High Order Boundary Conditions for High Order Finite Difference Schemes on Curvilinear Coordinates Solving Compressible Flows

Abstract

In the present paper, high order boundary conditions for high order finite difference schemes on curvilinear coordinates are implemented and analyzed. The finite difference scheme employed in the present paper is a conservative scheme on a finite volume type grid arrangement for which the implementation of the boundary conditions is characterized by a proper evaluation of the numerical fluxes at the boundaries. Simple extrapolation procedures dealing with both the inviscid and viscous fluxes at the boundary are proposed. Specifically, for the inviscid fluxes, a characteristic-variable-based extrapolation procedure is adopted, which effectively reduces the numerical reflection at the far-field boundaries. The emphasis of the present paper is to design straightforward and efficient numerical boundary closure procedures which are capable of achieving high order of accuracy near the boundaries. Special attention is paid to the influence of the extrapolation accuracy on the overall accuracy of the scheme. Numerical results show that the finite difference scheme can achieve its nominal order accuracy when the extrapolation accuracy is sufficiently high.