A Sparse and High-Order Accurate Line-Based Discontinuous Galerkin Method for Unstructured Meshes

摘要

We present a new line-based discontinuous Galerkin (DG) discretization schemefor first- and second-order systems of partial differential equations. Thescheme is based on fully unstructured meshes of quadrilateral or hexahedralelements, and it is closely related to the standard nodal DG scheme as well asseveral of its variants such as the collocation-based DG spectral elementmethod (DGSEM) or the spectral difference (SD) method. However, our motivationis to maximize the sparsity of the Jacobian matrices, since this directlytranslates into higher performance in particular for implicit solvers, whilemaintaining many of the good properties of the DG scheme. To achieve this, ourscheme is based on applying one-dimensional DG solvers along each coordinatedirection in a reference element. This reduces the number of connectivitiesdrastically, since the scheme only connects each node to a line of nodes alongeach direction, as opposed to the standard DG method which connects all nodesinside the element and many nodes in the neighboring ones. The resulting schemeis similar to a collocation scheme, but it uses fully consistent integrationalong each 1-D coordinate direction which results in different properties fornonlinear problems and curved elements. Also, the scheme uses solution pointsalong each element face, which further reduces the number of connections withthe neighboring elements. Second-order terms are handled by an LDG-typeapproach, with an upwind/downwind flux function based on a switch function ateach element face. We demonstrate the accuracy of the method and compare it tothe standard nodal DG method for problems including Poisson's equation, Euler'sequations of gas dynamics, and both the steady-state and the transientcompressible Navier-Stokes equations.