A Well-Balanced Fifth-Order A-WENO Scheme Based on Flux Globalization

TL;DR

This paper tackles accurate, well-balanced numerical simulation of nonconservative hyperbolic systems in quasi-conservative form $\bm U_t+\bm F(\bm U)_x=\bm B(\bm U)\bm U_x$. It introduces fifth-order flux globalization based WB A-WENO PCCU schemes, employing higher-order corrections $ (\bm K_{xx})$ and $(\bm K_{xxxx})$ and Ai-WENO-Z reconstructions to obtain the interface flux $\bm{\cal K}_{j+\frac12}$. Applied to the nozzle flow system and the two-layer shallow water equations, the method preserves a broad range of steady states and achieves higher resolution than the second-order counterpart. Numerical tests show improved accuracy and robustness using SSP Runge-Kutta time stepping and fifth-order quadrature, supporting its applicability to nonconservative models with variable topography.

Abstract

We construct a new fifth-order flux globalization based well-balanced (WB) alternative weighted essentially non-oscillatory (A-WENO) scheme for general nonconservative systems. The proposed scheme is a higher-order extension of the WB path-conservative central-upwind (PCCU) scheme recently proposed in [A. Kurganov, Y. Liu and R. Xin, J. Comput. Phys., 474 (2023), Paper No. 111773]. We apply the new scheme to the nozzle flow system and the two-layer shallow water equations. We conduct a series of numerical experiments, which clearly demonstrate the advantages of using the fifth-order extension of the flux globalization based WB PCCU scheme.

Figures (7)

• Figure 1: Example 1 (Convergent Nozzle): The difference $\rho(x,t)-\rho_{eq}$ (top row) and zoom at the perturbations (bottom row) at times $t=0.1$ (left column), 0.3 (middle column), and 0.5 (right column).
• Figure 2: Example 1: Same as in Figure \ref{['fig5']}, but for the divergent nozzle.
• Figure 3: Example 2: The difference $\rho(x,t)-\rho_{eq}$ (top row) and zoom at the perturbations (bottom row) at times $t=0.2$ (left column), 0.6 (middle column), and 1 (right column).
• Figure 4: Example 4: The difference $h_1(x,t)-(h_1)_{\rm eq}$ (top row) and zoom at the perturbations (bottom row) at times $t=0.02$ (left column), 0.05 (middle column), and 0.08 (right column).
• Figure 5: Example 5: The difference $h_1(x,t)-(h_1)_{\rm eq}$ (top row) and zoom at the perturbations (bottom row) at times $t=0.2$ (left column), 0.6 (middle column), and 1 (right column).