New Adaptive Low-Dissipation Central-Upwind Schemes

TL;DR

This paper presents new second-order adaptive low-dissipation central-upwind schemes for hyperbolic conservation laws, using adaptive limiters and smoothness indicators to improve shock resolution and reduce artificial artifacts.

Contribution

Introduction of novel adaptive LDCU schemes with nonlinear limiters and smoothness indicators for enhanced shock capturing in hyperbolic systems.

Findings

Outperform original LDCU schemes in numerical tests.

Effectively resolve shock and contact waves.

Reduce artificial kinks and staircase artifacts.

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.