Lagrangian Particle Classification and Lagrangian Flux Identities for a Moving Hypersurface

For a moving hypersurface in the flow of a nonautonomous ordinary differential equation in $n$-dimensional Euclidean spaces, the fluxing index of a passively-advected Lagrangian particle is the total number of times it crosses the moving hypersurface within a time interval. The problem of Lagrangian particle classification is to decompose the phase space into flux sets, equivalence classes of Lagrangian particles at the initial time. In the context of scalar conservation laws, the problem of Lagrangian flux calculation (LFC) is to find flux identities that relate the Eulerian flux of a scalar through the moving hypersurface, a spatiotemporal integral over the moving surface in a given time interval, to spatial integrals over donating regions at the initial time of the interval. In this work, we implicitly characterize flux sets via topological degrees, explicitly construct donating regions, prove the equivalence of flux sets and donating regions, and establish two flux identities; these analytical results constitute our solutions to the aforementioned problems. Based on a flux identity suitable for numerical calculation, we further proposed a new LFC algorithm, proved its convergence, and demonstrated its efficiency, good conditioning, and high-order accuracy by results of various numerical tests.