A second order finite-volume scheme on cartesian grids for Euler equations

International audienceWe present a second-order finite-volume scheme for compressible Euler flows in complex geometries, with a discretization on a cartesian grid which in general does not fit to the geometry. In the fluid domain, away from the solid boundary, we use a classical finite-volume method based on an approximate Riemann solver. At the cells located on the boundary with the solid, we solve an ad hoc Riemann problem taking into account the relevant boundary condition for the convective fluxes by an appropriate definition of the contact discontinuity speed. To avoid pressure oscillations near the boundary, we weight the boundary condition with a fluid extrapolation, as a function of the angle between the normal to the boundary and the cartesian mesh. The scheme is simple to implement and it is formally second order accurate. Error convergence rates with respect to several exact test cases are investigated, as well as comparisons with other numerical methods of the literature. Examples of flow solutions in one, two and three dimensions are presented