发表机构
Nanjing University of Aeronautics and Astronautics; Uppsala University; KTH Royal Institute of Technology（南京航空航天大学; 乌普萨拉大学; 皇家理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出基于间断伽辽金框架的高阶切割有限元方法，结合宏单元稳定化与通量限制，在复杂域上求解标量双曲守恒律，保持最大值原理并实现无振荡的激波捕捉。
AI 中文摘要
本文提出了一族基于间断伽辽金（DG）框架的高阶切割有限元方法，用于在复杂域上求解标量双曲守恒律。在我们先前工作的基础上，我们发展了一种多维格式，将宏单元稳定化与通量限制相结合，以获得保持最大值原理且对背景网格的任意边界切割具有鲁棒性的方案。物理域嵌入在规则背景网格中，这可能会产生任意小的切割单元。为避免通常与此类单元相关的严苛时间步长限制，我们在宏单元的内部面上添加了鬼罚稳定项。所得方法在拟合网格上表现出与标准DG方法相似的稳定性和精度特性。在半离散格式下，我们推导了周期边界条件和流入-流出边界条件下的$L^2$稳定性结果。为强制最大值原理并抑制非物理振荡，我们通过定义宏单元上的限制参数，将标准DG方法中的限制器技术适配到CutFEM设置中。特别地，我们提出了一种基于宏单元的参数化通量限制器，以及对Zhang-Shu保界限制器和Barth-Jespersen斜率限制器的适配。二维和三维空间中的数值实验表明，即使面对涉及极小切割单元相交的挑战性切割构型，该方法也能实现最优收敛阶、保持最大值原理，并准确捕捉激波而无伪振荡。
英文摘要
In this paper, we present a family of high-order cut finite element methods based on the discontinuous Galerkin (DG) framework for scalar hyperbolic conservation laws on complex domains. Building on our previous work, we develop a multidimensional formulation that combines macro-element stabilization with flux limiting to obtain a scheme that preserves the maximum principle and remains robust with respect to arbitrary boundary cuts of the background mesh. The physical domain is embedded in a regular background mesh, which may produce arbitrarily small cut cells. To avoid the severe time step restrictions typically associated with such cells, ghost penalty stabilization terms are added on interior facets of macro-elements. The resulting method exhibits stability and accuracy properties similar to those of standard DG methods on fitted meshes. An $L^2$-stability result is derived for the semi-discrete scheme under both periodic and inflow-outflow boundary conditions. To enforce the maximum principle and suppress nonphysical oscillations, we adapt limiter techniques from standard DG methods to the CutFEM setting by defining limiting parameters on macro-elements. In particular, we present a macro-element-based parameterized flux limiter together with adaptations of the Zhang-Shu bound-preserving limiter and the Barth-Jespersen slope limiter. Numerical experiments in two and three spatial dimensions demonstrate optimal convergence orders, preservation of the maximum principle, and accurate shock capturing without spurious oscillations, even for challenging cut configurations involving very small-cut cell intersections.
Comments30 pages
