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A well-balanced finite volume scheme for the Euler equations with gravitation
(2015)Research ReportContext <br/> Many problems in astrophysics feature flows which are close to hydrostatic equilibrium. However, standard numerical schemes for compressible hydrodynamics may be deficient in approximating this stationary state, in which the pressure gradient is nearly balanced by gravitational forces. <br/> Aims <br/> We aim to develop a second-order well-balanced scheme for the Euler equations. The scheme is designed to mimic a discrete ...Report -
Structure preserving schemes
(2014)SAM Research ReportWe present two novel structure preserving numerical schemes for the Euler equations of hydrodynamics. The first method is concerned with the exact preservation of certain hydrostatic equilibria. This is achieved by a hydrostatic preserving reconstruction procedure and a well-balanced discretization of the gravitational source term. The second method treats the deficiency of angular momentum conservation in standard Eulerian Godunov-type ...Report -
Construction of approximate entropy measure valued solutions for hyperbolic systems of conservation laws
(2014)Research ReportNumerical evidence is presented to demonstrate that state of the art numerical schemes need not converge to entropy solutions of systems of hyperbolic conservation laws in several space dimensions. Combined with recent results on the lack of stability of these solutions, we advocate the more general notion of entropy measure valued solutions as the appropriate paradigm for solutions of such multi-dimensional systems. We propose a detailed ...Report -
Well-balanced schemes for gravitationally stratified media
(2014)Research ReportWe present a well-balanced scheme for the Euler equations with gravitation. The scheme is capable of maintaining exactly (up to machine precision) a discrete hydrostatic equilibrium without any assumption on a thermodynamic variable such as specific entropy or temperature. The well-balanced scheme is based on a local hydrostatic pressure reconstruction. Moreover, it is computationally efficient and can be incorporated into any existing ...Report