AI 中文总结
针对复杂几何中双曲守恒律求解，提出加权逆拉克斯 - 温德罗夫边界处理的间断伽辽金方法。通过特定原理和重构方式施加边界条件，消除时间步限制，提高精度与效率，经理论分析和数值实验验证了稳定性、有效性及鲁棒性。
AI 中文摘要
本文提出一种加权逆拉克斯 - 温德罗夫（WILW）边界处理方法，用于非拟合网格上的间断伽辽金（DG）方法，以有效求解复杂几何中的双曲守恒律。该方法对内部单元采用标准DG格式，通过ILW原理为边界附近的切割单元重构高阶近似多项式来施加数值边界条件，消除小切割单元导致的时间步限制。为解决基本ILW格式中数值误差对切割单元几何尺寸的敏感性，提高边界重构阶数，确保精度与切割单元大小无关。此外，采用加权最小二乘重构减少构造时对复杂高阶边界导数的需求。该方法保持高阶精度，显著提高多维问题计算效率。最后，通过线性稳定性分析从理论上验证了该方法的稳定性，并通过一系列标量和系统方程的一维及二维数值实验，数值验证了该方案的有效性和鲁棒性。
英文摘要
In this paper, we propose a weighted inverse Lax-Wendroff (WILW) boundary treatment for the discontinuous Galerkin (DG) method on unfitted meshes to efficiently solve hyperbolic conservation laws in complex geometries. The proposed method employs the standard DG scheme for interior cells and reconstructs high-order approximation polynomials via the ILW principle for cut cells near boundaries to impose numerical boundary conditions, effectively eliminating the time-step restriction typically caused by small cut cells. In particular, to address the sensitivity of numerical errors to the geometric size of cut cells in the basic ILW scheme, we raise the reconstruction order at the boundary, ensuring that accuracy becomes independent of the cut-cell size. Furthermore, it incorporates a weighted least-squares reconstruction to reduce the need for complex high-order boundary derivatives during construction. As a result, the method maintains high-order accuracy while significantly improving computational efficiency for multi-dimensional problems. Finally, the stability of the proposed method is theoretically validated through linear stability analysis, and the effectiveness and robustness of the proposed scheme are numerically verified through a series of one-dimensional and two-dimensional numerical experiments for scalar and system equations.
