Improving Efficiency in the Discontinuous Galerkin Spectral Element Method for Complex Geometry Physics

Abstract

The discontinuous Galerkin spectral element method (DGSEM) is a numerical scheme that has risen in popularity, largely due to its high-order accuracy and geometrical flexibility. It is well suited for convection dominated problems such as high-speed flows and wave propagation. Despite these benefits, it still suffers from the drawback of a high computational cost. In this thesis, we aim to model physics in complex geometries using DGSEM with a focus on efficiency. We report on four different approaches to improve efficiency.

First, we use a transfinite mapping to extend a Cartesian grid DGSEM solver for the acoustic wave equation. This code possesses adaptive mesh refinement (AMR) capabilities through both h-refinement where the elements are split, and p-refinement where the polynomial order of the element is increased. hp-Adaptivity increases the capabilities of simulating complex physics as resolution can be provided in select regions of the domain where necessary. To address the load imbalance caused by AMR in parallel computing, the Cartesian code is dynamically load balanced and was shown to scale well in a high performance computing (HPC) environment. We were able to successfully model the acoustic wave equation in complex geometries using transfinite mapping while retaining the high accuracy of the method. We tested the scaling performance of the code after applying the transfinite mapping and compared to the Cartesian grid performance. The transfinite mapped code scales relatively well but does incur some computational overhead that affects the scaling. Based on the results, we pursued the immersed boundary method as an alternative approach to model complex geometries.

The immersed boundary method (IBM) is a technique where an object is modeled, without conforming to the computational grid. We specifically use the volume penalty method to model objects as porous obstacles for acoustic wave propagation problems. The IBM promises time savings for the end user by simplifying the mesh generation process, as only Cartesian grids are required. However, it does come with the tradeoff of a loss in accuracy and oscillations. We address these issues with hp-adaptivity and a low porosity parameter, where we observed an increase in accuracy and a reduction in oscillatory behavior. In addition to the time savings during the mesh generation process, computational savings can be obtained as the metrics associated with the transfinite mapping are not required.

Next, we explore the half-closed discontinuous Galerkin (HCDG) method to model incompressible flows. In the HCDG method, "half-closed" nodes are used which allow for more efficient operator construction and good sparsity patterns. Complex geometries are represented using unstructured grids. We successfully simulate the incompressible Navier-Stokes equations for a variety of problems at low Reynolds numbers.

Finally, we improve efficiency by developing a new "eliminated" linear solver. We use static condensation to eliminate degrees of freedom based on the local discontinuous Galerkin (LDG) switch function, and apply p-multigrid on the eliminated linear system. We apply this eliminated p-multigrid solver on elliptic problems and the incompressible Navier-Stokes equations for both structured and unstructured grids. We observed significant reductions in the number of degrees of freedom and number of non-zero entries for such problems using our elimination procedure. In addition to the benefits of operating on the smaller linear systems, we also observed reductions in the number of V-multigrid-cycles required for convergence, up to a factor of two for elliptic problems and a 20% reduction for the incompressible Navier-Stokes equations.