An extended seventh-order compact nonlinear scheme with positivity-preserving property

Abstract

A seventh-order accurate high-resolution compact nonlinear scheme is developed based on the fifth-order scheme, TCNS-II (Zhang et al., 2019), such that the stencil used for the interpolation is extended to include more endpoints in the smooth flowfield, thus maximizing the order of accuracy, while in the vicinity of discontinuities, the scheme can fall back to the original five-point scheme, thus maintaining strong stability. In the proposed extension strategy, the evaluation procedure of the smoothness measurement is simplified by following the method of the typical five-point scheme, i.e., no more complex local or global smoothness indicators are involved. The extension strategy is further optimized through the reuse of the available smoothness information, which significantly reduces the computational effort. A positivity-preserving flux limiting technique is incorporated into the flux differencing of the proposed high-order scheme for the compressible Euler system of equations in both one and two dimensions. Positive values of density and pressure are strictly enforced even for flow problems involving severe discontinuities. A variety of benchmark cases is used to demonstrate the improved performance of the present scheme particularly in shock-capturing and multi-scale wave-resolving capabilities.