Hyperbolic partial differential equations (PDEs) cover a wide range of interesting phenomena, from human and hearth-sciences up to astrophysics: this unavoidably requires the treatment of many space and time scales in order to describe at the same time observer-size macrostructures, multi-scale turbulent features, and also zero-scale shocks. Moreover, numerical methods for solving hyperbolic PDEs must reliably handle different families of waves: smooth rarefactions, and discontinuities of shock and contact type. In order to achieve these goals, an effective approach consists in the combination of space-time-based high-order schemes, very accurate on smooth features even on coarse grids, with Lagrangian methods, which, by moving the mesh with the fluid flow, yield highly resolved and minimally dissipative results on both shocks and contacts. However, ensuring the high quality of moving meshes is a huge challenge that needs the development of innovative and unconventional techniques. The scheme proposed here falls into the family of Arbitrary-Lagrangian-Eulerian (ALE) methods, with the unique additional freedom of evolving the shape of the mesh elements through connectivity changes. We aim here at showing, by simple and very salient examples, the capabilities of high-order ALE schemes, and of our novel technique, based on the high-order space-time treatment of topology changes.

翻译：超球部分偏差方程式(PDEs)涵盖从人类和听觉科学到天体物理学等一系列令人感兴趣的现象:这不可避免地需要处理许多空间和时间尺度,以便同时描述观察规模宏观结构、多尺度动荡特征和零尺度冲击。此外,解决超曲线部分方程式的数字方法必须可靠地处理波系不同的波系:平滑的稀释动作以及冲击和接触类型的不连续性。为了实现这些目标,有效的方法包括基于时空的高秩序组合,非常精确的光滑功能,甚至粗糙的电网,以及拉格朗格方法,通过移动流体流的网状组合,产生高度的分辨率和最小的分解效果。然而,确保高质量的移动模类是需要开发创新和非常规技术的巨大挑战。为了实现这些目标,这里提出的办法属于基于任意-劳改-奥利安(ALE)高阶系统(ALEE)的组合,甚至是粗粗网格的网格,非常精确的特征,使用拉格方法,这些方法,通过流流流流流的网状,产生高度的高度自由,通过我们高空系的系统结构,展示我们高端的高度的图图图图图图图图图图图图式,展示,通过我们高端的高度的高度的系统,显示我们高端图图图图图图图图图图图图的高度的转变,展示了我们高端的高度的系统。