Efficient Solution Techniques for Discontinuous Galerkin Discretizations of the Navier-Stokes Equations on Hybrid Anisotropic Meshes

Abstract

The goal of this paper is to investigate and develop fast and robust solution techniques for high-order accurate Discontinuous Galerkin discretizations of non-linear systems of conservation laws on unstructured meshes. Previous work was focused on the development of hp-multigrid techniques for inviscid flows and the current work concentrates on the extension of these solvers to steady-state viscous flows including the effects of highly anisotropic hybrid meshes. Efficiency and robustness are improved through the use of mixed triangular and quadrilateral mesh elements, the formulation of local order-reduction techniques, the development of a line-implicit Jacobi smoother, and the implementation of a Newton-GMRES solution technique. The methodology is developed for the two-and three-dimensional Navier-Stokes equations on unstructured anisotropic grids, using linear multigrid schemes. Results are presented for a flat plate boundary layer and for flow over a NACA0012 airfoil and a two-element airfoil. Current results demonstrate convergence rates which are independent of the degree of mesh anisotropy, order of accuracy (p) of the discretization and level of mesh resolution (h). Additionally, preliminary results of ongoing work for the extension to the Reynolds Averaged Navier-Stokes(RANS) equations, the development of a Gas-Kinetic (BGK) inter-cell flux function implementation, and the extension to three dimensions are given.

An Efficient Correction Method to Obtain a Formally Third-Order Accurate Flow Solver for Node-Centered Unstructured Grids

“…Perhaps the most common high-order scheme in the literature is the so-called discontinuous Galerkin (DG) method [4,5,14,16]. The DG method obtains high-order accuracy by employing high-degree polynomial approximations within each cell using internal degrees of freedom.…”

Computing Shocked Flows with High-order Accurate Discontinuous Galerkin Methods

“…9,15,19 The governing equations are discretized using modal basis functions to approximate the vector of conserved flow field variables u. The integrals are computed using Gaussian quadrature rules that integrate polynomials of degree 2p exactly for both surface and volume integrations.…”

“…The size of the local timestep is controlled by the two-norm of the residual such that the local time-step is increased as steady-state residual becomes closer to machine zero. The linear equations of the damped Newton solver are solved using a GMRES method [19][20][21] with line-implicit colored Gauss-Seidel preconditioning. 15…”

Solutions of High-order Methods for Three-dimensional Compressible Viscous Flows

“…In the present DG methods, a symmetric interior penalty (SIP) method 3,19 is implemented due to the fact that the scheme does not require introduction of any auxiliary variables, and moreover, it maintains a compact element-based stencil that simplifies the linearization of the full discretized system for developing implicit methods. In the SUPG methods, on the other hand, the solution is assumed to be continuous across the computational domain and the surface flux required in DG methods vanishes at elemental boundaries.…”

“…Moreover, a great deal of effort has been devoted to the versatility, robustness and efficiency of high-order flow solvers, including adaptive mesh refinement techniques, [8][9][10][11][12] solution limiting and shock capturing methods, 9,[13][14][15][16] hybrid methodologies and multigrid solution strategies. 2,[17][18][19] To this end, this paper continues on the development of high-order discretization methods, consisting of discontinuous Galerkin (DG) [1][2][3]6,18,20 and streamline/upwind Petrov-Galerkin (SUPG) [21][22][23][24] discretizations, to further expand the capability of high-order schemes in solving a wide range of viscous flow problems for complex geometries and to compare the accuracy of the high-order DG and SUPG methods. In particular, applications of the present methods for studying the flow around bluff bodies at a sub-critical Reynolds number and simulations of two-dimensional turbulent flows are considered.…”