Geometric discretization of diffeomorphisms

Many partial differential equations in mathematical physics describe the evolution of time-dependent (smooth) vector fields on a fixed domain. Examples include compressible fluid dynamics, shape analysis, optimal transport, and shallow water equations. The flow of the vector field generates a diffeomorphism, which in turn can be used to act on for instance functions or densities. Here, we consider a geometric discretization of diffeomorphisms on the sphere, based on quantization theory. We provide numerical examples and discuss potential applications of the discretization method.