Free-stream preserving linear-upwind and WENO schemes on curvilinear grids

Abstract

Applying high-order finite-difference schemes, like the extensively used linear-upwind or WENO schemes, to curvilinear grids can be problematic. If the scheme doesn't satisfy the geometric conservation law, the geometrically induced error from grid Jacobian and metrics evaluation can pollute the flow field, and degrade the accuracy or cause the simulation failure even when uniform flow imposed, i.e. free-stream preserving problem. In order to address this issue, a method for general linear-upwind and WENO schemes preserving free-stream on stationary curvilinear grids is proposed. Following Lax-Friedrichs splitting, this method rewrites the numerical flux into a central term, which achieves free-stream preserving by using symmetrical conservative metric method, and a numerical dissipative term with a local difference form of conservative variables for neighboring grid-point pairs. In order to achieve free-stream preservation for the latter term, the local differences are modified to share the same Jacobian and metric terms evaluated by high order schemes. In addition, this method allows a simple hybridization switching between linear-upwind and WENO schemes is proposed for improving computational efficiency and reducing numerical dissipation. A number of testing cases including free-stream, isentropic vortex convection, double Mach reflection, flow past a cylinder and supersonic wind tunnel with a step are computed to verify the effectiveness of this method.