A Flux Identity of Scalar Conservation Laws for Lagrangian Flux Calculation via Donating Regions

For a simple planar curve in a nonautonomous flow, the fluxing index of a passively advected Lagrangian particle is the net number of its crossing the curve within a given time interval. Such indices naturally induce flux sets, equivalence classes of the particles at the initial time. Previously, we proposed donating regions as an explicit geometric construction of flux sets. In this work, we strengthen our earlier results by removing an assumption on the equivalence of flux sets and donating regions. More importantly, we propose and prove for scalar conservation laws a flux identity, which establishes the equivalence of the traditional Eulerian flux integral (a double integral in time as well as in space) to a Lagrangian flux integral at the initial time (a spatial integral independent of time). Thus the evolution of the scalar function is no longer needed in expressing its total flux through the curve over any time interval of finite length. To numerically exploit this identity, we also propose a simple algorithm of Lagrangian flux calculation, which is demonstrated by results of a variety of numerical tests to be highly accurate and highly efficient.