An accurate and robust finite volume scheme based on the spline interpolation for solving the Euler and Navier-Stokes equations on non-uniform curvilinear grids

Spline schemes are proposed to simulate compressible flows on non-uniform structured grid in the framework of finite volume methods. The cubic spline schemes in the present paper can achieve fourth and third order accuracy on the uniform and non-uniform grids respectively. Due to the continuity of cubic spline polynomial function, the inviscid flux can be computed directly from the reconstructed spline polynomial without using the Riemann solvers or other flux splitting techniques. Isotropic and anisotropic artificial viscosity models are introduced to damp high frequency numerical disturbances and to enhance the numerical stability. The first derivatives that are used to calculate the viscous flux are directly obtained from the cubic spline polynomials and preserve second order accuracy on both uniform and non-uniform grids. A hybrid scheme, in which the spline scheme is blended with shock-capturing WENO scheme, is developed to deal with flow discontinuities. Benchmark test cases of inviscid/viscous flows are presented to demonstrate the accuracy, robustness and efficiency of the proposed schemes.