Highly accurate Lagrangian flux calculation via algebraic quadratures on spline-approximated donating regions

Lagrangian flux through a fixed curve segment within a time interval (tn,tn+k) can be formulated as an integral at the initial time tn over a compact point set called donating region, in which each particle will pass through the curve during the time interval and contribute to the flux. Based an explicit, constructive, and analytical solution of the donating region, the author proposes algorithms of Lagrangian flux calculation (LFC) as algebraic quadratures over the spline-approximated donating regions. This generic formulation yields high accuracies up to the 8th-order both in time and space. As another benefit, LFC leads to a conservative semi-Lagrangian method whose time step size is free of the Eulerian stability constraint Cr⩽1. The high accuracy and robustness of the proposed LFC algorithms are demonstrated by various numerical tests with Courant numbers ranging from 1 to 10,000 and with accuracies from the 2nd order to the 8th order. LFC might also be useful in designing hybrid Eulerian methods for complex geometries.