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978-3-8439-0797-2, Reihe Strömungsmechanik
Explicit Discontinuous Galerkin Methods for Magnetohydrodynamics
108 Seiten, Dissertation Universität Stuttgart (2012), Softcover, A5
In this work, the explicit space-time expansion discontinuous Galerkin (STE-DG) method is adapted and applied to unsteady ideal and viscous magnetohydrodynamic (MHD) computations. With a special emphasis on shock-capturing and divergence correction of the magnetic field, enhancements to the STE-DG method are proposed that are necessary within the MHD context.
Discontinuous Galerkin schemes enjoy continuously growing popularity, since they combine the flexibility in handling complex geometries, a variable adaptivity to the calculated problem and efficiency of parallel implementations. These are big advantages for modern numerical calculations of various fields of interest, also for MHD calculations e.g. in astrophysics or plasma physics. The presented STE-DG scheme can further enhance explicit computations by its local timestepping functionality, allowing each cell to run with its own determined timestep.
The necessary local formulation adds additional constraints to the implementation of new equation systems and numerical ingredients and not every method is suitable. On the other hand it enables mechanisms that would generally be considered to be ineffective for explicit numerical schemes, since they would drastically decrease the timestep of the calculation. The proposed use of artificial viscosity for shock capturing falls in this category: Artificial viscosity is used to capture shocks with a high order scheme. The thereby caused strong influence on the scheme's timestep is substantially reduced by the local timestepping. For this purpose, suitable oscillation indicators were found and evaluated. For the divergence correction of the magnetic field, the local timesteps enable a sub-cycling feature to increase the correction efficiency.
In addition, postprocessing and data reduction techniques are presented, that are especially of interest for high order schemes. Further more, the parallel efficiency of th STE-DG implementation and code development strategies are considered.
To validate the STE-DG implementation for MHD and the proposed ingredients, several multi-dimensional test cases have been set up, including convergence studies and shock tube tests. The scheme is then applied to two- and three-dimensional more complex astrophysical test cases of larger computational scale.