LAGRANGIAN FLUX CALCULATION THROUGH A FIXED PLANAR CURVE FOR SCALAR CONSERVATION LAWS

For a scalar function conserved by an unsteady flow, its flux through a simple curve is usually expressed as an Eulerian flux, a double integral both in time and in space. We show that this Eulerian flux equals a Lagrangian flux, a line integral in space with no time dependence. We further exploit this flux identity to propose a new algorithm for Lagrangian flux calculation (LFC), in which the Eulerian flux through the static curve is estimated only from the given velocity field and the initial distribution of the scalar function. In particular, the time interval of LFC can be arbitrarily long, yet the scalar conservation law is never numerically solved. As such, LFC complements traditional finite volume methods both theoretically and computationally. Compared to previous LFC algorithms, the new LFC algorithm is simple and well-conditioned with second-, fourth-, and sixth-order convergence. Results of numerical tests demonstrate the flexibility, accuracy, and efficiency of the proposed LFC algorithm.