Entropy-stable discontinuous Galerkin difference methods for hyperbolic conservation laws
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The paper describes the construction of entropy-stable discontinuous Galerkin difference (DGD) discretizations for hyperbolic conservation laws on unstructured grids. The construction takes advantage of existing theory for entropy-stable summation-by-parts (SBP) discretizations. In particular, the paper shows how DGD discretizations – both linear and nonlinear – can be constructed by defining the SBP trial and test functions in terms of interpolated DGD degrees of freedom. In the case of entropy-stable discretizations, the entropy variables rather than the conservative variables must be interpolated to the SBP nodes. A fully-discrete entropy-stable scheme is obtained by adopting the relaxation Runge-Kutta version of the midpoint method. In addition, DGD matrix operators for the first derivative are shown to be dense-norm SBP operators. Numerical results are presented to verify the accuracy and entropy-stability of the DGD discretization in the context of the Euler equations. The results suggest that DGD and SBP solution errors are similar for the same number of degrees of freedom. Finally, an investigation of the DGD spectra shows that spectral radius is relatively insensitive to discretization order; however, the high-order methods do suffer from the linear instability reported for other entropy-stable discretizations.
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