Robust numerical methods for shallow water flows and advective transport simulation on unstructured grids

The two-dimensional (2D) shallow water equations (SWEs) are extensively used for hydrodynamic simulations in hydraulic and environmental engineering. The transport process inside shallow water, such as the transport of contaminant and sediment, can be modeled by solving the transport equation numerically. When solving the advective transport equation and SWEs, second order numerical schemes are widely used to reduce numerical diffusion caused by first order schemes. However, numerical oscillations may be induced by second order schemes without proper limiters. Second order total variation diminishing based flux limiting schemes (TVD schemes) are able to get rid of such numerical oscillations. In this cumulative dissertation, second order TVD schemes derived on 1D grids are extended to 2D unstructured grids, within the framework of the cell-centered finite volume method, to comfort to complex geometry. Moreover, an efficient treatment for slope source terms of SWEs and a robust approach handling wetting and drying are devised. This dissertation is on the basis of four papers in peer reviewed international journals and four conference contributions. To extend second order TVD schemes to 2D unstructured grids, three methods are developed step by step. The first method adopts the TVD schemes which take the variation of cell size into account, for unstructured grids. In the second method, the first one is improved by applying more dominant upwind information perpendicular to the considered face. As a result, both the accuracy and efficiency are higher than the first one. Since an approximation is used, the accuracy of the second method is affected. By extrapolating the values of variables at the midpoint of the considered face, the third TVD method can produce more accurate results than the first two methods. The amelioration of each method in simulating linear advection is illustrated by the test cases in the corresponding papers and by a new test case in this dissertation. In addition, the third method is also employed to solve the SWEs. A new treatment for the slope source terms of the SWEs is devised for unstructured grids. This treatment together with the hydrostatic non-negative water depth reconstruction method and the HLLC approximate Riemann solver, constitute a well-balanced scheme, which satisfies the conservation property. In the case with the occurrence of wet-dry fronts, the very small water depths near wet-dry fronts may lead to unphysical high velocities and in turn to negative water depths. To preserve numerical stability, a new adaptive approach is proposed, by means of switching to first order scheme in such sensitive areas. In this dissertation, the third method extending TVD schemes to 2D unstructured grids incorporated with the well-balanced scheme and the adaptive approach are proposed finally to simulate shallow water flows and advective transport inside. This model is able to get rid of numerical oscillations, to preserve the C-property and mass conservation, to achieve good convergence to steady state, to capture discontinuous flows and to handle complex flows involving wetting and drying over uneven beds, on unstructured grids with poor connectivity, in an accurate, efficient and robust way. These capabilities are verified against analytical solutions, numerical results of alternative models and experimental and field data.