1210.5122 (Harald Höller)
Harald Höller
A generalization of implicit conservative numerics to multiple dimensions requires advanced concepts of tensor analysis and differential geometry and hence a more thorough dedication to mathematical fundamentals than maybe expected at first glance. Hence we begin to discuss fundamental mathematics and physics of RHD with special focus on differential geometric consistency and study numerical methods for nonlinear conservation laws to gain a solid definition of the term conservative. The efforts in tensor analysis will be needed when applying Vinokurs theorem to gain the strong conservation form for conservation laws in general curvilinear coordinates. Moreover, it will be required to slightly reformulate the artificial viscosity for such nonlinear coordinates. Astronomical objects are characterized by fast flows and high propagation speeds on the one hand but astronomical length and time scales on the other hand. Implicit numerical schemes are not affected by the Courant Friedrichs Levy condition which limits explicit schemes to rather impracticably small time steps. Implicit methods however produce algebraic problems that require matrix inversion which is computationally expensive. In order to achieve viable resolution, adaptive grid techniques have been developed. It is desired to treat processes on small length scales like shocks and ionization fronts as well as physics at the extent of the objects dimension itself like large scale convection flows and pulsations. The combination of implicit schemes and adaptive grids allows to resolve astrophysics appropriately at various scales. In the last chapter of this paper we study problem oriented adaptive grid generation in 2D and 3D. We establish three main postulations for an ideal grid and analyze several feasible approaches.
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http://arxiv.org/abs/1210.5122
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