Multigrid Solution for High-Order Discontinuous Galerkin Discretizations of the Compressible Navier-Stokes Equations

While CFD has achieved significant maturity during the past decades, computational costs are extremely large for aerodynamic simulations of aerospace vehicles. In this applied aerodynamics context, the discretization of the Euler or Navier-Stokes equations is performed almost exclusively by finite volume algorithms. The accuracy of many of these methods is at best second order; i.e. the error decreases as O(h) where h is a measure of the grid spacing. Recent studies have shown that this rate may not be adequate for modern engineering applications [4, 8]. The development of a practical higher-order solution method could alleviate this problem by significantly decreasing the computational time required to achieve an acceptable error level. Numerous reasons exist for why current finite-volume algorithms are not practical at higher-order. A root cause of many of these difficulties lies in the extended stencils that these algorithms employ. These extended stencils lead to high memory requirements due to matrix fill-in and difficulties in achieving stable iterative algorithms [10]. By contrast, finite element formulations introduce higher-order effects compactly within the element. For discontinuous Galerkin (DG) formulations, the element-to-element coupling exists only through the fluxes at the shared boundaries between elements. This limited coupling is an enabling feature which permits the development of an efficient higher-order solver and potentially significant improvements in the turn-around time for reliably accurate aerodynamic simulations.