Transparent Boundary Conditions

in cooperation with Ivan Sofronov

Magnetic flux tubes are strong local magnetic field concentrations which are thought to rise from the lower levels of the solar convection zone to the photosphere. For the simulation of these phenomena the ideal compressible magnetohydrodynamic (MHD) equations have to be solved. We use an explicit finite volume scheme based on Riemann solvers.

A major problem arises from the fact that even with strategies like locally-refined grids and parallelization it is still impossible to perform the simulations on the full domain. This makes it necessary to introduce artificial boundaries without physical meaning. To close the system of PDEs we have to find suitable boundary conditions at these open boundaries. These artificial boundary conditions have to guarantee that the solutions on the truncated domain are as close as possible to those obtained on the full domain. In other words, the boundary conditions on the horizontal boundaries should lead to solutions which are (practically) independent of the height of the computational domain; waves generated in the interior of the computational domain must be allowed to pass through the top and bottom boundary.

We obtain our boundary conditions by linearizing the MHD equations about a background atmosphere, thus assuming that the perturbations at the boundary are sufficiently small and smooth. We can prove that these conditions are exact for the linearized system, i.e., a solution computed on a truncated domain is identical to the solution on the full infinite domain. Our boundary condition involves a non-local convolution term, which we can nevertheless implement in a time stepping fashion by a special approximation of the convolution kernel. With this technique the cost of approximating the hydrodynamic quantities on the boundary is almost negligible: in a 2D example this computation took less than 1% of the overall CPU time. We have compared these boundary conditions with other more direct approaches (Dirichlet, Neumann). Our examples show how strongly the structure of the solution is influenced by the choice of the boundary conditions. Moreover, we find that - up to a certain extent - even large perturbations are hardly reflected at the artificial boundaries.

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