Perfect derivatives, conservative differences and entropy stable computation of hyperbolic conservation laws

Eitan Tadmor

doi:10.3934/dcds.2016.36.4579

Article Contents

2016, Volume 36, Issue 8: 4579-4598. Doi: 10.3934/dcds.2016.36.4579

This issue Previous Article The relative entropy method for the stability of intermediate shock waves; the rich case Next Article Numerical algorithms for stationary statistical properties of dissipative dynamical systems

Perfect derivatives, conservative differences and entropy stable computation of hyperbolic conservation laws

Eitan Tadmor^1,

1.
Department of Mathematics, Institute for Physical Science & Technology and Center of Scientific Computation and Mathematical Modeling (CSCAMM), University of Maryland, College Park, MD 20742

Received: April 30, 2015

Revised: December 31, 2015

Published: May 2017

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Abstract

Entropy stability plays an important role in the dynamics of nonlinear systems of hyperbolic conservation laws and related convection-diffusion equations. Here we are concerned with the corresponding question of numerical entropy stability --- we review a general framework for designing entropy stable approximations of such systems. The framework, developed in [28,29] and in an ongoing series of works [30,6,7], is based on comparing numerical viscosities to certain entropy-conservative schemes. It yields precise characterizations of entropy stability which is enforced in rarefactions while keeping sharp resolution of shocks.
We demonstrate this approach with a host of second-- and higher--order accurate schemes, ranging from scalar examples to the systems of shallow-water, Euler and Navier-Stokes equations. We present a family of energy conservative schemes for the shallow-water equations with a well-balanced description of their steady-states. Numerical experiments provide a remarkable evidence for the different roles of viscosity and heat conduction in forming sharp monotone profiles in Euler equations, and we conclude with the computation of entropic measure-valued solutions based on the class of so-called TeCNO schemes --- arbitrarily high-order accurate, non-oscillatory and entropy stable schemes for systems of conservation laws.

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Mathematics Subject Classification: Primary: 65M12, 35L65; Secondary: 65M06, 35R06.

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