A Third-Order Weighted Essentially Non-Oscillatory Compact Least-Squares Scheme for Hyperbolic Conservation Laws on Non-Uniform Grids

TL;DR

This paper introduces a third-order weighted essentially non-oscillatory compact least-squares scheme for hyperbolic conservation laws on non-uniform grids, achieving high accuracy and robustness in smooth and discontinuous regions.

Contribution

It develops a novel third-order WENO compact least-squares scheme with an improved shock detector for structured non-uniform grids, extending to Euler equations.

Findings

High accuracy on non-uniform grids

Effective shock detection and dissipation control

Robust performance on Euler and Navier-Stokes equations

Abstract

A third-order weighted essentially non-oscillatory compact least-squares scheme is developed for the finite volume method on structured curvilinear non-uniform grids. The proposed scheme features compact least-squares reconstruction with optimal linear weights for broad-spectrum accuracy and non-linear weights for essentially non-oscillatory property. Through explicit second-order polynomials given for each control volume, the scheme maintains high accuracy on structured meshes with non-uniform grids. By integrating an improved shock detector developed in this work, coefficients with adaptive levels of dissipation are applied to achieve both the high resolution in smooth regions and high robustness in discontinuous regions. Furthermore, the proposed scheme is extended to the Euler equations through a characteristic decomposition technique. Numerical examples including both linear convection equation and nonlinear Euler/Navier-Stokes equations demonstrate the robustness and high-resolution of the proposed method.