Slope Source Term Research Articles

Analysis of a new Kolgan-type scheme motivated by the shallow water equations

This paper presents the analysis of two different finite volume schemes for hyperbolic conservation laws: the Kolgan high-resolution scheme, and a new Kolgan-type scheme in which the high-order extrapolation of the conservative variables is used just in the upwind contribution of the numerical flux and source terms. Both schemes are compared in terms of the local truncation error, the stability conditions and the C-property. The schemes are applied to different hyperbolic conservation equations, including the one-dimensional scalar transport equation, the Burgers equation and the 2D shallow water equations, in order to compute the observed order of accuracy and to verify the C-property. When applied to the 2D shallow water equations, the new approach avoids spurious oscillations in the solution without the need of using high-order corrections in the definition of the bed slope source term.

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.

Discontinuous Galerkin Method for 1D Shallow Water Flow with Water Surface Slope Limiter

A water surface slope limiting scheme is tested and compared with the water depth slope limiter for the solution of one dimensional shallow water equations with bottom slope source term. Numerical schemes based on the total variation diminishing Runge- Kutta discontinuous Galerkin finite element method with slope limiter schemes based on water surface slope and water depth are used to solve one-dimensional shallow water equations. For each slope limiter, three different Riemann solvers based on HLL, LF, and Roe flux functions are used. The proposed water surface based slope limiter scheme is easy to implement and shows better conservation property compared to the slope limiter based on water depth. Of the three flux functions, the Roe approximation provides the best results while the LF function proves to be least suitable when used with either slope limiter scheme.

Well-Balanced Bottom Discontinuities Treatment for High-Order Shallow Water Equations WENO Scheme

A finite volume well-balanced weighted essentially nonoscillatory (WENO) scheme, fourth-order accurate in space and time, for the numerical integration of shallow water equations with the bottom slope source term, is presented. The main novelty introduced in this work is a new method for managing bed discontinuities. This method is based on a suitable reconstruction of the conservative variables at the cell interfaces, coupled with a correction of the numerical flux based on the local conservation of total energy. Further changes regard the treatment of the source term, based on a high-order extension of the divergence form for bed slope source term method, and the application of an analytical inversion of the specific energy-depth relationship. Two ad hoc test cases, consisting of a steady flow over a step and a surge crossing a step, show the effectiveness of the method of treating bottom discontinuities. Several standard one-dimensional test cases are also used to verify the high-order accuracy, the C-...

A Conservative Coupled Flow/Transport Model with Zero Mass Error

A fully conservative form applied to a coupled system of two-dimensional water flow and solute motion is presented. A cell-centred finite volume method based on Roe's approximate Riemann solver with unstructured grids is formulated. The bed slope source terms are discretized following an upwind approach and a semi-implicit treatment is used for the friction source terms. The centered discretization of the diffusion terms is in an implicit way. It is shown that this numerical technique reproduces almost exactly the steady state of still water and enables to achieve zero numerical errors in unsteady flow over configurations with strong variations on bed slope. The model ensures a global conservation and positive values of both water level and solute concentration. Numerical results show the effectiveness of the model in solute transport over real complex geometries.

Numerical resolution of well-balanced shallow water equations with complex source terms

This paper presents a well-balanced numerical scheme for simulating frictional shallow flows over complex domains involving wetting and drying. The proposed scheme solves, in a finite volume Godunov-type framework, a set of pre-balanced shallow water equations derived by considering pressure balancing. Non-negative reconstruction of Riemann states and compatible discretization of slope source term produce stable and well-balanced solutions to shallow flow hydrodynamics over complex topography. The friction source term is discretized using a splitting implicit scheme. Limiting value of the friction force is derived to ensure stability. This new numerical scheme is validated against four theoretical benchmark tests and then applied to reproduce a laboratory dam break over a domain with irregular bed profile.

Time step restrictions for well‐balanced shallow water solutions in non‐zero velocity steady states

AbstractNumerical models based on the conservative formulation of the shallow water equations in finite volumes have recently paid special attention to the convenience of a unified discretization of bed slope source terms and pressure momentum fluxes in order to ensure a correct representation of both in steady states. In cases of steady shallow water flow with non‐zero velocity, the discrete balance must include the friction term and the good equilibrium of the discrete solution must be revisited. Besides, there is the question of stability. The presence of relatively large friction terms reduces considerably the stability region of the explicit schemes (with either separate or unified discretization). Implicit formulations strongly help to relax the friction related stability restrictions but are only feasible in the context of separated discretization. Therefore, a strategy to blend both approaches (unified explicit and separated implicit) is presented that ensures both a perfect balance in steady state and numerical stability in unsteady cases in the presence of friction terms. Copyright © 2008 John Wiley & Sons, Ltd.

Closure to “Divergence Form for Bed Slope Source Term in Shallow Water Equations” by Alessandro Valiani and Lorenzo Begnudelli

Closure to “Divergence Form for Bed Slope Source Term in Shallow Water Equations” by Alessandro Valiani and Lorenzo Begnudelli

Discussion of “Divergence Form for Bed Slope Source Term in Shallow Water Equations” by Alessandro Valiani and Lorenzo Begnudelli

Discussion of “Divergence Form for Bed Slope Source Term in Shallow Water Equations” by Alessandro Valiani and Lorenzo Begnudelli

Discussion of “Divergence Form for Bed Slope Source Term in Shallow Water Equations” by Alessandro Valiani and Lorenzo Begnudelli

Discussion of “Divergence Form for Bed Slope Source Term in Shallow Water Equations” by Alessandro Valiani and Lorenzo Begnudelli

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.

Finite Volume Model for Shallow Water Equations with Improved Treatment of Source Terms

A simple yet precise relation between the flux gradient and the bed slope source term is presented, which produces a net force within the cell with an inclined water surface, but ensures still water condition when there is no flow across the boundaries. The proposed method consists of calculating the pressure term based on the water depths at the cell vertices, which may be computed by a higher order scheme and the bed slope source term by a centered discretization technique. The methodology is demonstrated with a Godunov-type upwind finite volume formulation. The inviscid fluxes are calculated using Roe’s approximate Riemann solver and a second-order spatial accuracy is obtained by implementing multidimensional gradient reconstruction and slope limiting techniques. The accuracy and applicability of the numerical model is verified with a couple of test problems and a real flow example of tidal water movement in a stretch of River Hooghly in India.

2D modelling of erosion/deposition processes with suspended load using

This work describes a finite volume model applied to solve the coupled shallow water/load transport equations, where low concentrations of sediment load are assumed so that the concentration of the solid fraction is assumed not to influence the equations governing the dynamics. The unsteady simulation of erodible bed requires special attention in the discretization of the bed and friction source terms as important instabilities can arise in some situations. The upwind discretization of the time variable bed slope source terms is presented to provide an exact balance of the numerical fluxes and to guarantee steady-state solutions contrary to the pointwise formulation that leads to instabilities destroying the computation. Robust numerical schemes are presented for both moving and fixed boundaries. In the last case a numerical technique is provided to keep the concentration load bounded for values of CFL (Courant–Friedrichs–Lewy) greater than one and therefore decreasing the necessary computational cost.

High order upwind scheme for modelling turbulent shallow water flow in hydraulic structures

AbstractAn unstructured finite volume model for quasi‐2D free surface flow with wet‐dry fronts and turbulence modelling is presented. The convective flux is discretised with either a an hybrid second‐order/first‐order scheme, or a fully second order scheme, both of them upwind Godunov's schemes based on Roe's average. The hybrid scheme uses a second order discretisation for the two unit discharge components, whilst keeping a first order discretisation for the water depth [2]. In such a way the numerical diffusion is much reduced, without a significant reduction on the numerical stability of the scheme, obtaining in such a way accurate and stable results. It is important to keep the numerical diffusion to a minimum level without loss of numerical stability, specially when modelling turbulent flows, because the numerical diffusion may interfere with the real turbulent diffusion.In order to avoid spurious oscillations of the free surface when the bathymetry is irregular, an upwind discretisation of the bed slope source term [4] with second order corrections is used [2]. In this way a fully second order scheme which gives an exact balance between convective flux and bed slope in the hydrostatic case is obtained. The k – ε equations are solved with either an hybrid or a second order scheme.In all the numerical simulations the importance of using a second order upwind spatial discretisation has been checked [1]. A first order scheme may give rather good predictions for the water depth, but it introduces too much numerical diffusion and therefore, it excessively smooths the velocity profiles. This is specially important when comparing different turbulence models, since the numerical diffusion introduced by a first order upwind scheme may be of the same order of magnitude as the turbulent diffusion. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Evaluation of well‐balanced bore‐capturing schemes for 2D wetting and drying processes

AbstractWe consider numerical solutions of the two‐dimensional non‐linear shallow water equations with a bed slope source term. These equations are well‐suited for the study of many geophysical phenomena, including coastal engineering where wetting and drying processes are commonly observed. To accurately describe the evolution of moving shorelines over strongly varying topography, we first investigate two well‐balanced methods of Godunov‐type, relying on the resolution of non‐homogeneous Riemann problems. But even if these schemes were previously proved to be efficient in many simulations involving occurrences of dry zones, they fail to compute accurately moving shorelines. From this, we investigate a new model, called SURF_WB, especially designed for the simulation of wave transformations over strongly varying topography. This model relies on a recent reconstruction method for the treatment of the bed‐slope source term and is able to handle strong variations of topography and to preserve the steady states at rest. In addition, the use of the recent VFRoe‐ncv Riemann solver leads to a robust treatment of wetting and drying phenomena. An adapted ‘second order’ reconstruction generates accurate bore‐capturing abilities.This scheme is validated against several analytical solutions, involving varying topography, time dependent moving shorelines and convergences toward steady states. This model should have an impact in the prediction of 2D moving shorelines over strongly irregular topography. Copyright © 2006 John Wiley & Sons, Ltd.

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.

Divergence Form for Bed Slope Source Term in Shallow Water Equations

A novel technique is presented for the treatment of the bed slope source terms within the numerical solution of the shallow water equations. The proposed method consists of writing the bed slope source term as the divergence of a proper matrix, related to the static force due to bottom slope. Such a technique is founded on analytical reasoning and is physically based, so that it can be easily applied to a wide range of numerical models, as it is completely independent of any adopted discretization technique, and requires a minimum computational effort. Herein, we show an application to a Godunov-type model, second order accurate both in space and time, based on the finite-volume method. The presented technique leads to a strong improvement in the source terms numerical treatment, especially for steady states, in which flux gradients are exactly balanced by source terms. A surprising degree of simplicity is maintained, with respect to different existing methods. The numerical model has been applied to a set of classical test cases and to a selected laboratory experiment, in order to verify its stability, accuracy, and applicability to practical real-world cases.

Fourth-order balanced source term treatment in central WENO schemes for shallow water equations

The aim of this work is to develop a well-balanced central weighted essentially non-oscillatory (CWENO) method, fourth-order accurate in space and time, for shallow water system of balance laws with bed slope source term. Time accuracy is obtained applying a Runge–Kutta scheme (RK), coupled with the natural continuous extension (NCE) approach. Space accuracy is obtained using WENO reconstructions of the conservative variables and of the water-surface elevation. Extension of the applicability of the standard CWENO scheme to very irregular bottoms, preserving high-order accuracy, is obtained introducing two original procedures. The former involves the evaluation of the point-values of the flux derivative, coupled with the bed slope source term. The latter involves the spatial integration of the source term, analytically manipulated to take advantage from the regularity of the free-surface elevation, usually smoother than the bottom elevation. Both these procedures satisfy the C-property, the property of exactly preserving the quiescent flow. Several standard one-dimensional test cases are used to verify high-order accuracy, exact C-property, and good resolution properties for smooth and discontinuous solutions.

Simulation of shallow flows over variable topographies using unstructured grids

AbstractSimulation of shallow flows over variable topographies is a challenging case for most available shock‐capturing schemes. This problem arises because the source terms and flux gradients are not balanced in the numerical computations. Treatments for this problem generally work well on structured grids, but they are usually too expensive, and most of them are not directly applicable to unstructured grids. In this paper we propose two efficient methods to treat the source terms without upwinding and to satisfy the compatibility condition on unstructured grids. In the first method, the calculation of the bed slope source term is performed by employing a compatible approximation of water depth at the cell interfaces. In the second one, different components of the bed slope term are considered separately and a compatible discretization of the components is proposed. The present treatments are applicable for most schemes including the Roe's method without changing the performance of the original scheme for smooth topographies. Copyright © 2006 John Wiley & Sons, Ltd.

Coupling between shallow water and solute flow equations: analysis and management of source terms in 2D

A two-dimensional model for the simulation of solute transport by convection and diffusion into shallow water flow over variable bottom is presented. It is based on a finite volume method over triangular unstructured grids. A first order upwind technique is applied to solve the flux terms in both the flow and solute equations and the bed slope source terms and a centred discretization is applied to the diffusion and friction terms. The convenience of considering the fully coupled system of equations is indicated and the methodology is well explained. Three options are suggested and compared in order to deal with the diffusion terms. Some comparisons are carried out in order to show the performance in terms of accuracy and computational effort of the different options. Copyright © 2005 John Wiley & Sons, Ltd.