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ENO reconstruction and ENO interpolation are stable
(2011)SAM Research ReportWe prove stability estimates for the ENO reconstruction and ENO interpolation procedures. In particular, we show that the jump of the reconstructed ENO pointvalues at each cell interface has the same sign as the jump of the underlying cell averages across that interface. We also prove that the jump of the reconstructed values can be upper-bounded in terms of the jump of the underlying cell averages. Similar sign properties hold for the ...Report -
Existence of solutions to a model of two-phase flow in porous media
(2011)SAM Research ReportWe consider the flow of two-phases in a porous medium and propose a modi ed version of the fractional flow model for incompressible, two-phase flow based on a Helmholtz regularization of the Darcy phase velocities. We show the existence of global-in-time entropy solutions for this model with suitable assumptions on the boundary conditions. Numerical experiments demonstrating the approximation of the classical two-phase flow equations with ...Report -
Arbitrarily high order accurate entropy stable essentially non-oscillatory schemes for systems of conservation laws
(2011)SAM Research ReportWe design arbitrarily high-order accurate entropy stable schemes for systems of conservation laws. The schemes, termed TeCNO schemes, are based on two main ingredients: (i) high-order accurate entropy conservative uxes, and (ii) suitable numerical di usion operators involving ENO reconstructed cell-interface values of scaled entropy variables. Numerical experiments in one and two space dimensions are presented to illustrate the robust ...Report -
Accurate numerical schemes for approximating intitial-boundary value problems for systems of conservation law
(2011)SAM Research ReportSolutions of initial-boundary value problems for systems of conservation laws depend on the underlying viscous mechanism, namely different viscosity operators lead to different limit solutions. Standard numerical schemes for approximating conservation laws do not take into account this fact and converge to solutions that are not necessarily physically relevant. We design numerical schemes that incorporate explicit information about the ...Report -
Scaling pattern with size by a morphogen directed cell division rule
(2011)CMA Collected Preprint seriesReport -
Entropy conservative and entropy stable schemes for non-conservative hyperbolic systems
(2011)Research ReportThe vanishing viscosity limit of non-conservative hyperbolic systems depends heavily on the speci c form of the viscosity. Numerical approximations, such as the path consistent schemes of [16], may not converge to the physically relevant solutions of the system. We construct entropy stable path consistent (ESPC) schemes to approximate non-conservative hyperbolic systems by combining entropy conservative discretizations with numerical ...Report -
Multi-level Monte Carlo finite volume methods for nonlinear systems of conservation laws in multi-dimensions
(2011)SAM Research ReportWe extend the Multi-Level Monte Carlo (MLMC) algorithm of [19] in order to quantify uncertainty in the solutions of multi-dimensional hyperbolic systems of conservation laws with uncertain initial data. The algorithm is presented and several issues arising in the massively parallel numerical implementation are addressed. In particular, we present a novel load balancing procedure that ensures scalability of the MLMC algorithm on massively ...Report -
Multi-level Monte Carlo finite volume methods for shallow water equations with uncertain topography in multi-dimensions
(2011)SAM Research ReportThe initial data and bottom topography, used as inputs in shallow water models, are prone to uncertainty due to measurement errors. We model this uncertainty statistically in terms of random shallow water equations. We extend the Multi-Level Monte Carlo (MLMC) algorithm to numerically approximate the random shallow water equations efficiently. The MLMC algorithm is suitably modified to deal with uncertain (and possibly uncorrelated) data ...Report -
Static load balancing for multi-level Monte Carlo finite volume solvers
(2011)SAM Research ReportThe Multi-Level Monte Carlo finite volumes (MLMC-FVM) algorithm was shown to be a robust and fast solver for uncertainty quantification in the solutions of multi- dimensional systems of stochastic conservation laws. A novel load balancing procedure is used to ensure scalability of the MLMC algorithm on massively parallel hardware. We describe this procedure together with other arising challenges in great detail. Finally, numerical experiments ...Report