标题： 高阶移动曲边界处理：具有移位边界多项式修正的任意拉格朗日-欧拉方法 显示英文标题

标题： High order treatment of moving curved boundaries: Arbitrary-Lagrangian-Eulerian methods with a shifted boundary polynomials correction

摘要： 本文提出了一种针对可压缩气体动力学欧拉方程在弯曲移动域上求解时高阶边界条件设定的新方法。处理弯曲边界时，边界条件的一致性是一个真正的挑战，而在使用高阶任意拉格朗日-欧拉（ALE）格式离散的移动域中，这一挑战变得更加严峻。ALE公式特别适合处理移动和变形域，从而允许复杂流固耦合问题的模拟。然而，如果不适当地处理，边界条件的施加可能会导致数值解中出现显著误差，这会破坏基础数学模型的高阶离散化。为了解决这个问题，我们提出了一种基于最近发展的偏移边界多项式校正的新方法，该方法最初是在固定网格上提出的。新方法被集成到直接ALE有限体积法的空间时间校正步骤中，通过利用有限体积控制体的高阶重构多项式来考虑移动边界的局部曲率。它依赖于基于单元多项式在真实几何形状上评估的外推值的校正，因此不需要显式计算高阶泰勒级数。这种方法极大地简化了移动弯曲边界的处理，因为它允许使用标准的单纯形网格，这些网格比曲线网格更容易生成和移动，特别是在三维瞬态问题中。本文还展示了新方法在移动弯曲域中可压缩流动的背景下具有高阶收敛特性，其中域仍然由分段线性单元逼近的几个数值实验结果。 显示英文摘要

摘要： In this paper we present a novel approach for the prescription of high order boundary conditions when approximating the solution of the Euler equations for compressible gas dynamics on curved moving domains. When dealing with curved boundaries, the consistency of boundary conditions is a real challenge, and it becomes even more challenging in the context of moving domains discretized with high order Arbitrary-Lagrangian-Eulerian (ALE) schemes. The ALE formulation is particularly well-suited for handling moving and deforming domains, thus allowing for the simulation of complex fluid-structure interaction problems. However, if not properly treated, the imposition of boundary conditions can lead to significant errors in the numerical solution, which can spoil the high order discretization of the underlying mathematical model. In order to tackle this issue, we propose a new method based on the recently developed shifted boundary polynomial correction, which was originally proposed on fixed meshes. The new method is integrated into the space-time corrector step of a direct ALE finite volume method to account for the local curvature of the moving boundary by only exploiting the high order reconstruction polynomial of the finite volume control volume. It relies on a correction based on the extrapolated value of the cell polynomial evaluated at the true geometry, thus not requiring the explicit evaluation of high order Taylor series. This greatly simplifies the treatment of moving curved boundaries, as it allows for the use of standard simplicial meshes, which are much easier to generate and move than curvilinear ones, especially for 3D time-dependent problems. Several numerical experiments are presented demonstrating the high order convergence properties of the new method in the context of compressible flows in moving curved domains, which remain approximated by piecewise linear elements.