Balancing the source terms in a SPH model for solving the shallow water equations

A shallow flow generally features complex hydrodynamics induced by complicated domain topography and geometry. A numerical scheme with well-balanced flux and source term gradients is therefore essential before a shallow flow model can be applied to simulate real-world problems. The issue of source term balancing has been exhaustively investigated in grid-based numerical approaches, e.g. discontinuous Galerkin finite element methods and finite volume Godunov-type methods. In recent years, a relatively new computational method, smooth particle hydrodynamics (SPH), has started to gain popularity in solving the shallow water equations (SWEs). However, the well-balanced problem has not been fully investigated and resolved in the context of SPH. This work aims to discuss the well-balanced problem caused by a standard SPH discretization to the SWEs with slope source terms and derive a corrected SPH algorithm that is able to preserve the solution of lake at rest. In order to enhance the shock capturing capability of the resulting SPH model, the Monotone Upwind-centered Scheme for Conservation Laws (MUSCL) is also explored and applied to enable Riemann solver based artificial viscosity. The new SPH model is validated against several idealized benchmark tests and a real-world dam-break case and promising results are obtained.     

A 2D well-balanced shallow flow model for unstructured grids with novel slope source term treatment

Within the framework of the Godunov-type cell-centered finite volume (CCFV) scheme, this paper proposes a 2D well-balanced shallow water model for unstructured grids. In this model, the face-based van Albada limiting scheme is employed in conjunction with a directional correction to reconstruct second order spatial values at the midpoint of the considered face. The Harten, Lax and van Leer approximate Riemann solver with the Contact wave restored (HLLC) is applied to compute the fluxes of mass and momentum, while the splitting implicit method is utilized to solve the friction source terms. The novel aspects of the model include the new limited directional correction with which the new local extrema caused by the unlimited correction are prevented efficiently, the simplified non-negative water depth reconstruction used to get rid of numerical instabilities and in turn to preserve mass conservation at wet–dry interfaces and the novel slope source term treatment which suits complex unstructured grids well by transforming the slope source of a cell into fluxes at its faces. This model is able 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.     

High-order balanced CWENO scheme for movable bed shallow water equations

In this work the numerical integration of 1D shallow water equations (SWE) over movable bed is performed using a well-balanced central weighted essentially non-oscillatory (CWENO) scheme, fourth-order accurate in space and in time. Time accuracy is obtained following a Runge–Kutta (RK) procedure, coupled with its natural continuous extension (NCE). Spatial accuracy is obtained using WENO reconstructions of conservative variables and of flux and bed derivatives. An original treatment for bed slope source term, which maintains the established order of accuracy and satisfies the property of exactly preserving the quiescent flow (C-property), is introduced in the scheme. This treatment consists of two procedures. The former involves the evaluation of the point-values of the flux derivative, considered as a whole with the bed slope source term. The latter involves the spatial integration of the source term, analytically manipulated to take advantage from the expected regularity of the free surface elevation. The high accuracy of the scheme allows to obtain good results using coarse grids, with consequent gain in terms of computational effort. The well-balancing of the scheme allows to reproduce small perturbations of the free surface and of the bottom otherwise of the same order of magnitude of the numerical errors induced by the non-balancing. The accuracy, the well-balancing and the good resolution of the model in reproducing free surface flow over movable bed are tested over analytical solutions and over numerical results available in literature.     

A robust well-balanced finite volume model for shallow water flows with wetting and drying over irregular terrain

An unstructured Godunov-type finite volume model is developed for the numerical simulation of geometrically challenging two-dimensional shallow water flows with wetting and drying over convoluted topography. In the framework of sloping bottom model, a modified formulation of shallow water equations is used to preserve mass conservation during flooding and recession. The key ingredient of the model is the use of this combination of the sloping bottom model and the modified shallow water equations to provide a robust technique for wet/dry fronts tracking and, together with centered discretization of the bed slope source term, to exactly preserve the static flow on irregular topographies. The variable reconstruction technique ensures nonnegative reconstructed water depth and reasonable reconstructed velocity, and the friction terms are solved by semi-implicit scheme that does not invert the direction of velocity components. The robustness and accuracy of the proposed model are assessed by comparing numerical and reference results of extensive test cases. Moreover, the results of a dam-break flooding over real topography are presented to show the capability of the model on field-scale application.     

High-order finite volume WENO schemes for the shallow water equations with dry states

The shallow water equations are used to model flows in rivers and coastal areas, and have wide applications in ocean, hydraulic engineering, and atmospheric modeling. These equations have still water steady state solutions in which the flux gradients are balanced by the source term. It is desirable to develop numerical methods which preserve exactly these steady state solutions. Another main difficulty usually arising from the simulation of dam breaks and flood waves flows is the appearance of dry areas where no water is present. If no special attention is paid, standard numerical methods may fail near dry/wet front and produce non-physical negative water height. A high-order accurate finite volume weighted essentially non-oscillatory (WENO) scheme is proposed in this paper to address these difficulties and to provide an efficient and robust method for solving the shallow water equations. A simple, easy-to-implement positivity-preserving limiter is introduced. One- and two-dimensional numerical examples are provided to verify the positivity-preserving property, well-balanced property, high-order accuracy, and good resolution for smooth and discontinuous solutions.     

Two-dimensional numerical model of two-layer shallow water equations for confluence simulation

This study presents a finite-volume explicit method to solve 2D two-layer shallow water equations. This numerical model is intended to describe two-layer shallow flows in which the superposed layers differ in velocity, density and rheology in a two-dimensional domain. The rheological behavior of mudflow or debris flow is called the Bingham fluid. Thus, the shear stress on rigid bed can be derived from the constitutive equation. The computational approach adopts the HLL scheme, a novel approach for the purpose of computing a Godunov flux and solving the Riemann problem approximately proposed by Harten, Lax and van Leer, as a basic building block, treats the bottom slope by lateralizing the momentum flux, and refines the scheme using the Strang splitting to manage the frictional source term. This study successfully performed 2D two-layer shallow water computations on a rigid bed. The proposed numerical model can describe the variety of depths and velocities of substances including water and mud, when the hyperconcentrated tributary flows into the main river. The analytical results in this study will be valuable for further advanced research and for designing or planning hydraulic engineering structures.     

Well-balanced numerical modelling of non-uniform sediment transport in alluvial rivers

The last two decades have witnessed the development and application of well-balanced numerical models for shallow flows in natural rivers.However,until now there have been no such models for flows with non-uniform sediment transport.This paper presents a 1D well-balanced model to simulate flows and non-capacity transport of non-uniform sediment in alluvial rivers.The active layer formulation is adopted to resolve the change of bed sediment composition.In the framework of the finite volume Slope Llmiter Centred(SLIC) scheme,a surface gradient method is incorporated to attain well-balanced solutions to the governing equations.The proposed model is tested against typical cases with irregular topography,including the refilling of dredged trenches,aggradation due to sediment overloading and flood flow due to landslide dam failure.The agreement between the computed results and measured data is encouraging.Compared to a non-well-balanced model,the well-balanced model features improved performance in reproducing stage,velocity and bed deformation.It should find general applications for non-uniform sediment transport modelling in alluvial rivers,especially in mountain areas where the bed topography is mostly irregular.     

Well-balanced high-order centred schemes for non-conservative hyperbolic systems. Applications to shallow water equations with fixed and mobile bed

This paper concerns the development of high-order accurate centred schemes for the numerical solution of one-dimensional hyperbolic systems containing non-conservative products and source terms. Combining the PRICE-T method developed in [Toro E, Siviglia A. PRICE: primitive centred schemes for hyperbolic system of equations. Int J Numer Methods Fluids 2003;42:1263–91] with the theoretical insights gained by the recently developed path-conservative schemes [Castro M, Gallardo J, Parés C. High-order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products applications to shallow-water systems. Math Comput 2006;75:1103–34; Parés C. Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM J Numer Anal 2006;44:300–21], we propose the new PRICE-C scheme that automatically reduces to a modified conservative FORCE scheme if the underlying PDE system is a conservation law. The resulting first-order accurate centred method is then extended to high order of accuracy in space and time via the ADER approach together with a WENO reconstruction technique. The well-balanced properties of the PRICE-C method are investigated for the shallow water equations. Finally, we apply the new scheme to the shallow water equations with fix bottom topography and with variable bottom solving an additional sediment transport equation.     

Well-balanced high-order centered schemes on unstructured meshes for shallow water equations with fixed and mobile bed

In this paper, we study the numerical approximation of the two-dimensional morphodynamic model governed by the shallow water equations and bed-load transport following a coupled solution strategy. The resulting system of governing equations contains non-conservative products and it is solved simultaneously within each time step. The numerical solution is obtained using a new high-order accurate centered scheme of the finite volume type on unstructured meshes, which is an extension of the one-dimensional PRICE-C scheme recently proposed in Canestrelli et al. (2009) [5]. The resulting first-order accurate centered method is then extended to high order of accuracy in space via a high order WENO reconstruction technique and in time via a local continuous space–time Galerkin predictor method. The scheme is applied to the shallow water equations and the well-balanced properties of the method are investigated. Finally, we apply the new scheme to different test cases with both fixed and movable bed. An attractive future of the proposed method is that it is particularly suitable for engineering applications since it allows practitioners to adopt the most suitable sediment transport formula which better fits the field data.     

Two-dimensional numerical modeling of dam-break flows over natural terrain using a central explicit scheme

A two-dimensional (2D) numerical model has been developed to solve shallow water equations for simulation of dam-break flows. The spatial derivatives are discretized using a well-balanced explicit central upwind conservative scheme. The scheme is Riemann solver free and guarantees the positivity of the flow depth over complex topography if the Courant number is kept less than 0.25. The time integration is performed by Euler’s scheme. The model is verified against analytical results for water surface elevation and discharge for three benchmark test cases. A good agreement between analytical solutions and computed results is observed. The property of well-balancing in still water over an uneven bottom is also confirmed. The model is then validated by simulating a laboratory experiment in which a dam break flow propagates over a triangular obstacle. The model performance was found to be satisfactory. A dam break laboratory experimental test case on a frictionless horizontal bottom is also simulated for 2D validation of the model, and good agreement between simulation and the experimental data is observed. The suitability of the proposed model for real life applications is demonstrated by simulating the Malpasset dam-break event, which occurred in 1959 in France. The computed arrival time of the flood wave front and the maximum flow depths at various observation points matched well with the measurements on a 1/400 scale physical model. The overall performance indicates that this model can be applied for simulation of dam-break waves in real life cases.     

Analytical Solution for Testing Debris Avalanche Numerical Models

—We present here the analytical solution of a one-dimensional dam-break problem over inclined planes. This solution is used to test a numerical model developed for debris avalanches. We consider a dam with infinite length in one direction where material is released from rest at the initial instant. We solve analytically and numerically the depth-averaged long-wave equations derived in a topography-linked coordinate system. The numerical and analytical solutions provide for a Coulomb-type friction law at the base of the flow. The analytical solution is obtained by using the method of characteristics and describes the flow over a constant slope, provided that the angle is higher than the friction angle. The numerical model utilizes a finite-difference method based on a Godunov-type scheme. Comparison between analytical and numerical results illustrates the remarkable stability and precision of the numerical method as well as its ability to deal with strong discontinuities.     

A limiter-based well-balanced discontinuous Galerkin method for shallow-water flows with wetting and drying: One-dimensional case

An important part in the numerical simulation of tsunami and storm surge events is the accurate modeling of flooding and the appearance of dry areas when the water recedes. This paper proposes a new algorithm to model inundation events with piecewise linear Runge–Kutta discontinuous Galerkin approximations applied to the shallow water equations. This study is restricted to the one-dimensional case and shows a detailed analysis and the corresponding numerical treatment of the inundation problem.The main feature is a velocity based “limiting” of the momentum distribution in each cell, which prevents instabilities in case of wetting or drying situations. Additional limiting of the fluid depth ensures its positivity while preserving local mass conservation. A special flux modification in cells located at the wet/dry interface leads to a well-balanced method, which maintains the steady state at rest. The discontinuous Galerkin scheme is formulated in a nodal form using a Lagrange basis. The proposed wetting and drying treatment is verified with several numerical simulations. These test cases demonstrate the well-balancing property of the method and its stability in case of rapid transition of the wet/dry interface. We also verify the conservation of mass and investigate the convergence characteristics of the scheme.     

A weighted surface-depth gradient method for the numerical integration of the 2D shallow water equations with topography

A finite volume MUSCL scheme for the numerical integration of 2D shallow water equations is presented. In the framework of the SLIC scheme, the proposed weighted surface-depth gradient method (WSDGM) computes intercell water depths through a weighted average of DGM and SGM reconstructions, in which the weight function depends on the local Froude number. This combination makes the scheme capable of performing a robust tracking of wet/dry fronts and, together with an unsplit centered discretization of the bed slope source term, of maintaining the static condition on non-flat topographies (C-property). A correction of the numerical fluxes in the computational cells with water depth smaller than a fixed tolerance enables a drastic reduction of the mass error in the presence of wetting and drying fronts. The effectiveness and robustness of the proposed scheme are assessed by comparing numerical results with analytical and reference solutions of a set of test cases. Moreover, to show the capability of the numerical model on field-scale applications, the results of a dam-break scenario are presented.     

A mass conservative scheme for simulating shallow flows over variable topographies using unstructured grid

Most available numerical methods face problems, in the presence of variable topographies, due to the imbalance between the source and flux terms. Treatments for this problem generally work well for structured grids, but most of them are not directly applicable for unstructured grids. On the other hand, despite of their good performance for discontinuous flows, most available numerical schemes (such as HLL flux and ENO schemes) induce a high level of numerical diffusion in simulating recirculating flows. A numerical method for simulating shallow recirculating flows over a variable topography on unstructured grids is presented. This mass conservative approach can simulate different flow conditions including recirculating, transcritical and discontinuous flows over variable topographies without upwinding of source terms and with a low level of numerical diffusion. Different numerical tests cases are presented to show the performance of the scheme for some challenging problems.     

An automated routing methodology to enable direct rainfall in high resolution shallow water models

Recent high profile flood events have highlighted the need for hydraulic models capable of simulating pluvial flooding in urban areas. This paper presents a constant velocity rainfall routing scheme that provides this ability within the LISFLOOD‐FP hydraulic modelling code. The scheme operates in place of the shallow water equations within cells where the water depth is below a user‐defined threshold, enabling rainfall‐derived water to be moved from elevated features such as buildings or curbstones without causing instabilities in the solution whilst also yielding a reduction in the overall computational cost of the simulation. Benchmarking against commercial modelling packages using a pluvial and point‐source test case demonstrates that the scheme does not impede the ability of LISFLOOD‐FP to match both predicted depths and velocities of full shallow water models. The stability of the scheme in conditions unsuitable for traditional two‐dimensional hydraulic models is then demonstrated using a pluvial test case over a complex urban digital elevation model containing buildings. Deterministic single‐parameter sensitivity analyses undertaken using this test case show limited sensitivity of predicted water depths to both the chosen routing speed within a physically plausible range and values of the depth threshold parameter below 10 mm. Local instabilities can occur in the solution if the depth threshold is >10 mm, but such values are not required even when simulating extreme rainfall rates. The scheme yields a reduction in model runtime of ~25% due to the reduced number of cells for which the hydrodynamic equations have to be solved. Copyright © 2012 John Wiley & Sons, Ltd.     

Numerical modelling of two-dimensional morphodynamics with applications to river bars and bifurcations

We study the numerical approximation of the two-dimensional morphodynamic model governed by the shallow water and Exner equations to simulate reach-scale two-dimensional morphodynamics of bedload-dominated alluvial rivers. The solution strategy relies on a full coupling of the governing equations within each time step. The resulting system of governing equations contains nonconservative products related to the longitudinal and lateral bed slopes and source terms related to friction. The full problem is solved numerically on unstructured triangular grids, simultaneously updating the principal part and adding the source terms (friction) using a splitting technique.The principal part is solved by means of a novel second-order accurate upwind-biased centred scheme of the finite volume type, while the source terms are added to the problem by solving a system of ordinary differential equations. A new algorithm for treating the wetting-and-drying is also proposed.The model is applied to well-established test problems in order to verify the accuracy of the proposed method, the robustness of the wetting-and-drying algorithm and the ability of the model in dealing with transcritical flows. Finally we test the model ability to reproduce two dimensional morphodynamic processes occurring at the scale of tens of channel widths in bedload dominated alluvial rivers with homogeneous grain size. This is achieved by comparing model outcomes with those of analytical theories and flume experiments on the same morphodynamic processes. These selected “benchmarks” include migrating free bars spontaneously developing in straight reaches, steady bars forced by abrupt river planform changes and the dynamics of channel bifurcations.     

Weakly-reflective boundary conditions for two-dimensional shallow water flow problems

In this paper weakly-reflective boundary conditions are derived for the two-dimensional shallow water equations, including bottom friction and Coriolis force. The essential aspects of the derivation are given. Zeroth and first order approximations are applied to the test problem of an initially Gaussian-shaped free surface elevation. For the numerical solution a finite element program is used and various aspects of the numerical implementation are discussed. For small scale practical problems a rather simple (one parameter) formulation might be sufficient. The influence of this parameter is discussed on the weakly-reflectiveness of the boundary condition.     

Positive solution of two-dimensional solute transport in heterogeneous aquifers

The transport of contaminants in aquifers is usually represented by a convection-dispersion equation. There are several well-known problems of oscillation and artificial dispersion that affect the numerical solution of this equation. For example, several studies have shown that standard treatment of the cross-dispersion terms always leads to a negative concentration. It is also well known that the numerical solution of the convective term is affected by spurious oscillations or substantial numerical dispersion. These difficulties are especially significant for solute transport in nonuniform flow in heterogeneous aquifers. For the case of coupled reactive-transport models, even small negative concentration values can become amplified through nonlinear reaction source/sink terms and thus result in physically erroneous and unstable results. This paper includes a brief discussion about how nonpositive concentrations arise from numerical solution of the convection and cross-dispersion terms. We demonstrate the effectiveness of directional splitting with one-dimensional flux limiters for the convection term. Also, a new numerical scheme for the dispersion term that preserves positivity is presented. The results of the proposed convection scheme and the solution given by the new method to compute dispersion are compared with standard numerical methods as used in MT3DMS.     

Numerical modeling of the propagation and morphological changes of turbidity currents using a cost-saving strategy of solution updating

Existing layer-averaged numerical models for turbidity currents have mostly adopted the global minimum time step (GMiTS) for solution updating, which confines their computational efficiency and limits their attractiveness for field applications. This paper presents a highly efficient layer-averaged numerical model for turbidity currents by implementing the combined approach of the local graded-time-step (LGTS) and the global maximum-time-step (GMaTS). The governing equations are solved for unstructured triangular meshes by the shock-capturing finite volume method along with a set of well-balanced evaluations of the numerical flux and geometrical slope source terms. The quantitative accuracy of the model, given reasonably estimated empirical and model parameters (e.g., bed friction, water entrainment, sediment deposition and erosion coefficients), is demonstrated by comparing the numerical solutions against laboratory data of the current front positions and deposition profiles, as well as field data of the current front positions. The improved computational efficiency is demonstrated by comparing the computational cost of the present model against that of a traditional model that uses a GMiTS. For the present simulated cases, the maximum reduction of the computational cost is approximately 80% (e.g., a simulation that cost 1 h before will only require 12 min with the new model).     

An approximate Riemann solver for sensitivity equations with discontinuous solutions

The sensitivity of a model output (called a variable) to a parameter can be defined as the partial derivative of the variable with respect to the parameter. When the governing equations are not differentiable with respect to this parameter, problems arise in the numerical solution of the sensitivity equations, such as locally infinite values or instability. An approximate Riemann solver is thus proposed for direct sensitivity calculation for hyperbolic systems of conservation laws in the presence of discontinuous solutions. The proposed approach uses an extra source term in the form of a Dirac function to restore sensitivity balance across the shocks. It is valid for systems such as the Euler equations for gas dynamics or the shallow water equations for free surface flow. The method is first detailed and its application to the shallow water equations is proposed, with some test cases such as dike- or dam-break problems with or without source terms. An application to a two-dimensional flow problem illustrates the superiority of direct sensitivity calculation over the classical empirical approach.