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.

翻译：我们针对一维和二维双曲守恒律系统，提出了新的二阶自适应低耗散中心迎风（LDCU）格式。新自适应LDCU格式采用基于[{\sc A. Kurganov 与 R. Xin}，J. Sci. Comput.，96（2023），论文编号56]中最新提出的LDCU数值通量，该通量通过借助自适应选取的非线性限制器重构的点值进行计算。为此，我们使用光滑度指标检测计算解中的“粗糙”部分，在此区域采用过压缩限制器进行分段线性重构，从而实现对激波和接触间断的超锐利分辨率。在“光滑”区域，则使用更具耗散性的限制器，以防止出现人工扭结和阶梯状结构。为避免振荡，我们利用局部特征分解获得的局部特征变量进行重构。我们测试了两种不同的光滑度指标，并将所提格式应用于一维和二维欧拉气体动力学方程组。数值结果表明，新的自适应LDCU格式显著优于原始格式。