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Total Variation Diminishing Implicit Runge‐Kutta Methods for Dissipative Advection‐Diffusion Problems in Astrophysics
Author(s) -
Happenhofer Natalie,
Koch Othmar,
Kupka Friedrich,
Zaussinger Florian
Publication year - 2011
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.201110377
Subject(s) - runge–kutta methods , dissipative system , total variation diminishing , nonlinear system , mathematics , integrator , advection , space (punctuation) , flow (mathematics) , diffusion , stability (learning theory) , spacetime , mathematical analysis , physics , computer science , numerical analysis , geometry , quantum mechanics , voltage , machine learning , thermodynamics , operating system
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)