WLS-ENO: Weighted-Least-Squares Based Essentially Non-Oscillatory Schemes for Finite Volume Methods on Unstructured Meshes

TL;DR

The paper introduces WLS-ENO, a new non-oscillatory finite-volume scheme for hyperbolic PDEs on unstructured meshes, offering improved flexibility and robustness over existing methods like WENO.

Contribution

WLS-ENO is a novel scheme based on weighted least squares that does not require sub-stencils, enhancing flexibility and mesh insensitivity for unstructured mesh applications.

Findings

WLS-ENO achieves high-order accuracy and stability on unstructured meshes.

Numerical results demonstrate superior performance compared to traditional WENO schemes.

The scheme is effective in 1-D, 2-D, and 3-D benchmark problems.

Abstract

ENO (Essentially Non-Oscillatory) and WENO (Weighted Essentially Non-Oscillatory) schemes are widely used high-order schemes for solving partial differential equations (PDEs), especially hyperbolic conservation laws with piecewise smooth solutions. For structured meshes, these techniques can achieve high order accuracy for smooth functions while being non-oscillatory near discontinuities. For unstructured meshes, which are needed for complex geometries, similar schemes are required but they are much more challenging. We propose a new family of non-oscillatory schemes, called WLS-ENO, in the context of solving hyperbolic conservation laws using finite-volume methods over unstructured meshes. WLS-ENO is derived based on Taylor series expansion and solved using a weighted least squares formulation. Unlike other non-oscillatory schemes, the WLS-ENO does not require constructing sub-stencils, and hence it provides more flexible framework and is less sensitive to mesh quality. We present rigorous analysis of the accuracy and stability of WLS-ENO, and present numerical results in 1-D, 2-D, and 3-D for a number of benchmark problems, and also report some comparisons against WENO.