# Discontinuous Galerkin solution of incompressible flows

## Francesco Bassi

Dipartimento di Ingegneria Industriale
Universita di Bergamo
Viale Marconi 5
24044 Dalmine (BG), Italy

In this talk we will present our recent work about the development of a discontinuous Galerkin (DG) method for incompressible flows.

Application of the DG discretization to the incompressible flow equations has been considered in [1] for the two dimensional vorticity streamfunction formulation of the equations, and more recently in [2-3-4] for the Stokes, Oseen and Navier-Stokes equations in primitive variables formulation.

In a recent paper [5] we also apply the DG discretization to the primitive variables form of the incompressible Navier-Stokes equations, but we propose a novel formulation of the inviscid numerical flux which relies on the solution of a Riemann problem for the incompressible Euler equations with a suitably relaxed incompressibility constraint.

In our method the old idea of relaxing the incompressibility constraint by adding an artificial compressibility term to the continuity equation is exploited in a new way in that it is employed only for the construction of the interface fluxes. This entails that the DG discretization here introduced is always a consistent approximation of the incompressible Euler or Navier-Stokes equations since, independently of the amount of artificial compressibility added, the interface flux reduces to the physical one for vanishing interface jumps, i.e. for a continuous solution.

Besides presenting the main features of the method, in the talk we will give numerical evidence of the accuracy and versatility of the method by showing the numerical results of higher order computations of steady and unsteady, inviscid and viscous incompressible flows.

• [1] J.-G. Liu and C.-W. Shu. A high-order discontinuous Galerkin method for 2D incompressible flows. J. Comput. Phys., 160:577--596, 2000.
• [2] B. Cockburn, G. Kanschat, D. Sch?tzau, and C. Schwab. Local discontinuous Galerkin methods for the Stokes system. SIAM J. Numer. Anal., 40(1):319--343, 2002.
• [3] B. Cockburn, G. Kanschat, and D. Sch?tzau. The local discontinuous Galerkin method for the Oseen equations. Math. Comp., 73(246):569--593, 2003.
• [4] B. Cockburn, G. Kanschat, and D. Sch?tzau. A locally conservative LDG method for the incompressible Navier-Stokes equations. Math. Comp., 2005. Article electronically published on October 5, 2004.
• [5] F. Bassi, A. Crivellini, D. A. Di Pietro, and S. Rebay. An Artificial Compressibility Flux for the Discontinuous Galerkin Solution of the Incompressible Navier-Stokes Equations. J. Comput. Phys., 2005. submitted.