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

Per-Olof Persson
Department of Mathematics
University of California
California, USA


We present a new line-based discontinuous Galerkin (DG) discretization scheme for first- and second-order systems of partial differential equations. The scheme uses fully unstructured meshes of quadrilateral or hexahedral elements, and it attempts to maximize the sparsity of the Jacobian matrices, since this directly translates into higher performance in particular for implicit solvers. Our scheme is based on applying one-dimensional DG solvers along each coordinate direction in a reference element. This reduces the number of connectivities drastically, since the scheme only connects each node to a line of nodes along each direction, as opposed to the standard DG method which connects all nodes inside the element and many nodes in the neighboring ones. The resulting scheme is similar to a collocation scheme (e.g. the DGSEM/SD methods), but it uses fully consistent integration along each 1-D coordinate direction which results in different properties for nonlinear problems and curved elements. Also, the scheme uses solution points along each element face, which further reduces the number of connections with the neighboring elements. Second-order terms are handled by an LDG-type approach, with an upwind/downwind flux function based on a switch function at each element face. We demonstrate the accuracy and performance of the method on several problems in fluid and solid dynamics. We also show how to use Newton-Krylov solvers without impairing the high sparsity of the matrices, by a splitting of the matrix-vector products and a block-Jacobi preconditioner. We integrate in time using a high-order diagonally implicit Runge-Kutta (DIRK) scheme. Using a Newton approach with reused Jacobians, this leads to an implicit scheme with a computational cost comparable to that of an explicit one, without stability-based timestep restrictions.