Abstract
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 (diagonal-norm) 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 a 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 entropy-stability of the DGD discretization in the context of the Euler equations. Accuracy studies reveal that the DGD method is efficient; indeed, like tensor-product DGD schemes, the unstructured DGD method exhibits superconvergent solution error for periodic problems. An investigation of the DGD spectra shows that the spectral radius is relatively insensitive to discretization order. Finally, the DGD scheme is applied to a one-dimensional Riemann problem, and global conservation and convergence in the L1 norm are observed.
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