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We describe in this paper the improvement on the numerical resolution of a fluid dynamics system by means of an Adaptive Mesh Refinement algorithm in order to handle an infinitely thin interface. This model is derived from the compressible Navier-Stokes equations in the case of diphasic flows for which both phases have a low Mach number. It consists of a coupled hyperbolic-elliptic system. The first part is numerically treated thanks to a hierarchical grid structure whereas we use the Local Defect Correction method to solve the second part.

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We present in this paper two Euler/Navier-Stokes couplings for multiscale aeroacoustic problems based on a Discontinuous Galerkin method: the first coupling concerns an interface coupling between adjacent domains, the second coupling concerns a coupling in volume and interface between overlapping domains. In both cases, each domain provide a donor field to the other domain. After a numerical study of the convergence of the precision of the method, these couplings are compared on a 2D test case of a flow around a cylinder and the noise generated.

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.