A high order Discontinuous Galerkin Finite Element solver for the incompressible Navier-Stokes equations

The paper presents an unsteady high order Discontinuous Galerkin (DG) solver that has been developed, verified and validated for the solution of the two-dimensional incompressible Navier-Stokes equations. A second order stiffly stable method is used to discretise the equations in time. Spatial discretisation is accomplished using a modal DG approach, in which the inter-element fluxes are approximated using the Symmetric Interior Penalty Galerkin formulation. The non-linear terms in the Navier-Stokes equations are expressed in the convective form and approximated through the Lesaint-Raviart fluxes modified for DG methods. Verification of the solver is performed for a series of test problems; purely elliptic, unsteady Stokes and full Navier-Stokes. The resulting method leads to a stable scheme for the unsteady Stokes and Navier-Stokes equations when equal order approximation is used for velocity and pressure. For the validation of the full Navier-Stokes solver, we consider unsteady laminar flow past a square cylinder at a Reynolds number of 100 (unsteady wake). The DG solver shows favourably comparisons to experimental data and a continuous Spectral code. (C) 2010 Elsevier Ltd. All rights reserved.