A robust well-balanced model on unstructured grids for shallow water flows with wetting and drying over complex topography

Preface Special Issue “Advances in Numerical Modelling of Hydrodynamics”

“…A systematic and diagnostic 1D analysis is presented crossing aspects of accuracy-order, conservation properties, continuous vs. discontinuous terrain shapes, steady and unsteady flows.  Theoretical and sensitivity analysis of the divergence discretisation of bed slope source term: The divergence formulation of the bed slope source term (Valiani and Begnudelli 2006) within the numerical solution of the shallow water equations has emerged as physically-based technique to easily, efficiently and accurately integrate the discretisation of the topography in a wide range of numerical models and on different mesh structures (Kim et al 2014, Hou et al 2013. In this issue, Bruwier et al present a theoretical analysis of the divergence bed slope formulation going beyond the classical assumption of locally-constant free-surface elevation and considering the influence on energy balance.…”

Novel Slope Source Term Treatment for Preservation of Quiescent Steady States in Shallow Water Flows

“…Their method was based on the splitting of a cell into sub-cells and approximating the flux by adding fluxes through the sub-cells. Hou et al [16] further improved the scheme proposed in [26] through local modification of the bed elevation in slope source terms and through a better approximation of water depths at the interface by hydrostatic reconstruction.…”

“…It is worth mentioning some of the classical well-balanced numerical approaches. Popular approaches include the upwind discretization of slope terms [18,19], the surface water gradient (SGM) methods in which the water surface level is used as a conservative variable instead of water depth [7,20,21]; the hydrostatic reconstruction method proposed by [22] is another popular method among researchers [16,[23][24][25]. The upwind method performs well for first-order schemes, but needs additional source term treatment for higher-order schemes, rendering this method both complicated and expensive.…”

“…Some of the popular methods of establishing second-order accuracy are Monotonic Upwind Scheme for Conservation Laws (MUSCL) scheme [1,[7][8][9], the Weighted Essentially Non-Oscillatory (WENO) scheme [10][11][12], and a high-resolution reconstruction procedure developed by [13] which has been widely employed for shallow water flows [5,14,15]. Reconstruction by MUSCL scheme may result in negative water depths at the interface which consequently give unphysical high velocities because of discharge divided by small water depths [16]. Application of the WENO scheme give accurate results but it is expensive and complicated.…”

“…Therefore, in this study, we propose a novel slope source term treatment method which possesses all the features of a robust numerical scheme without any compromise on accuracy, simplicity and computational efficiency. We convert the slope terms into fluxes at the interface as previously done in [16,26], but with a different mathematical formulation to average water depth in convective flux and slope source terms. Unlike [16] where the scheme needs to switch to first-order accuracy at wet-dry interfaces to avoid numerical instabilities, the proposed method, paired with the multidimensional reconstruction method, is able to maintain second-order accuracy throughout the domain including at the wet-dry interfaces.…”