Simple smoothness indicator WENO-Z scheme for hyperbolic conservation laws

TL;DR

This paper introduces a WENO-Z type scheme with a novel global smoothness indicator that improves accuracy and reduces dissipation near discontinuities for hyperbolic conservation laws.

Contribution

A new WENO-Z scheme with a global smoothness indicator is proposed, enhancing convergence and accuracy at critical points and smooth regions.

Findings

Achieves the desired order of convergence in smooth regions

Reduces dissipation near discontinuities

Performs well on Euler equations in 1D and 2D

Abstract

The advantage of WENO-JS5 scheme [ J. Comput. Phys. 1996] over the WENO-LOC scheme [J. Comput. Phys.1994] is that the WENO-LOC nonlinear weights do not achieve the desired order of convergence in smooth monotone regions and at critical points. In this article, this drawback is achieved with the WENO-LOC smoothness indicators by constructing a WENO-Z type nonlinear weights which contains a novel global smoothness indicator. This novel smoothness indicator measures the derivatives of the reconstructed flux in a global stencil, as a result, the proposed numerical scheme could decrease the dissipation near the discontinuous regions. The theoretical and numerical experiments to achieve the required order of convergence in smooth monotone regions, at critical points, the essentially non-oscillatory (ENO), the analysis of parameters involved in the nonlinear weights like $\epsilon$ and $p$ are studied. From this study, we conclude that the imposition of certain conditions on $\epsilon$ and $p$, the proposed scheme achieves the global order of accuracy in the presence of an arbitrary number of critical points. Numerical tests for scalar, one and two-dimensional system of Euler equations are presented to show the effective performance of the proposed numerical scheme.