Hybrid Compact Least-Squares and Central Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws on Structured Curvilinear Grids

TL;DR

This paper introduces hybrid high-order schemes combining compact least-squares and WENO methods for solving hyperbolic conservation laws on structured curvilinear grids, effectively capturing shocks and preserving multiscale features.

Contribution

The paper proposes a novel hybrid scheme that integrates compact least-squares and WENO techniques, with optimized parameters and a robust shock detector, enhancing accuracy and efficiency on complex grids.

Findings

Effective shock capturing with minimal oscillations.

High resolution of multiscale structures in smooth regions.

Validated on diverse 1D and 2D flow problems.

Abstract

A series of third- and fifth-order hybrid compact least-squares central weighted essentially non-oscillatory schemes are proposed and applied to curvilinear structured grids for the finite volume method. In smooth regions, compact least-squares schemes based on interfacial differences of derivatives are utilized to keep the high resolution of multiscale structures, whereas in discontinuous regions, central weighted essentially non-oscillatory schemes are included to enable the oscillation free shock capturing capability of the hybrid schemes. Free parameters in the proposed compact least-squares schemes are firstly optimized to reach a broad range of resolved bandwidths with different levels of dissipation. In addition, a shock detector proposed in our previous work is introduced and validated to be not only able to detect smooth first-order extrema but also robust in detecting discontinuities. Through the solution of block tridiagonal reconstruction linear systems, the resulting schemes can give an explicit polynomial for each control volume and are efficient with an acceptable computational overhead when compared to the pure central weighted essentially non-oscillatory schemes. Eventually, benchmarks including one-dimensional and two-dimensional, linear and nonlinear, inviscid and viscous problems on uniform and nonuniform curvilinear grids demonstrate the proposed schemes' promising applicability in compressible flows which have both multiscale structures and discontinuities.