固定与运动域上高阶弯曲边界的基于极小化的多项式修正：在有限体积和间断伽略金格式上的评估
Minimization-based polynomial corrections for high-order curved boundaries on fixed and moving domains: assessment on finite volume and discontinuous Galerkin schemes
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中文总结 AI 辅助
本研究提出无需线性系统求解的多项式修正策略，将其应用于两类数值框架，在固定与运动弯曲域的可压缩流中实现最高五阶边界条件收敛精度，简化了基于极小化的边界处理方法。
中文摘要 AI 辅助
本工作提出两种新策略，用于在由分段仿射三角剖分近似的固定与运动弯曲域上施加高阶边界条件。在弯曲域上实现高阶精度需要同时处理偏微分方程（PDE）离散误差和几何误差：前者可通过采用有限体积、间断伽略金等高阶数值方法降低，后者则要求物理域的高阶参数化或边界条件的一致近似。基于极小化的方法，如场外数据重构（ROD）方法，可通过在不匹配物理边界的计算边界上定义高阶一致边界条件，跳过高阶曲线网格的构建；标准ROD方法通过在每个边界单元中重构修正多项式，使边界条件在物理边界上精确满足，从而缓解二阶几何误差，但需求解局部线性系统，其成本随多项式次数和网格加密程度增长。受近期一维分析启发，本研究表明ROD类方法可重构为无需任何线性系统求解的简单多项式修正，这极大简化了基于极小化的边界处理方法的开发并降低了相关计算成本。为验证该策略的广泛适用性，本研究将其应用于用于固定与运动域上可压缩流的Runge-Kutta间断伽略金框架和ADER任意拉格朗日-欧拉有限体积框架中；并开展多项数值实验，涉及Dirichlet边界条件和滑移壁边界条件，收敛性分析显示在二维和三维中均达到最高五阶精度。
英文摘要
In this work, we present two novel strategies to impose high-order boundary conditions on fixed and moving curved domains, approximated with piecewise affine triangulations. Achieving high-order accuracy on curved domains requires tackling both the PDE discretization error and the geometrical error simultaneously. While the former can be reduced by employing high-order numerical methods such as finite volume and discontinuous Galerkin, the latter demands either a high-order parametrization of the physical domain or a consistent approximation of the boundary conditions. Minimization-based approaches like the Reconstruction for Off-site Data (ROD) method allow one to skip the construction of high-order curvilinear meshes by defining high-order consistent boundary conditions on a computational boundary that does not match the physical one. The ROD approach mitigates the second-order geometrical error by retrieving a modified polynomial in each boundary cell, which enforces the boundary conditions exactly on the physical boundary. However, the standard ROD method requires the inversion of a local linear system, whose cost grows with the polynomial degree and mesh refinement. Inspired by a recent one-dimensional analysis, we show that ROD-type approaches can be recast as simple polynomial corrections, applicable without any linear system inversion. This greatly simplifies the development of minimization-based boundary treatments and reduces the associated computational cost. To prove the wide applicability of our strategy, we develop it within a Runge-Kutta discontinuous Galerkin framework and an ADER arbitrary-Lagrangian-Eulerian finite volume framework for compressible flows on fixed and moving domains. Several numerical experiments with Dirichlet and slip-wall boundary conditions are presented, with convergence analysis up to fifth order in both 2D and 3D.
