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Total Variation Diminishing Implicit Runge-Kutta Methods for Dissipative Advection-Diffusion Problems in Astrophysics |
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| Authors: | Natalie Happenhofer Othmar Koch Friedrich Kupka Florian Zaussinger |
| Institution: | 1. University of Vienna, Faculty of Mathematics, Nordbergstrasse 15, A-1090 Wien, Austria
Natalie Happenhofer, Supported by the Austrian Science Fund (FWF), project P20973 Othmar Koch, phone +43 1 58801 10156, fax +43 1 58801 10196. Supported by the Austrian Science Fund (FWF), project P21742-N16 Friedrich Kupka, Supported by the Austrian Science Fund (FWF), project P21742-N16;2. University of Vienna, Faculty of Mathematics, Nordbergstrasse 15, A-1090 Wien, Austria;3. Brandenburg University of Technology Cottbus, Department of Aerodynamics and Fluid Mechanics, D-03046 Cottbus, Germany |
| Abstract: | We investigate the properties of dissipative full discretizations for the equations of motion associated with models of flow and radiative transport inside stars. We derive dissipative space discretizations and demonstrate that together with specially adapted total-variation-diminishing (TVD) or strongly stable Runge-Kutta time discretizations with adaptive step-size control this yields reliable and efficient integrators for the underlying high-dimensional nonlinear evolution equations. For the most general problem class, fully implicit SDIRK methods are demonstrated to be competitive when compared to popular explicit Runge-Kutta schemes as the additional effort for the solution of the associated nonlinear equations is compensated by the larger step-sizes admissible for strong stability and dissipativity. For the parameter regime associated with semiconvection we can use partitioned IMEX Runge-Kutta schemes, where the solution of the implicit part can be reduced to the solution of an elliptic problem. This yields a significant gain in performance as compared to either fully implicit or explicit time integrators. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) |
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