Keywords: locally conservative flux, finite element, parallel computing, software toolkit, face/edge library.

Conserving mass in computing water flow and contaminant transport is essential. Passing conservative water flux from flow models to transport models is critical for accurate simulation and analysis. At the U.S. Army Engineer Research and Development Center (ERDC), most water flow models employ the finite element (FE) method, while many contaminant transport models use the finite volume (FV) method. Because the continuous Galerkin FE method does not yield locally conservative flux approximation directly, the computation of conservative water flux through each elemental face has become necessary for passing FE-based water flux to FV-based contaminant transport models. The computed conservative flux can also be used to set up boundary conditions for inset models when desired.

This paper presents two algorithms to compute local mass conserved flux. The local approach wins over the global approach, and these approaches were derived based on the Larson-Niklasson method. We detailed the derivation based on different situations, such as for an internal node, and a boundary node adjacent to different types of boundary conditions. Obviously the global approach requires more memory consumption for the linear system, compared with the local approach, whose linear system has only the size of the total number of adjacent elements to a node. The global approach may require an efficient linear solver, while the local approach can simply use a direct solver.

Our software development aims to enable the CCFlux (consistent/inconsistent conservative flux computation) toolkit to support the solution of the local approach. In the CCFlux toolkit, functionalities include (1) a unique edges/faces set, (2) local renumbering of adjacent elements around a node on the index space, and (3) an efficient edge/face manipulation for divergent-free operation. The toolkit is intended to be incorporated into different application codes, e.g. ADH (C code), pWASH123D (mixed C and Fortran code), or even the old-fashioned Fortran-77 FEMATER code. Parallel mesh manipulation for flux calculation is embedded in CCFlux. A set of light-weight application interface functions facilitates easy incorporation to different applications. The aforementioned local approach was implemented in the two-dimensional overland flow module of ADH. The pseudo of the implementation was described in the paper. An experimental case solving the San-Diego Bay area was presented for demonstration. The result was verified by computing the difference, within an element, between the total flux across all the elemental faces and the sum of boundary fluxes described in the governing equation. After checking every element throughout the entire domain, the correct implementation of the local conserved flux computation and the CCFlux toolkit was then proved. More cases, including three-dimensional groundwater flow, three-dimensional CFD problems, and three-dimensional shallow water flow, will be built for demonstration in the near future.

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