We present a novel framework based on semi-bounded spatial operators for analyzing and discretizing initial boundary value problems on moving and deforming domains. This development extends an existing framework for well-posed problems and energy stable discretizations from stationary domains to the general case including arbitrary mesh motion. In particular, we show that an energy estimate derived in the physical coordinate system is equivalent to a semi-bounded property with respect to a stationary reference domain. The continuous analysis leading up to this result is based on a skew-symmetric splitting of the material time derivative, and thus relies on the property of integration-by-parts. Following this, a mimetic energy stable arbitrary Lagrangian-Eulerian framework for semi-discretization is formulated, based on approximating the material time derivative in a way consistent with discrete summation-by-parts. Thanks to the semi-bounded property, a method-of-lines approach using standard explicit or implicit time integration schemes can be applied to march the system forward in time. The same type of stability arguments applies as for the corresponding stationary domain problem, without regards to additional properties such as discrete geometric conservation. As an additional bonus we demonstrate that discrete geometric conservation, in the sense of exact free-stream preservation, can still be achieved in an automatic way with the new framework. However, we stress that this is not necessary for stability.

翻译：本文提出了一种基于半有界空间算子的新型框架，用于分析移动和变形域上的初边值问题及其离散化。该工作将现有的适定问题与能量稳定离散化框架从静态域推广至包含任意网格运动的通用情形。特别地，我们证明在物理坐标系中导出的能量估计等价于相对于固定参考域的半有界性质。这一连续分析结果基于物质时间导数的斜对称分解，因此依赖于分部积分性质。继而，通过以与离散求和分部积分一致的方式近似物质时间导数，构建了一种用于半离散化的模仿能量稳定的任意拉格朗日-欧拉框架。借助半有界性质，可采用标准显式或隐式时间积分格式的线方法对系统进行时间推进。此时，稳定性论证与相应静态域问题具有相同形式，无需考虑离散几何守恒等额外性质。作为附加优势，我们证明该新框架仍能自动实现精确自由流保持意义上的离散几何守恒，但需强调这对稳定性并非必要条件。