A sixth-order WCNS based on nonpolynomial interpolation with enhanced accuracy and resolution

The classic fifth-order weighted compact nonlinear scheme (WCNS) suffers from excessive numerical dissipation and an accuracy mismatch between its nonlinear interpolation and flux differences. Although the sixth-order central/upwind WCNS (WCNS-CU6) resolves the accuracy mismatch, it compromises stability. In this paper, an alternative sixth-order WCNS based on nonpolynomial interpolation (WCNS-NP6) is proposed to enhance accuracy and resolution while maintaining stability. The basic framework of WCNS-NP6 relies on the nonlinear weighting of three-point substencils, similar to the classic fifth-order WCNS. However, in WCNS-NP6, a radial basis function (RBF) is used to interpolate variables from point-based stencils to midpoints, and information from a global six-point stencil is integrated through the shape parameter of the RBF to achieve sixth-order accuracy. A novel measurement function is constructed to assess the smoothness of the six-point stencil. Near discontinuities, the measurement function adaptively removes the shape parameter, reverting WCNS-NP6 to the classic fifth-order WCNS and thereby ensuring stability. In smooth regions, the measurement function confines the active range of the nonlinear weights, thereby mitigating the impact of nonlinear mechanisms on spectral properties. Furthermore, a stencil rotation method is presented to ensure that WCNS-NP6 maintains its nominal sixth-order accuracy for solutions containing arbitrary numbers and orders of critical points. The numerical tests demonstrate that WCNS-NP6 outperforms classic fifth-order and sixth-order WCNSs in terms of numerical dissipation, resolution, and accuracy, particularly at high-order critical points. Notably, the WCNS-NP6 scheme demonstrates better stability than the classical sixth-order WCNS-CU6 scheme, while the computational cost increases by only 19% in 2D benchmark inviscid cases and remains below 10% in a 3D viscous case in engineering.