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Efficient computation of all speed flows using an entropy stable shock-capturing space-time discontinuous Galerkin method
(2014)Research ReportWe present a shock-capturing space-time Discontinuous Galerkin method to approximate all speed flows modeled by systems of conservation laws with multiple time scales. The method provides a very general and computationally efficient framework for approximating such systems on account of its ability to incorporate large time steps. Numerical examples ranging from computing the incompressible limit (robustness with respect to Mach number) ...Report -
Multi-level Monte Carlo finite volume method for shallow water equations with uncertain parameters applied to landslides-generated tsunamis
(2014)SAM Research ReportTwo layer Savage-Hutter type shallow water PDEs model flows such as tsunamis generated by rockslides. On account of heterogeneities in the composition of the granular matter, these models contain uncertain parameters like the ratio of densities of layers, Coulomb and interlayer friction. These parameters are modeled statistically and quantifying the resulting solution uncertainty (UQ) is a crucial task in geophysics. We propose a novel ...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 -
Computation of measure valued solutions for the incompressible Euler equations
(2014)Research ReportWe combine the spectral (viscosity) method and ensemble averaging to propose an algorithm that computes admissible measure valued solutions of the incompressible Euler equations. The resulting approximate young measures are proved to converge (with increasing numerical resolution) to a measure valued solution. We present numerical experiments demonstrating the robustness and efficiency of the proposed algorithm, as well as the appropriateness ...Report -
Entropy stable schemes on two-dimensional unstructured grids
(2014)Research ReportWe propose an entropy stable high-resolution finite volume scheme to approximate systems of two-dimensional symmetrizable conservation laws on unstructured grids. The scheme is constructed using a judicious combination of entropy conservative fluxes and entropy-stable numerical dissipation operators. High resolution is achieved based on a piecewise linear reconstruction procedure satisfying a suitable sign property. The proposed scheme ...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 -
Multi-Level Monte Carlo Finite Volume methods for uncertainty quantification of acoustic wave propagation in random heterogeneous layered medium
(2014)Research ReportWe consider the very challenging problem of efficient uncertainty quantification for acoustic wave propagation in a highly heterogeneous, possibly layered, random medium, characterized by possibly anisotropic, piecewise log-exponentially distributed Gaussian random fields. A multi-level Monte Carlo finite volume method is proposed, along with a novel, bias-free upscaling technique that allows to represent the input random fields, generated ...Report