Numerical Solution of the Navier–Stokes Equations by Semi–Implicit Schemes

In this paper we deal with a numerical solution of the compressible Navier-Stokes equations with the aid of higher order schemes. We use the discontinuous Galerkin finite element method for the space semi-discretization and a backward difference formula for the time discretization. Moreover, a linearization of inviscid/viscous fluxes and a suitable explicit extrapolation for nonlinear terms lead to the solution of a linear algebraic problem at each time step. Then we obtain an efficient numerical scheme which has a higher degree of approximation with respect to the space and time coordinates. The whole derivation of these semi– implicit schemes is briefly described in this paper and one preliminary numerical example is presented. Introduction Our goal is to develop an efficient, robust and accurate numerical scheme for a solution of a viscous compressible flow, which is described by the system of the Navier-Stokes equations. We use the discontinuous Galerkin method (DGM) for the space discretization. DGM is based on a piecewise polynomial, but discontinuous approximation and there are several variants of DGM for the solution of a viscous flow. We prefer the discontinuous Galerkin finite element (DGFE) method with a nonsymmetric variant of stabilization and an interior and boundary penalty. This scheme is usually denoted as NIPG (nonsymetric interior penalty Galerkin) method, see [1], [2]. In case of the time discretization, the method of lines is used. Runge-Kutta methods are very popular for their simplicity and a high order of accuracy, but they suffer from a strong restriction of the choice of the time step. To avoid their drawback it is suitable to use an implicit time discretization, but a fully implicit scheme leads to a system of highly nonlinear algebraic equations whose solution is rather complicated. Therefore, a semi-implicit method for the simulation of inviscid compressible flow was proposed in [4]. In this paper we extend the approach of semi-implicit scheme to the viscous case, where the diffusive terms are discretized by NIPG approach presented in [5]. Moreover, we apply a backward difference formula (BDF) to the time discretization which allows us the choice of a longer time step. The described method is applied to the numerical simulation of viscous flow around a flat plate. System of the Navier–Stokes equations Let Ω ⊂ IR be a bounded domain and T > 0. We set QT = Ω × (0, T ) and by ∂Ω denote the boundary of Ω which consists of several disjoint parts. We distinguish inlet ΓI , outlet ΓO and impermeable walls ΓW on ∂Ω. The system of the Navier-Stokes equations describing a 2D motion of a viscous compressible fluid can be written in the dimensionless form