Extensions of the discontinuous galerkin difference method

Abstract

This thesis explores extensions to the discontinuous Galerkin difference (DGD) method as a means of simulating hyperbolic conservation laws in a stable and efficient way. The first part of the thesis focuses on an entropy-stable discontinuous Galerkin difference (DGD) method for hyperbolic conservation laws on unstructured grids. The entropy-stable DGD method takes advantage of existing theory for entropy-stable (diagonal-norm) summation-by-parts (SBP) discretizations. 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. Furthermore, the DGD scheme is applied to a one-dimensional Riemann problem, and global conservation and convergence in the L1 norm are observed. The second part of the thesis presents a generalized DGD (GDGD) method, where the basis functions are not associated with the element centers. In addition, generalized DGD basis that are enriched with non-polynomial functions are investigated. Two practical issues that cause the GDGD stencils to produce ill-conditioned Vandermonde matrices are addressed. Numerical experiments verify the accuracy of the GDGD interpolation operators and demonstrate that the GDGD spatial discretization exhibits superconvergent solution error. Finally, the flexibility of the GDGD method is exploited for r-adaptation. This adaptation method uses an optimization approach in which the objective function is the 2-norm of the discontinuous Galerkin residual and the optimization variables are the basis center locations. A gradient-based optimization algorithm is adopted, and details on the derivation and computation of the gradient using adjoint method are discussed. Numerical experiments demonstrate that the GDGD r-adaptivity method is capable of reducing solution and/or functional error by aligning the GDGD basis distribution with solution features.