We present a systematic study of the novel entropy stable approximations of a variety of nonlinear conservation laws, from the scalar Burger's equation to 1D Navier–Stokes and 2D shallow water equations. This new family of second-order difference schemes avoids using artificial numerical viscosity, in the sense that their entropy dissipation is dictated solely by physical dissipation terms. The numerical results of 1D compressible Navier–Stokes equations provide us a remarkable evidence for different roles of viscosity and heat conduction in forming sharp monotone profiles in the immediate neighborhoods of shocks and contacts. Further implementation in 2D shallow water equations is realized dimension by dimension.
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Tadmor, E., Zhong, W. (2008). Novel Entropy Stable Schemes for 1D and 2D Fluid Equations. In: Benzoni-Gavage, S., Serre, D. (eds) Hyperbolic Problems: Theory, Numerics, Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-75712-2_119
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