Performance and Comparison of Cell-Centered and Node-Centered Unstructured Finite Volume Discretizations for Shallow Water Free Surface Flows

Finite volume (FV) methods for solving the two-dimensional (2D) nonlinear shallow water equations (NSWE) with source terms on unstructured, mostly triangular, meshes are known for some time now. There are mainly two basic formulations of the FV method: node-centered (NCFV) and cell-centered (CCFV). In the NCFV formulation the finite volumes, used to satisfy the integral form of the equations, are elements of the mesh dual to the computational mesh, while for the CCFV approach the finite volumes are the mesh elements themselves. For both formulations, details are given of the development and application of a second-order well-balanced Godunov-type scheme, developed for the simulation of unsteady 2D flows over arbitrary topography with wetting and drying. The popular approximate Riemann solver of Roe is utilized to compute the numerical fluxes, while second-order spatial accuracy is achieved with a MUSCL-type reconstruction technique. The Green-Gauss (G-G) formulation for gradient computations is implemented for both formulations, in order to maintain a common framework. Two different stencils for the G-G gradient computations in the CCFV formulation are implemented and tested. An edge-based limiting procedure is applied for the control of the total variation of the reconstructed field. This limiting procedure is proved to be effective for the NCFV scheme but inadequate for the CCFV approach. As such, a simple but very effective modification to the reconstruction procedure is introduced that takes into account geometrical characteristics of the computational mesh. In addition, consistent well-balanced second-order discretizations for the topography source term treatment and the wet/dry front treatment are presented for both FV formulations, ensuring absolute mass conservation, along with a stable friction term treatment. Using a controlled environment for a fair comparison, a complete assessment of both FV formulations is attempted through rigorous individual and relative performance comparisons to the approximation of analytical benchmark solutions, as well as to experimental and field data. To this end, an extensive evaluation is performed using different time dependent and steady-state test cases that incorporate topography, wetting and drying process, different types of boundary conditions as well as friction. These test cases are chosen as to compare the performance and robustness of each formulation under certain conditions and evaluate the effectiveness of the proposed modifications. Emphasis to grid convergence studies is given, with the grids used to range from regular grids to irregular ones with random perturbations of nodes. The results indicate that the quality of the mesh has a major impact on the convergence performance of the CCFV method while the NCFV method exhibits a uniform behavior on the different grid types used. The proposed correction in the computation of the reconstructed values for the CCFV formulation greatly improves the convergence behavior to the formal order of accuracy.