A Simplified Artificial Compressibility Flux for the Discontinuous Galerkin Solution of the Incompressible Navier-Stokes Equations (Communications in Computational Physics, 2018)
The discontinuous Galerkin (DG) method has attained increasing popularity for solving the incompressible Navier-Stokes (INS) equations in recent years. Inthis work, we present a novel DG discretization for solving the two-dimensional INSequations in which the inviscid term of the INS equations is split into two parts, theStokes operator and the nonlinear convective term, and treated separately. The Stokesoperator is discretized using the artificial compressibility flux which is provided bythe (exact) solution of a Riemann problem associated with a local artificial compressibility perturbation of the Stokes system, while the nonlinear term is discretized indivergency form by using the local Lax-Friedrichs fluxes; thus, local conservativity isinherent. Unlike the existing artificial compressibility flux for the DG discretizationof the INS equations which needs to solve a Riemann problem for a nonlinear systemby numerical iteration, the separate treatment of the nonlinear term from the Stokesoperator makes the Riemann problem become linear and can be solved explicitly andstraightforwardly, therefore, no iterative procedure is further required. A number oftest cases with a wide range of Reynolds number are presented to assess the performance of the proposed method, which demonstrates its potential to be an alternativeapproach for high order numerical simulations of incompressible flows.
Keywords: Discontinuous Galerkin methods; Navier-Stokes equations; incompressible flows; artificial compressibility flux.