Discontinuous Galerkin schemes for the compressible Navier-Stokes equations

Abstract

The discretization of the viscous terms in a space discontinuous Galerkin method is investigated through a theoretical analysis and numerical calculations.Two formulations are considered : the first one uses a shifted cell and the other one auxiliary variables. Both are second order accurate on Cartesian meshes. They are applied to the interaction of a reflected shock with a boundary layer in a shock tube and compared to a higher order TVD scheme. Discontinuous Galerkin methods are found to provide accurate solutions for the unsteady compressible Navier-Stokes equations. 1 Introduction Among the various methods to devise spatial discretizations, the space discontinuous Galerkin methods, i.e. weighted residual methods using a local basis, are now famous for their high accuracy. Based on ideas due to Lesaint and Raviart (1974) and van Leer (1977), they have been first applied to the compressible Euler equations by Allmaras and Giles (1987) and Cockburn et hi.. Borrel and Berde (1995) developed a limiter-free p1 approach with a great accuracy on irregular structured meshes. The present authors [Drozo (97)] proved it to be third-order accurate for linear problems. More recently, the space discontinuous Galerkin method have been extended to the compressible Navier-Stokes equations [Borrel (95), Bassi (96)]. In this paper two formulations for the discretization of the shear stress and the heat transfer terms are studied. Their accuracy and stability are analyzed for a scalar advection-diffusion equation. Then both approaches are applied to the direct simulation of a complex unsteady problem studied in Daru (1998) : the interaction of a reflected shock with a boundary layer in a shock tube. Numerical results are compared to those given by a third-order TVD scheme.

A Problem-Independent Limiter for High-Order Runge–Kutta Discontinuous Galerkin Methods

“…A very simple example of the projection effects on two configurations is proposed, in order to understand the action of (19). The projection's results for an extremum are explained by referring to Fig.…”

“…Biswas et al [6] used the present method to achieve parallel adaptive refinement for conservation laws. For more details on the use of RKDG methods, see for example the introduction of [11] and [7,19,20,24].…”