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<p>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 second-order total variation diminishing Runge–Kutta method. The rationale for the improved performance of the finite volume WENO-Z scheme 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 a 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 bring substantial extra computational expenses.</p>

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A Spatial–Temporal Weight Analysis and Novel Nonlinear Weights of Weighted Essentially Non-oscillatory Schemes for Hyperbolic Conservation Laws

  • 期刊论文

  • 发表日期：

    2025年1月3日

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  • Xinjuan Chen,
  • Jiaxi Gu,
  • Jae-Hun Jung

摘要

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 second-order total variation diminishing Runge–Kutta method. The rationale for the improved performance of the finite volume WENO-Z scheme 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 a 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 bring substantial extra computational expenses.