High-order Navier-Stokes simulations using a sparse line-based discontinuous Galerkin method,” Paper AIAA-2012-436

We study some of the properties of a line-based discontinuous Galerkin (DG) scheme for the compressible Euler and Navier-Stokes equations. The scheme is based on fully unstructured meshes of quadrilateral or hexahedral elements, and it is closely related to the standard nodal DG scheme as well as several of its variants such as the collocation-based DG spectral element method (DGSEM) or the spectral difference (SD) method. However, our motivation is to maximize the sparsity of the Jacobian matrices, since this directly translates into higher performance in particular for implicit solvers. The scheme is based on applying one-dimensional DG solvers along each coordinate direction in a reference element, which reduces the number of connectivities drastically in a line-wise fashion. The resulting scheme is similar to the DGSEM and the SD methods, but it uses fully consistent integration along each 1-D coordinate direction. This gives the method almost the same accuracy and nonlinear stability properties as the standard nodal DG method, at a cost similar to that of the collocation-based methods. 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 of the method and compare it to the standard nodal DG method for model problems of an inviscid vortex problem, flow over a cylinder, and laminar flow around an airfoil. 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, and apply it on a problem of transient laminar flow around an airfoil. Using a quasi-Newton approach, the leads to an implicit scheme with a computational cost comparable to that of an explicit one, without stability-based timestep restrictions. I