New More Efficient A-WENO Schemes

TL;DR

The paper introduces new, more efficient fifth-order A-WENO schemes for both conservative and nonconservative hyperbolic systems. The core idea is to compute high-order correction terms using precomputed FV fluxes rather than repeatedly evaluating point fluxes, enabling substantial computational savings while preserving accuracy and stability. The authors extend the approach to flux globalization based well-balanced path-conservative formulations for nonconservative systems, and demonstrate the method in both 1-D and 2-D settings, including Euler and multifluid models. Numerical experiments show that the new schemes achieve comparable accuracy and resolution to existing schemes, with up to around 15% reductions in CPU time across a range of test problems. The work highlights the practicality of the approach for complex hyperbolic PDEs and provides a pathway to higher-order extensions and broader applications.

Abstract

We develop new more efficient A-WENO schemes for both hyperbolic systems of conservation laws and nonconservative hyperbolic systems. The new schemes are a very simple modification of the existing A-WENO schemes: They are obtained by a more efficient evaluation of the high-order correction terms. We conduct several numerical experiments to demonstrate the performance of the introduced schemes.

Figures (8)

• Figure 6.1: Example 1: Numerical solutions computed by the New and Old Schemes.
• Figure 6.2: Example 2: Numerical solutions computed by the New and Old Schemes with the zoom at the left part of the shock area.
• Figure 6.3: Example 3: Numerical solutions computed by the New and Old Schemes at different times.
• Figure 6.4: Example 4: Density $\rho$ computed by the New and Old Schemes with $\Delta x=1/20$.
• Figure 6.5: Example 5: Density computed by the New and Old Schemes (left) and zoom at $x\in[-2,0]$ (right).

Theorems & Definitions (5)

• Remark 2.1
• Remark 2.2
• Remark 3.1
• Remark 4.1
• Remark 5.1