An entropy stable spacetime discontinuous Galerkin method for the two-dimensional compressible Navier-Stokes equations
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Date
2017-07
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Abstract
In this paper, we present an entropy stable scheme for solving the compressible
Navier-Stokes equations in two space dimensions. Our scheme uses entropy variables
as degrees of freedom. It is an extension of an existing spacetime discontinuous
Galerkin method for solving the compressible Euler equations. The physical diffusion
terms are incorporated by means of the symmetric (SIPG) or nonsymmetric
(NIPG) interior penalty method, resulting in the two versions ST-SDSC-SIPG and
ST-SDSC-NIPG. The streamline diffusion and shock-capturing terms from the original
scheme have been kept, but have been adjusted appropriately. This guarantees
that the new scheme essentially reduces to the original scheme for the compressible
Euler equations in regions with underresolved physical diffusion. We show entropy
stability for both versions under suitable assumptions. We also present numerical
results confirming the accuracy and robustness of our schemes.
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discontinuous Galerkin method, compressible Navier-Stokes equations, entropy stability, entropy variables, interior penalty method, wall boundary conditions