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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 -
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 -
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 -
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