Flux-Form Eulerian-Lagrangian Method for Solving Advective Transport of Scalars in Free-Surface Flows

A two-dimensional (2D) flux-form Eulerian-Lagrangian method (FFELM) on unstructured grid is proposed for solving the advection equation in free-surface scalar transport models. The scalar concentrations of backtracking points are combined with time-interpolated cell-face velocities to evaluate cell-face advective fluxes. A G-correction is defined as an additional mechanism to eliminate potential nonphysical oscillations by correcting the cell-face advective fluxes. A flux-form cell update is finally carried out to obtain new cell concentrations. The role of the G-correction in the FFELM is clarified using a test of scalar transport in unsteady open-channel flows. A solid-body rotation test, a laboratory bend-flume test, and a real river test (using a 600-km river reach of the upper Yangtze River) are used to demonstrate the FFELM. The FFELM is revealed in tests to achieve almost the same accuracy as a pure Eulerian-type method [the subcycling finite-volume method (SCFVM)] and a conservative ELM [the finite-volume ELM (FVELM)]. Relative to explicit Eulerian methods, the FFELM uses the information of backtracking points over an extended upwind dependence domain in evaluating cell-face advective fluxes, and allows larger time steps for which the Courant-Friedrichs-Lewy number (CFL) is greater than 1. In the real river test, stable and accurate FFELM simulations can be achieved at a time step for which the CFL is as large as 5. Efficiency issues of the FFELM are clarified using the bend-flume test (193,536 cells) and the real river test (213,363 cells). In solving a transport problem (using 1-32 kinds of scalars and 16 cores), a parallel run using the FFELM is 1.0-3.3 times faster than a parallel run using the SCFVM. The FFELM has a computational cost slightly less (15%-17%) than that of the FVELM. Moreover, the implementation of the FFELM is much easier than that of the FVELM, and extension of the 2D FFELM to its one-dimensional (1D) and three-dimensional (3D) versions is straightforward.