In this paper we present a novel approach for the design of high order general boundary conditions when approximating solutions of the Euler equations on domains with curved boundaries, using meshes which may not be boundary conformal. When dealing with curved boundaries and/or unfitted discretizations, the consistency of boundary conditions is a well-known challenge, especially in the context of high order schemes. In order to tackle such consistency problems, the so-called Reconstruction for Off-site Data (ROD) method has been recently introduced in the finite volume framework: it is based on performing a boundary polynomial reconstruction that embeds the considered boundary treatment thanks to the implementation of a constrained minimization problem. This work is devoted to the development of the ROD approach in the context of discontinuous finite elements. We use the genuine space-time nature of the local ADER predictors to reformulate the ROD as a single space-time reconstruction procedure. This allows us to avoid a new reconstruction (linear system inversion) at each sub-time node and retrieve a single space-time polynomial that embeds the considered boundary conditions for the entire space-time element. Several numerical experiments are presented proving the consistency of the new approach for all kinds of boundary conditions. Computations involving the interaction of shocks with embedded curved boundaries are made possible through an a posteriori limiting technique.

翻译：本文提出了一种新颖方法，用于在使用非边界拟合网格求解带弯曲边界区域上的欧拉方程时，设计高阶通用边界条件。当处理弯曲边界和/或非贴体离散时，边界条件的相容性是一个众所周知的挑战，特别是在高阶格式的背景下。为解决此类相容性问题，近期在有限体积框架中引入了所谓的离位数据重建（ROD）方法：该方法通过实施约束最小化问题，执行嵌入所考虑边界处理的边界多项式重建。本研究致力于在间断有限元背景下发展ROD方法。我们利用局部ADER预测子的固有时空特性，将ROD重新表述为单一的时空重建过程。这使我们能够避免在每个子时间节点重新进行重建（线性系统求逆），从而为整个时空单元获取一个嵌入所考虑边界条件的单一时空多项式。多项数值实验证明了该方法对所有类型边界条件的相容性。通过后验限制技术，实现了冲击与嵌入弯曲边界相互作用问题的计算。