Astronomy and Astrophysics – Astronomy
Scientific paper
Apr 2001
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2001newar..45..425h&link_type=abstract
New Astronomy Reviews, Volume 45, Issue 6, p. 425-447.
Astronomy and Astrophysics
Astronomy
19
Scientific paper
Most of the numerical methods used in astrophysical fluid dynamics rely on explicit time-stepping schemes, whereas the higher robustness of implicit methods which constitute the core of modern computational fluid dynamics is rarely explored. In this paper, we survey some modern implicit solvers which are specially adapted to multi-dimensional problems and discuss their potential and range of application in comparison to common explicit methods. Special emphasis is put on the aspect of efficiency and robustness. Our reference set of equations are those corresponding to radiative magneto-hydrodynamics (MHD) modeling self-gravitating and partially and/or fully ionized flows. Explicit methods may be viewed as a very special class of so-called `defect-correction iterations' for solving an implicit discretization. Within this context one can design various implicit methods, ranging from weakly to fully implicit, which allow to follow evolutionary phases on much longer time scales than the dynamic one. We particularly present a new three-stages implicit numerical method for searching strongly time-dependent, quasi-stationary and steady-state solutions for the above-mentioned equations. Preconditioned Krylov-space and multilevel techniques are employed for enhancing the efficiency and robustness of the computation. The spatial discretization is on highly nonuniform tensor-product meshes and uses cartesian, cylinder or spherical coordinates depending on the geometrical structure of the problem. The accuracy is of second-order in space and time and can easily be increased without modifying the structure of the scheme. The algebraic solver consists of a pre-conditioned transpose-free Krylov iteration for the conservation equations and optimized multigrid algorithms for solving the Poisson equation for the gravitational potential and the transport-diffusion equation for the radiation density in flow regions with dynamically varying optical depths. The existing implementation of the proposed method employs axis-symmetry in order to reduce the problem to two dimensions.
Hujeirat A. A.
Rannacher R.
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