Numerical Schemes for Nonlinear Heat Conduction and their Application to Astrophysical Flows

Description

Nonlinear heat conduction plays an important role for high energy astrophysical flows. For ionized plasma the thermal diffusivity K depends on temperature (K~T^5/2). This high nonlinearity affects the dynamics of strong shocks and is an essential process in governing the emission of X-ray photons. The numerical method must be able to deal with stiff equations, have second order accuracy, and require resonable computational time.

Our application, colliding winds in a particular binary star system, shows the appearance of isothermal shocks and steep heat fronts. Both properties are important physical aspects.

We use a fractional step method to solve the Euler equations, including cooling and thermal diffusion. This allows us to use an existing solver for the Euler equations plus cooling, and to investigate different numerical schemes for the diffusion part. The diffusion part must be solved by an implicit scheme as explicit time steps are far too small. We use the AMRCART code (Adaptive Mesh Refinement for Cartesian grid) (See http://www.astro.phys.ethz.ch/staff/walder).

We implemented different numerical schemes for the nonlinear diffusion part. The numerical scheme by Dai & Woodward with a fixpoint solver and the L stable TR-BDF2 scheme and the A-stable Crank-Nicolson scheme with a Newton solver. For high density jumps the A-stable Crank-Nicolson scheme with a Newton solver is an appropriate scheme as it is an accurate method and the Newton solver allows larger time steps than the fixpoint solver. If the density jump is not too high the L-stable formula may allow larger time steps than the A-stable Crank-Nicolson scheme.

Adaptive mesh refinement would help to have a better spatial solution in steep fronts and reduce CPU time. Thus the implementation of an adaptive mesh refinement would be desirable.

In collaboration with:

• Dr. Rolf Walder (walder@astro.phys.ethz.ch), Institute of Astronomy, ETH Zürich

Contacts

Prof. Jürg Marti (marti@sam.math.ethz.ch) and Simin Motamen (motamen@sam.math.ethz.ch).