A fast, p-Multigrid Discontinuous Galerkin Method for Compressible Flows at All Speeds

A p-multigrid (p=polynomial degree) discontinuous Galerkin method is presented for the solution of the compressible Euler equations on unstructured grids. The method operates on a sequence of solution approximations of different polynomial orders. A distinct feature of this p-multigrid method is the application of an explicit smoother on the higher level approximations (p > 0) and an implicit smoother on the lowest level approximation (p = 0), resulting in a fast as well as low storage method that can be efficiently used to accelerate the convergence to a steady state solution. Furthermore, this p-multigrid method can be naturally applied to compute the flows with discontinuities, where a monotonic limiting procedure is usually required for discontinuous Galerkin methods. An accurate representation of the boundary normals based on the definition of the geometries is used for imposing slip boundary conditions for curved geometries. A variety of compressible flow problems for a wide range of flow conditions, from low Mach number to supersonic, in both 2D and 3D configurations are computed to demonstrate the performance of this p-multigrid method. The numerical results obtained strongly indicate the order independent property of this p-multigrid method and demonstrate that this method is orders of magnitude faster than its explicit counterpart. The performance comparison with a finite volume method shows that using this p-multigrid method, the discontinuous Galerkin method provides a viable, attractive, competitive and probably even superior alternative to the finite volume method for computing compressible flows at all speeds.