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:我们提出了一个基于半封闭空间操作器的新框架,用于分析和分解移动和变形领域的初始边界值问题;这一发展扩展了从固定领域到一般情况(包括任意网状运动)的良好存在问题和能源稳定离散的现有框架,包括任意网状运动;特别是,我们表明,在物理协调系统中得出的能源估计相当于一个固定参考域的半封闭属性。导致这一结果的连续分析基于物质时间衍生物的对称分解,从而依赖整合的特性。在此之后,根据与离散和逐个比较的方式对物质时间衍生物的近似性计算物质时间衍生物。由于半封闭属性,使用标准明确或隐含的时间集成方法方法方法方法方法方法可以用来推动系统向前发展。同样类型的稳定性论据适用于相应的离散性拉格-欧莱尔内分解框架,根据与离散式对物质时间衍生物的近性衍生物相匹配的方式制定。我们所实现的离式稳定性理论可以用来在地球离散域框架中展示新的稳定性。我们所实现的稳定性,在精确的稳定性方面可以证明我们所实现的稳定性。