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

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.

Figures (20)

• Figure 1: The WENO approximation of $v(x)$ at the cell boundaries $v^-_{i+1/2}$ and $v^+_{i-1/2}$ depends on the cell averages over the stencil $S^5 = \{ I_{i-2}, \cdots, I_{i+2} \}$, as well as the substencils $S_0, S_1, S_2$.
• Figure 2: The nonlinear weights $\omega^\mathcal{N}_0$ (left), $\omega^\mathcal{N}_1$ (middle) and $\omega^\mathcal{N}_2$ (right) in Stencil \ref{['case:10']} by different WENO techniques.
• Figure 3: $e^{\mathcal{N}\!,\, 2}_0$ (left), $e^{\mathcal{N}\!,\, 2}_1$ (middle) and $\left| e^{\mathcal{N}\!,\, 2}_0/e^{\mathcal{N}\!,\, 2}_1 \right|$ (right).
• Figure 4: The nonlinear weights $\omega^\mathcal{N}_0$ (left), $\omega^\mathcal{N}_1$ (middle) and $\omega^\mathcal{N}_2$ (right) in Stencil \ref{['case:19']} by different WENO techniques.
• Figure 5: The nonlinear weights $\omega^\mathcal{N}_0$ (left), $\omega^\mathcal{N}_1$ (middle) and $\omega^\mathcal{N}_2$ (right) in Stencil \ref{['case:20']} by different WENO techniques.

Theorems & Definitions (10)

• Example 5.1
• Example 5.2
• Example 5.3
• Example 5.4
• Example 5.5
• Example 5.6
• Example 5.7
• Example 5.8
• Example 5.9
• Example 5.10