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

Abstract

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.