Radiation Hydrodynamics

For our simulations of the solar photosphere the energy transport by radiation has to be taken into account. This requires the coupling of the equations of magnetohydrodynamics (MHD) and the stationary radiation transport (RT) equation. In each evolution step of our explicit finite volume scheme based on Riemann solvers the radiation intensity for many different propagation directions has to be computed. On the one hand the solution algorithm for the intensity should be of higher order to minimize numerical diffusion. On the other hand the development of a very fast solver is crucial to reach results in an acceptable time. We have therefore developed a new solver for the RT equation

We are also interested in analytically justifying the coupled system and the numerical approximation used. In the course of our work we have studied the local existence of solutions and the convergence of the numerical scheme.

Numerical Schemes for the Radiation Transport Equation

in cooperation with Peter Vollmöller.

By embedding the well-known and often used short characteristics approach into the finite element framework, we have developed a class of solution algorithms for linear equations of Boltzmann type. This very general approach allows us to construct higher-order schemes on arbitrary grids in two or three space dimensions. The main advantage lies in the fact that the major part of the scheme is independent of the underlying grid structure and the desired order, so that very little recoding is necessary when changing the order of the scheme or the underlying grid structure.

We have compared the method with the classical short characteristics approach and the Discontinuous Galerkin finite element scheme in the case of the stationary radiation transport equation. We have focused on the error to runtime ratio of the schemes and have shown that our method can reduce the computational cost by up to two orders of magnitude. Furthermore, the efficiency of the scheme is demonstrated in the case of a challenging problem from the field of solar physics. We have studied the possibility of locally varying the order of the scheme and of the size of the grid cells.

Related Publications:

Analysis for the coupled MHD+RT system

Consider the Cauchy problem in 3D for the MHD-system coupled with the nonstationary radiation transport equation. If the initial datum is three times weakly differentiable and lies in a compact subset of the state space we obtain (under appropriate compatibility conditions)

Theorem: [Zajaczkowski & Rohde]
There is a constant , such that, for , there is classical solution for the Cauchy problem in .
The solution is unique in the class of classical solutions.

Theorem: [Rohde 2001]
The classcial solution is unique in the (bigger) class of entropy solutions.

Related Publications: