Numerical models solving the full 2‐D shallow water equations (SWEs) have been increasingly used to simulate overland flows and better understand the transient flow dynamics of flash floods in a catchment. However, there still exist key challenges that have not yet been resolved for the development of fully dynamic overland flow models, related to (1) the difficulty of maintaining numerical stability and accuracy in the limit of disappearing water depth and (2) inaccurate estimation of velocities and discharges on slopes as a result of strong nonlinearity of friction terms. This paper aims to tackle these key research challenges and present a new numerical scheme for accurately and efficiently modeling large‐scale transient overland flows over complex terrains. The proposed scheme features a novel surface reconstruction method (SRM) to correctly compute slope source terms and maintain numerical stability at small water depth, and a new implicit discretization method to handle the highly nonlinear friction terms. The resulting shallow water overland flow model is first validated against analytical and experimental test cases and then applied to simulate a hypothetic rainfall event in the 42 km2 Haltwhistle Burn, UK.

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SUMMARY The requirements for flux limiter functions preserving total variation diminishing (TVD) are derived based on a 1D nonuniform grid, and a new TVD region is determined to fit arbitrary 1D grids. Some second‐order TVD schemes called improved TVD schemes are developed, such as modified Van Leer scheme, modified Van Albada scheme, and modified SUPERBEE scheme. Then, they are extended to 2D grids. Because the element sizes and face positions are taken into account, good behaviors are observed in the implementations in both 1D and 2D cases for pure advection simulation. That is, good conservation, better monotonicity, and higher accuracy are maintained by the improved TVD schemes compared with the present ones deduced from uniform grids, and they keep superiorities even when implemented on poor grids. Among all the improved TVD schemes, the modified SUPERBEE is only recommended for poor 1D grids, but the modified Van Leer scheme can suit both poor 1D and 2D grids. Copyright © 2011 John Wiley & Sons, Ltd.

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This paper presents a numerical model suitable for a broad range of surface flow problems such as overland flow, wetting and drying processes, varying flow conditions and shock waves. It is based on solution of two-dimensional fully dynamic shallow water equations using a cell-centred finite-volume method. Numerical fluxes are computed with a Harten, Lax and van Leer approximate Riemann solver with a contact wave restored. The scheme is second-order accurate in space, and a total variation diminishing method is used to avoid spurious oscillations in the solution. For extending the model to rainfall-runoff applications, infiltration is considered as a constant runoff coefficient and by the Green-Ampt model. The model is implemented in the Hydroinformatics Modelling System, an object-oriented framework that enables the implementation and simulation of single and multiple processes in different spatial and temporal resolutions, as well as their interactions. The capabilities of the model are shown by comparison with analytical solutions and experimental data of a flash flood and a surface runoff experiment. Finally, a real rainfall-runoff event in a small alpine catchment is investigated. Overall, good agreement of numerical and analytical results, as well as measurements, has been obtained. A area of the considered cell m 2 À Á C Chézy coefficient s m À1=6 À Á c concentration (À) D diffusion coefficient m 2 s À1 À Á d water depth (m) ε vegetation drag related coefficient η water elevation above datum: η ¼ z B þ d (m) f flux vector g standard gravity 9:81 m s À2 À Á Γ boundary of control volume (m) I cumulative infiltration (m) i infiltration rate m s À1 À Á K average residence time (s) k index of a face of the considered cell l k length of face k (m) m c contaminant source/sink term s À1 À Á m w water source/sink term m s À1 À Á n k normal vector pointing outwards of face k n Manning coefficient (s m À1=3 ) n time step index ∇ del operator: ∇ ¼ @ @x , @ @y T ν t turbulent kinematic viscosity m 2 s À1 À Á Ψ runoff coefficient (-) Q discharge m 3 s À1 À Á q vector of conserved flow variables r rainfall intensity m s À1 À Á ρ density of water kg m À3 À Á S storage (m 3 ) s source vector t time (s) Δt time step (s) u velocity component in x-direction m s À1 À Á v velocity component in y-direction m s À1 À Á v velocity vector V w specific soil water volume (m) x X coordinate (m) y Y coordinate (m) z B bottom elevation above datum (m) Ω control volume (m 2 )

SUMMARYFor advection schemes with flux limiters derived on one‐dimensional grids to two‐dimensional (2D) unstructured triangular ones. In this method, the variables located normal to this face are taken into more account to compute the flux, by means of defining required nodes along the line through the center point of the considered face and perpendicular to it. Besides, the new method adopts the improved total variation diminishing schemes in, which consider the face position between the two neighboring cells as well as the size differences of the related cells; therefore, it fits 2D unstructured grids well. Compared with the present flux limiting methods for 2D unstructured grids, a higher accuracy and efficiency and a good monotonicity are achieved by the new method.Copyright © 2012 John Wiley & Sons, Ltd.

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