preserving Monotonicity in Anisotropic Diffusion
Heat is conducted from higher to lower temperatures by random thermal motion of electrons.
In dilute, magnetized plasmas thermal conduction is primarily along magnetic field lines.
In the collisional regime (Larmor radius << mean free path <~ length scales of interest)
thermal conductivity is given by Braginskii's formulae. However, heat flux has
the form of saturated conduction (heat flux proportional to temperature and not to
temperature gradient) in the collisionless regime.
Unlike isotropic conduction, there is a subtlety in numerical
implementation of anisotropic conduction. Heat is not guaranteed to flow from higher to
lower temperatures with standard finite differencing, if the grid is not aligned
with the magnetic field lines. One can use a field-aligned grid for avoiding such problems; this
is sometimes the case for modeling of fusion plasmas. However, magnetic fields are wildly
turbulent in most astrophysical plasmas and a field-aligned grid is unfeasible.
We recognized that simple finite differencing of anisotropic
conduction can result in non-monotonicity of temperature, which can lead to negative
temperatures (and associated numerical instabilities) in presence of large temperature
gradients. We came up with a numerical method that preserves monotonicity of anisotropic
thermal conduction. Our method, which conserves energy, uses limiters
(similar to those used for linear reconstruction
in finite volume methods for hyperbolic equations) and ensures that heat flows from hot to
cold temperatures.
The non-monotonicity of temperature with simple finite differencing can be seen from a simple test problem.
Static, circular magnetic fields are initialized in a cartesian grid. A hot patch initialized at r=0.6, theta=pi is
evolved for many diffusion times across the magnetic ring. The figure above shows 2-D plots of temperature
at late times with different schemes for anisotropic thermal conduction (asymmetric (left),
symmetric (center), and asymmetric with Monotonized-Central limiter (right)). Simple finite differencing
gives temperature below the initial temperature minimum (10 in code units), but finite differencing
with the limiter is monotonic (T>=10 everywhere, as it should be).
Use of limiters in anisotropic thermal conduction will be necessary in numerical simulations of magnetized plasmas
with large temperature gradients; e.g., X-ray emitting plasma in galaxy clusters with short cooling time,
accretion flows with hot corona on top of a colder accretion disk.
Recently we suggested an implicit method for treating anisotropic diffusion, which is unconditionally stable and does not violate the monotonicity constraint seriously.
References:
Preserving monotonicity in anisotropic diffusion
A Fast Semi-implicit Method for Anisotropic Diffusion