Fifth-order weighted essentially non-oscillatory schemes with new Z-type nonlinear weights for hyperbolic conservation laws

TL;DR

This paper introduces a new Z-type nonlinear weighting method for fifth-order WENO schemes that improves accuracy at critical points and sharpness near discontinuities in hyperbolic conservation law solutions.

Contribution

The paper develops a novel Z-type nonlinear weight based on the pth root of smoothness indicators, enhancing WENO scheme accuracy and shock capturing capabilities.

Findings

Achieves fifth order accuracy in smooth regions, including at critical points.

Provides sharper approximations around discontinuities.

Outperforms existing WENO schemes like WENO-JS, WENO-M, and WENO-Z in numerical tests.

Abstract

In this paper we propose new Z-type nonlinear weights of the fifth-order weighted essentially non-oscillatory (WENO) finite difference scheme for hyperbolic conservation laws. Instead of employing the classical smoothness indicators for the nonlinear weights, we take the $p$th root of the smoothness indicators and follow the form of Z-type nonlinear weights, leading to fifth order accuracy in smooth regions, even at the critical points, and sharper approximations around the discontinuities. We also prove that the proposed nonlinear weights converge to the linear weights as $p \to \infty$, implying the convergence of the resulting WENO numerical flux to the finite difference numerical flux. Finally, numerical examples are presented by comparing with other WENO schemes, such as WENO-JS, WENO-M and WENO-Z, to demonstrate that the proposed WENO scheme performs better in shock capturing.