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978-3-8439-2293-7, Reihe Strömungsmechanik

Jonatan Núñez de la Rosa
High-Order Methods for Computational Astrophysics

230 Seiten, Dissertation Universität Stuttgart (2014), Softcover, A5

Zusammenfassung / Abstract

In the present work, the development and application of three high-order numerical methods, namely, the conservative finite difference (FD) method, the finite volume (FV) method, and the discontinuous Galerkin spectral element method (DGSEM), is presented. These methods are then used for solving three equations systems arising in computational astrophysics on flat spacetimes, specifically, the ideal magnetohydrodynamics (MHD), relativistic hydrodynamics (SRHD) and relativistic magnetohydrodynamics (SRMHD). The FD and FV methods are extended to very high-order accuracy on regular Cartesian meshes by making use of the arbitrary high-order reconstruction WENO operator. The time discretization is carried out with a strong stability-preserving Runge-Kutta (SSPRK) method, and the solenoidal constraint of the magnetic field is enforced with the generalized Lagrange multiplier method. Because discontinuous solutions form part of the nature of the hyperbolic conservation laws, shock capturing strategies have to be devised. For the FD and FV schemes, a fallback approach is developed, where, in presence of discontinuities and shocks, the order of the reconstruction operator is lowered to the robust third order WENO scheme, while a very high-order operator is retained in those regions with smooth flows. For the DGSEM a robust and efficient hybrid DG/FV approach is devised as shock capturing. The hybrid DGSEM/FV is constructed in such a way that, in regions of smooth flows, the DGSEM method is employed, and those parts of the flow having shocks, the DGSEM elements are interpreted as quadrilateral/hexahedral subdomains. In each of these subdomains, the nodal DG solution values are used to build a new local domain composed now of finite volume subcells, which are evolved with a robust finite volume method with third order WENO reconstruction. This new numerical framework for computational astrophysics based on the hybridization of high-order methods brings very promising results. Our computational framework has been subject to the standard testbench in computational astrophysics. Numerical results of problems having smooth flows, and problems with shock-dominated flows, are also reported.