In this work, we present a novel family of high order accurate numerical schemes for the solution of hyperbolic partial differential equations (PDEs) which combines several geometrical and physical structure preserving properties. First, we settle our methods in the Lagrangian framework, where each element of the mesh evolves following as close as possible the local fluid flow, so to reduce the numerical dissipation at contact waves and moving interfaces and to satisfy the Galilean and rotational invariance properties of the studied PDEs system. In particular, we choose the direct Arbitrary-Lagrangian-Eulerian (ALE) approach which, in order to always guarantee the high quality of the moving mesh, allows to combine the Lagrangian motion with mesh optimization techniques. The employed polygonal tessellation is thus regenerated at each time step, the previous one is connected with the new one by space-time control volumes, including hole-like sliver elements in correspondence of topology changes, over which we integrate a space-time divergence form of the original PDEs through a high order accurate ADER discontinuous Galerkin (DG) scheme. Mass conservation and adherence to the GCL condition are guaranteed by construction thanks to the integration over closed control volumes, and robustness over shock discontinuities is ensured by the use of an a posteriori subcell finite volume (FV) limiting technique.

翻译：本文提出了一类新型高阶精度数值格式，用于求解双曲型偏微分方程，该格式兼具多种几何与物理结构保持特性。首先，我们将方法建立在拉格朗日框架下，网格单元随局部流体运动演化，以降低接触波与运动界面处的数值耗散，并保持所研究偏微分方程系统的伽利略不变性与旋转不变性。我们特别采用直接任意拉格朗日-欧拉方法，通过结合拉格朗日运动与网格优化技术，始终保证移动网格的高质量。因此，多边形网格在每个时间步被重新生成，前一时刻与当前时刻的网格通过时空控制体连接（包含拓扑变化时产生的孔状条带单元），并在这些控制体上对原始偏微分方程的时空散度形式进行高阶ADER间断伽辽金格式积分。由于在闭合控制体上积分，质量守恒与几何守恒律条件得到本征满足；通过后验子网格有限体积限制技术，可确保激波间断处的鲁棒性。