A new efficient implicit scheme for discretising the stiff friction terms in the shallow water equations

© 2018 The Authors Discretisation of the friction terms to ensure numerical stability and accuracy remains to be challenging for the development of robust numerical schemes to solve the shallow water equations (SWEs), particularly for applications involving very shallow flows (e.g. overland flows and wet/dry fronts) over complex domain topography. The key challenge is to ensure relaxation of the flow towards an equilibrium state characterised by the balance between friction and gravity in a computationally efficient way. To overcome this numerical challenge, this paper proposes a novel approach for discretising the friction source terms in the SWEs in the context of an explicit finite volume method. The overall numerical scheme adopts the HLLC Riemann solver and surface reconstruction method (SRM) to explicitly discretise the flux and bed slope source terms. Whilst a fully implicit scheme is used to handle the friction source terms, solution to the implicit formulation is analytically derived to explicitly update the flow variables. Compared with the existing approaches, the proposed scheme effectively resolves the issue associated with stiff relaxation without necessity to use an iteration method and it supports efficient simulation using time steps controlled only by the Courant–Friedrichs–Levy (CFL) condition. The current friction term discretisation scheme is not coupled with flux and bed slope calculation and therefore may be readily implemented in any other explicit finite volume SWE models. After being successfully validated against two benchmark tests with analytical solutions, the resulting new SWE model is applied to reproduce a rainfall-flooding event in the Upper Lee catchment in the UK

A new depth-averaged model for flow-like landslides over complex terrains with curvatures and steep slopes

Flow-like landslides are one of the most catastrophic types of natural hazards due to their high velocity and long travel distance. They travel like fluid after initiation and mainly fall into the ‘flow’ movement type in the updated Varnes classification (Hungr et al., 2014). In recent years, depth-averaged models have been widely reported to predict the velocity and run-out distance of flow-like landslides. However, most of the existing depth-averaged models present different shortcomings for application to real-world simulations. This paper presents a novel depth-averaged model featured with a set of new governing equations derived in a mathematically rigorous way based on the shallow flow assumption and Mohr-Coulomb rheology. Particularly, the new mathematical formulation takes into account the effects of vertical acceleration and curvature effects caused by complex terrain topographies. The model is derived on a global Cartesian coordinate system so that it is easy to apply in real-world applications. A Godunov-type finite volume method is implemented to numerically solve these new governing equations, together with a novel scheme proposed to discretise the friction source terms. The hydrostatic reconstruction approach is implemented and improved in the context of the new governing equations, providing well-balanced and non-negative numerical solutions for mass flows over complex domain topographies. The model is validated against several test cases, including a field-scale flow-like landslide. Satisfactory results are obtained, demonstrating the model's improved simulation capability and potential for wider applications

A two-dimensional hydro-morphological model for river hydraulics and morphology with vegetation

This work develops a two-dimensional hydro-morphological model which can be used to simulate river hydraulics and morphology with various vegetation covers. The model system consists of five modules, including a hydrodynamic model, a sediment transport model, a vegetation model, a bank failure model and a bed deformation model. The secondary flow effects are incorporated through additional dispersion terms. The core components of the model system solve the full shallow water equations; this is coupled with a non-equilibrium sediment transport model. The new integrated model system is validated against a number of laboratory-scale test cases and then applied to a natural river. The satisfactory simulation results confirm the model's capability in reproducing both stream hydraulics and channel morphological changes with vegetation. Several hypothetical simulations indicate that the model can be used not only to predict flooding and morphological evolution with vegetation, but also to assess river restoration involving vegetation

A novel 1D-2D coupled model for hydrodynamic simulation of flows in drainage networks

Drainage network modelling is often an essential component in urban flood prediction and risk assessment. Drainage network models most commonly use different numerical procedures to handle flows in pipes and junctions. Numerous numerical schemes and models of different levels of complexity have been developed and reported to predict flows in pipes. However, calculation of the flow conditions in junctions has received much less attention and has been traditionally achieved by solving only the continuity equation. This method is easy to implement but it neglects the momentum exchange in the junctions and cannot provide sufficient boundary conditions for the pipe calculation. In this work, a novel numerical scheme based on the finite volume solution to the two-dimensional (2D) shallow water equations (SWEs) is proposed to calculate flow dynamics in junctions, which directly takes into account both mass and momentum conservation and removes the necessity of implementing complicated boundary settings for pipe calculations. This new junction simulation method is then coupled with the widely used two-component pressure approach (TPA) for the pipe flow calculation, leading to a new integrated drainage network model. The new 1D-2D coupled drainage network model is validated against an experimental and several idealised test cases to demonstrate its potential for efficient and stable simulation of flow dynamics in drainage networks.<br

New prospects for computational hydraulics by leveraging high-performance heterogeneous computing techniques

In the last two decades, computational hydraulics has undergone a rapid development following the advancement of data acquisition and computing technologies. Using a finite-volume Godunov-type hydrodynamic model, this work demonstrates the promise of modern high-performance computing technology to achieve real-time flood modeling at a regional scale. The software is implemented for high-performance heterogeneous computing using the OpenCL programming framework, and developed to support simulations across multiple GPUs using a domain decomposition technique and across multiple systems through an efficient implementation of the Message Passing Interface (MPI) standard. The software is applied for a convective storm induced flood event in Newcastle upon Tyne, demonstrating high computational performance across a GPU cluster, and good agreement against crowd- sourced observations. Issues relating to data availability, complex urban topography and differences in drainage capacity affect results for a small number of areas

Accuracy of depth-integrated nonhydrostatic wave models

Depth-integrated nonhydrostatic models have been wildly used to simulate propagation of waves. Yet, there lacks a well-documented theoretical framework that can be used to assess the accuracy and scope of applications of these models and the related numerical approaches. In this work, we carry out Stokes-type Fourier and shoaling analyses to examine the linear and nonlinear properties of a popular one-layer depth-integrated nonhydrostatic model derived by Stelling and Zijlema (2003). The theoretical analysis shows that the model can satisfactorily interpret the dispersity for linear waves but presents evident divergence for nonlinear solutions even when kd → 0. A generalized depth-integrated nonhydrostatic formulation using arbitrary elevation as a variable is then derived and analyzed to examine the effects of neglecting advective and diffusive nonlinear terms in the previous studies and explore possible improvements in numerical solutions for wave propagation. Compared with the previous studies, the new generalized formulation exhibits similar dispersion relationship and improved shoaling effect. However, no significant improvement is presented for the nonlinear properties, indicating that retaining neglected nonlinear terms may not significantly improve the nonlinear performance of the nonhydrostatic model. Further analysis shows that the nonlinear properties of the depth-integrated nonhydrostatic formulation may be improved by defining variables at one-third of the still water level. However, such an improvement comes at the price of decreasing accuracy in describing dispersion and shoaling properties

A coupled hydrological and hydrodynamic model for flood simulation

This paper presents a new flood modelling tool developed by coupling a full 2D hydrodynamic model with hydrological models. The coupled model overcomes the main limitations of the individual modelling approaches, i.e. high computational costs associated with the hydrodynamic models and less detailed representation of the underlying physical processes related to the hydrological models. When conducting a simulation using the coupled model, the computational domain (e.g. a catchment) is first divided into hydraulic and hydrological zones. In the hydrological zones that have high ground elevations and relatively homogeneous land cover or topographic features, a conceptual lumped model is applied to obtain runoff/net rainfall, which is then routed by a group of pre-acquired ‘unit hydrographs’ to the zone borders. These translated hydrographs will then be used to drive the full 2D hydrodynamic model to predict flood dynamics at high resolution in the hydraulic zones that are featured with complex topographic settings, including roads, buildings, etc. The new coupled flood model is applied to reproduce a major flood event that occurred in Morpeth, northeast England in September 2008. While producing similar results, the new coupled model is shown to be computationally much more efficient than the full hydrodynamic model

An efficient and stable hydrodynamic model with novel source term discretization schemes for overland flow and flood simulations

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 km2Haltwhistle Burn, UK

Reply to comment by Lu et al. on “An efficient and stable hydrodynamic model with novel source term discretization schemes for overland flow and flood simulations”

This document addresses the comments raised by Lu et al. (2017). Lu et al. (2017) proposed an alternative numerical treatment for implementing the fully implicit friction discretization in Xia et al. (2017). The method by Lu et al. (2017) is also effective, but not necessarily easier to implement or more efficient. The numerical wiggles observed by Lu et al. (2017) do not affect the overall solution accuracy of the surface reconstruction method (SRM). SRM introduces an antidiffusion effect, which may also lead to more accurate numerical predictions than hydrostatic reconstruction (HR) but may be the cause of the numerical wiggles. As suggested by Lu et al. (2017), HR may perform equally well if fine enough grids are used, which has been investigated and recognized in the literature. However, the use of refined meshes in simulations will inevitably increase computational cost and the grid sizes as suggested are too small for real-world applications

A new multilayer nonhydrostatic formulation for surface water waves

This work presents a new multilayer nonhydrostatic formulation for surface water waves. The new governing equations define velocities and pressure at an arbitrary location of a vertical layer and only contain spatial derivatives of maximum second order. Stoke-type Fourier and shoaling analyses are carried out to scrutinize the mathematical properties of the new formulation, subsequently optimizing the representative interface and the location to define variables in each layer to improve model accuracy. Following the analysis, the one-layer model exhibits accurate linear and nonlinear characteristics up to kd = I, demonstrating similar solution accuracy to the existing second-order Boussinesq-type models. The two-layer model with optimized coefficients can maintain its linear and nonlinear accuracy up to kd = 4I, which boasts of better solution accuracy a larger application range than most existing fourth-order Boussinesq model and two-layer Boussinesq models. The three-layer model presents accurate linear and nonlinear characteristics up to kd = 10Ï, effectively removing any shallow water limitation. The current multilayer nonhydrostatic water wave model does not predefine the vertical flow structures, and more accurate vertical velocity distributions can be obtained by considering the velocity profiles in coefficient optimization