Towards optimal high-order compact schemes for simulating compressible flows

TL;DR

This paper introduces an improved high-order compact nonlinear scheme for simulating compressible flows, combining ENO-like stencil selection with optimal linear weights to enhance shock capturing and resolution in smooth flow regions.

Contribution

The work integrates ENO-like stencil selection into WCNS to fully eliminate oscillatory stencils and recover optimal resolution in smooth flows, advancing high-order compact schemes.

Findings

Enhanced shock capturing capability.

High resolution in smooth flow regions.

Validated on scalar and Euler equations.

Abstract

Weighted compact nonlinear schemes (WCNS) [Deng and Zhang, JCP 165(2000): 22-44] were developed to improve the performance of the compact high-order nonlinear schemes (CNS) by utilizing the weighting technique originally designed for WENO schemes, and excellent shock capturing capability and high resolution are achieved. Various work has been given for further improving the performance of WCNSs since then. In this work, the ENO-like stencil selection procedure of Targeted ENO schemes [Fu et al. JCP 305(2016):333-359] is introduced for interpolating midpoint variables, targeting compact nonlinear schemes which fully abandon the oscillatory stencils crossing discontinuities, and directly apply optimal linear weights when the flow field is smooth, such that the optimal numerical resolution is fully recovered in smooth flow field. Several canonical numerical cases of scalar equations and the Euler equations of gas dynamics are given to examine the performance of the presented method.