Discrete flow mapping: transport of phase space densities on triangulated surfaces

Energy distributions of high-frequency linear wave fields are often modelled in terms of flow or transport equations with ray dynamics given by a Hamiltonian vector field in phase space. Applications arise in underwater and room acoustics, vibroacoustics, seismology, electromagnetics and quantum mechanics. Related flow problems based on general conservation laws are used, for example, in weather forecasting or in molecular dynamics simulations. Solutions to these flow equations are often large-scale, complex and high-dimensional, leading to formidable challenges for numerical approximation methods. This paper presents an efficient and widely applicable method, called discrete flow mapping, for solving such problems on triangulated surfaces. An application in structural dynamics, determining the vibroacoustic response of a cast aluminium car body component, is presented.