A kind of third order central scheme based on mapping functions

High-order finite difference schemes have low numerical dissipation and dispersion error. They can capture flow structures in detail. In engineering applications, robustness should be considered, while high resolution is pursued. The method of mapping function was first proposed by Henrick to adjust nonlinear weights in 0, 1 for the fifth-order WENO scheme. With the help of this method, the requirement of convergence order is satisfied, and the performance of the scheme is improved. Different from Henrick's method, a concept of piecewise polynomial function is proposed by authors in this paper, and the corresponding improved WENO schemes have better resolution with the preservation of robustness. In this study, a fifth-order piecewise polynomial function (PPM5) is applied to a third order central scheme, which is referred as the SWENO3-PPM5. Henrick's mapping function is also applied to the third order central scheme, which is referred as the SWENO3-M. The performance of the SWENO3-PPM5 and the SWENO3-M are compared with that of the NND and the WENO3. The analyses of approximate dispersion relations (ADR) indicate that the dissipation and dispersion relations of SWENO3-PPM5 and SWENO3-M have an improvement compared with those of the WENO3. The performance of the new schemes is evaluated by testing typical problems, i.e. one-dimensional Shu-osher problem, two-dimensional ones including shock-boundary layer interaction problem, shock/vortex interaction problem, and hypersonic flows of the sharp double cone. The results indicate that the new schemes have better resolution than the WENO3 and the NND and can present a better description on complex flow structures, such as shock waves and contact discontinuity. Because the PPM5 can make the final weight closer to the idea weight of the scheme, the SWENO3-PPM5 possesses a little better resolution than the SWENO3-M.