New Adaptive Low-Dissipation Central-Upwind Schemes

TL;DR

The paper addresses accurate, high-resolution shock capturing for 1-D and 2-D hyperbolic conservation laws by marrying low-dissipation central-upwind fluxes with adaptive nonlinear limiters. It introduces two smoothness indicators, a minmod-based indicator and a weak local residual indicator, to automatically identify non-smooth regions and apply overcompressive SBM limiters there while using dissipative Minmod2 limiters in smooth regions, with reconstructions performed in local characteristic variables via LCD. The approach yields significantly reduced numerical dissipation and sharper resolution of shocks and contact discontinuities in 1-D and 2-D Euler tests, with MM-based adaptivity offering robust performance and WLR-based adaptivity delivering sharper results in some scenarios at the cost of tuning. Overall, the adaptive LDCU framework enhances accuracy and stability for gas-dynamics simulations, showing strong potential for robust, high-resolution computations of complex flow features.

Abstract

We introduce new second-order adaptive low-dissipation central-upwind (LDCU) schemes for the one- and two-dimensional hyperbolic systems of conservation laws. The new adaptive LDCU schemes employ the LDCU numerical fluxes (recently proposed in [{\sc A. Kurganov and R. Xin}, J. Sci. Comput., 96 (2023), Paper No. 56]) computed using the point values reconstructed with the help of adaptively selected nonlinear limiters. To this end, we use a smoothness indicator to detect ``rough'' parts of the computed solution, where the piecewise linear reconstruction is performed using an overcompressive limiter, which leads to extremely sharp resolution of shock and contact waves. In the ``smooth'' areas, we use a more dissipative limiter to prevent the appearance of artificial kinks and staircase-like structures there. In order to avoid oscillations, we perform the reconstruction in the local characteristic variables obtained using the local characteristic decomposition. We test two different smoothness indicators and apply the developed schemes to the one- and two-dimensional Euler equations of gas dynamics. The obtained numerical results clearly demonstrate that the new adaptive LDCU schemes outperform the original ones.

Figures (10)

• Figure 4.1: Example 1: Density $\rho$ computed by the LDCU, A-MM, and A-WLR schemes (left) and zoom at $x\in[-1,0]$ (right).
• Figure 4.2: Example 2: Density $\rho$ computed by the LDCU, A-MM, and A-WLR schemes (left) and zoom at $x\in[11.8,13.6]$ (right).
• Figure 4.3: Example 2: Density $\rho$ computed by the A-WLR scheme with $\texttt{C}=0.2$ and $\texttt{0.35}$ on a coarse (left) and fine (right) meshes. Zoom at $x\in[9,11]$.
• Figure 4.4: Example 3: Density $\rho$ computed by the LDCU, A-MM, and A-WLR schemes (left) and zoom at $x\in[0.55,0.85]$.
• Figure 4.5: Example 4: Density $\rho$ computed by the LDCU (left), A-MM (middle), and A-WLR (right) schemes.

Theorems & Definitions (5)

• Remark 2.1
• Remark 2.2
• Remark 4.1
• Remark 4.2
• Remark A.1