PhD defence | Ruibin Qin, The Discontinuous Galerkin Method on Cartesian Grids with Embedded Geometry: Spectrum Analysis and Implementation for Euler Equations

In this thesis, we analyze theoretical properties of the discontinuous Galerkin method (DGM) and propose novel approaches to implementation with the aim to increase its efficiency. First, we derive explicit expressions for the eigenvalues (spectrum) of the discontinuous Galerkin spatial discretization applied to the linear advection equation. We show that the eigenvalues are related to the subdiagonal [p/p + 1] Padé approximation of e−z when the p-th degree basis functions are used. We derive an upper bound on the eigenvalue with the largest magnitude as (p + 1)(p + 2). We demonstrate that this bound is not tight and prove that the asymptotic growth rate of the spectral radius is slower than quadratic in p. We also analyze the behavior of the spectrum near the imaginary axis to demonstrate that the spectral curves approach the imaginary axis although there are no purely imaginary eigenvalues.

Then, we extend the analysis to nonuniform meshes where both the size of elements and the composition of the mesh influence the spectrum. We show that the spectrum depends on the ratio of the size of the largest to the smallest cell as well as the number of cells of different types. We find that the spectrum grows linearly as a function of the proportion of small cells present in the mesh when the size of small cells is greater than some critical value. When the smallest cells are smaller than this critical value, the corresponding eigenvalues lie outside of the main spectral curve. We provide an easily computable estimate of the upper bound on the spectrum. Numerical examples on nonuniform meshes are presented to show the improvement on the time step restriction. In particular, this result can be used to improve the time step restriction on Cartesian grids.

Finally, we present a discontinuous Galerkin method for solutions of the Euler equations on Cartesian grids with embedded geometries. Cartesian grids are an alternative to a more popular unstructured triangular/tetrahedral approach. Their advantage is a simplified mesh generation and a computational efficiency resulting from the structure of a Cartesian grid. However, cutting an embedded geometry out of the grid creates cut cells, which are difficult to deal with for two reasons. One is the restrictive CFL number and the other is the integration on irregularly shaped cells. Both of these issues are more involved for the DGM than for finite volume methods, which most Cartesian grid techniques have been developed for. We use explicit time integration employing cell merging to avoid restrictively small time steps. We provide an algorithm for splitting complex cells into triangles and use standard quadrature rules on these for numerical integration. To avoid the loss of accuracy due to straight sided grids, we employ the curvature boundary conditions. We show that the proposed method is robust and high-order accurate.