Many partial differential equations in mathematical physics describe the evolution of a time-dependent vector field. Examples arise in compressible fluid dynamics, shape analysis, optimal transport and shallow water equations. The flow of such a vector field generates a diffeomorphism, which can be viewed as the Lagrangian variable corresponding to the Eulerian vector field. From both computational and theoretical perspectives, it is natural to seek finite-dimensional analogs of vector fields and diffeomorphisms, constructed in such a way that the underlying geometric and algebraic properties persist (in particular, the induced Lie--Poisson structure). Here, we develop such a geometric discretization of the group of diffeomorphisms on a two-dimensional K\"ahler manifold, with special emphasis on the sphere. Our approach builds on quantization theory, combined with complexification of Zeitlin's model for incompressible two-dimensional hydrodynamics. Thus, we extend Zeitlin's approach from the incompressible to the compressible case. We provide a numerical example and discuss potential applications of the new, geometric discretization.

翻译：许多数学物理中的偏微分方程描述依赖于时间的向量场的演化，这类方程出现在可压缩流体动力学、形状分析、最优输运和浅水方程等领域。该向量场的流生成一个微分同胚，可视为对应于欧拉向量场的拉格朗日变量。从计算与理论的双重视角来看，构造向量场与微分同胚的有限维模拟自然成为目标，其构造方式需确保底层几何与代数性质得以保持（特别是诱导的李-泊松结构）。本文针对二维凯勒流形（特别关注球面）上的微分同胚群，发展了一种几何离散化方法。我们的方法建立在量子化理论之上，并结合了不可压缩二维流体动力学Zeitlin模型的复化处理，从而将Zeitlin方法从不可压缩情形推广至可压缩情形。文中给出了数值示例，并讨论了新型几何离散化方法的潜在应用。