Nonlinear weights for shock capturing schemes with unconditionally optimal high order

Abstract

We develop in this paper nonlinear weights for shock capturing schemes, such that the design optimal high order is achieved regardless of any order of the critical point. In particular, we validate the proposed nonlinear weights by testing their performance for a fifth-order weighted compact nonlinear scheme (WCNS), which has advantages in terms of implementing numerical fluxes and keeping the freestream property on curvilinear grids. Theoretical analysis is performed to demonstrate the unconditionally optimal high order of the proposed method. In addition, the spectral property of the proposed nonlinear weights is compared with some existing nonlinear ones. Several one- and two-dimensional benchmark examples are presented to validate the advantages of the proposed method. It is worth noting that the developed nonlinear weights can be applied straightforwardly for widely used weighted essentially non-oscillatory (WENO) schemes.