A spatial-temporal weight analysis and novel nonlinear weights of weighted essentially non-oscillatory schemes for hyperbolic conservation laws

TL;DR

This paper analyzes existing WENO schemes for hyperbolic conservation laws, introduces novel nonlinear weights based on a logarithmic function, and demonstrates improved shock capturing with minimal additional computational cost.

Contribution

The paper proposes a new Z-type nonlinear weight for finite volume WENO schemes that reduces numerical dissipation while maintaining efficiency.

Findings

Enhanced accuracy of WENO schemes in the first time step.

Reduced numerical dissipation around discontinuities.

Effective shock capturing demonstrated through numerical examples.

Abstract

In this paper we analyze the weighted essentially non-oscillatory (WENO) schemes in the finite volume framework by examining the first step of the explicit third-order total variation diminishing Runge-Kutta method. The rationale for the improved performance of the finite volume WENO-M, WENO-Z and WENO-ZR schemes over WENO-JS in the first time step is that the nonlinear weights corresponding to large errors are adjusted to increase the accuracy of numerical solutions. Based on this analysis, we propose novel Z-type nonlinear weights of the finite volume WENO scheme for hyperbolic conservation laws. Instead of taking the difference of the smoothness indicators for the global smoothness indicator, we employ the logarithmic function with tuners to ensure that the numerical dissipation is reduced around discontinuities while the essentially non-oscillatory property is preserved. The proposed scheme does not necessitate substantial extra computational expenses. Numerical examples are presented to demonstrate the capability of the proposed WENO scheme in shock capturing.