Embedded, or immersed, approaches have the goal of reducing to the minimum the computational costs associated with the generation of body-fitted meshes by only employing fixed, possibly Cartesian, meshes over which complex boundaries can move freely. However, this boundary treatment introduces a geometrical error of the order of the mesh size that, if not treated properly, can spoil the global accuracy of a high order discretization, herein based on discontinuous Galerkin. The shifted boundary polynomial correction was proposed as a simplified version of the shifted boundary method, which is an embedded boundary treatment based on Taylor expansions to deal with unfitted boundaries. It is used to accordingly correct the boundary conditions imposed on a non-meshed boundary to compensate the aforementioned geometrical error, and reach high order accuracy. In this paper, the stability analysis of discontinuous Galerkin methods coupled with the shifted boundary polynomial correction is conducted in depth for the linear advection equation, by visualizing the eigenvalue spectrum of the high order discretized operators. The analysis considers a simplified one-dimensional setting by varying the degree of the polynomials and the distance between the real boundary and the closest mesh interface. The main result of the analysis shows that the considered high order embedded boundary treatment introduces a limitation to the stability region of high order discontinuous Galerkin methods with explicit time integration, which becomes more and more important when using higher order methods. The implicit time integration is also studied, showing that the implicit treatment of the boundary condition allows one to overcome such limitation and achieve an unconditionally stable high order embedded boundary treatment.

翻译：嵌入或浸入式方法旨在通过仅使用固定的（可能为笛卡尔）网格来最小化与贴体网格生成相关的计算成本，复杂边界可在这些网格上自由移动。然而，这种边界处理会引入与网格尺寸同阶的几何误差，若处理不当，可能破坏高阶离散化（本文基于间断伽辽金方法）的整体精度。移位边界多项式校正作为移位边界法的简化版本被提出，后者是一种基于泰勒展开处理非贴体边界的嵌入边界方法。该方法用于相应修正施加在非网格边界上的边界条件，以补偿上述几何误差并达到高阶精度。本文通过可视化高阶离散化算子的特征值谱，对线性对流方程中结合移位边界多项式校正的间断伽辽金方法进行了深入的稳定性分析。分析考虑简化的一维情形，通过改变多项式阶数以及真实边界与最近网格界面之间的距离进行研究。分析的主要结果表明，所考虑的高阶嵌入边界处理会限制显式时间积分的高阶间断伽辽金方法的稳定区域，且在使用更高阶方法时该限制作用愈发显著。研究同时探讨了隐式时间积分，表明边界条件的隐式处理能够克服此类限制，实现无条件稳定的高阶嵌入边界处理。