Numerical Simulation of Bed Load and Suspended Load Sediment Transport Using Well-Balanced Numerical Schemes

Sediment transport can be modelled using hydrodynamic models based on shallow water equations coupled with the sediment concentration conservation equation and the bed con-servation equation.The complete system of equations is made up of the energy balance law and the Exner equations.The numerical solution for this complete system is done in a seg-regated manner.First,the hyperbolic part of the system of balance laws is solved using a finite volume scheme.Three ways to compute the numerical flux have been considered,the Q-scheme of van Leer,the HLLCS approximate Riemann solver,and the last one takes into account the presence of non-conservative products in the model.The discretisation of the source terms is carried out according to the numerical flux chosen.In the second stage,the bed conservation equation is solved by using the approximation computed for the system of balance laws.The numerical schemes have been validated making comparisons between the obtained numerical results and the experimental data for some physical experiments.The numerical results show a good agreement with the experimental data.

A Well-Balanced Numerical Model for the Simulation of Long Waves over Complex Domains

This paper presents a well-balanced two-dimensional (2D) finite volume model to simulate the propagation, runup and rundown of long wave. Non-staggered grid is adopted to discretize the governing equation and the intercell flux is computed using a central upwind scheme, which is a Riemann-problem-solver-free method for hyperbolic conservation laws. The nonnegative reconstruction method for water depth is implemented in the present model to treat the appearance of wet/dry fronts, and the friction term is solved by a semi-implicit scheme to ensure the stability of the model. The Euler method is applied to update flow variable to the new time level. The model is verified against two experimental cases and good agreements are observed between numerical results and observed data.

A New Well-Balanced Finite Volume CWENO Scheme for Shallow Water Equations over Bottom Topography

In this article,we develop a new well-balanced finite volume central weighted essentially non-oscillatory(CWENO)scheme for one-and two-dimensional shallow water equations over uneven bottom.The well-balanced property is of paramount importance in practical applications,where many studied phenomena can be regarded as small perturbations to the steady state.To achieve the well-balanced property,we construct numerical fluxes by means of a decomposition algorithm based on a novel equilibrium preserving reconstruction procedure and we avoid applying the traditional hydrostatic reconstruction technique accordingly.This decomposition algorithm also helps us realize a simple source term discretization.Both rigorous theoretical analysis and extensive numerical examples all verify that the proposed scheme maintains the well-balanced property exactly.Furthermore,extensive numerical results strongly suggest that the resulting scheme can accurately capture small perturbations to the steady state and keep the genuine high-order accuracy for smooth solutions at the same time.

A Well-Balanced FVC Scheme for 2D Shallow Water Flows on Unstructured Triangular Meshes

This paper aims to present a new well-balanced,accurate and fast finite volume scheme on unstructured grids to solve hyperbolic conservation laws.It is a scheme that combines both finite volume approach and characteristic method.In this study,we consider a shallow water system with Coriolis effect and bottom friction stresses where this new Finite Volume Characteristics(FVC)scheme has been applied.The physical and mathematical properties of the system,including the C-property,have been well preserved.First,we developed this approach by preserving the advantages of the finite volume discretization such as conservation property and the method of characteristics,in order to avoid Riemann solvers and to enhance the accuracy without any complexity of the MUSCL reconstruction.Afterward,a discretization was applied to the bottom source term that leads to a well-balanced scheme satisfying the steady-state condition of still water.A semi-implicit treatment will also be presented in this study to avoid stability problems due to source terms.Finally,the proposed finite volume method is verified on several benchmark tests and shows good agreement with analytical solutions and experimental results;moreover,it gives a noteworthy accuracy and rapidity improvement compared to the original approaches.

Flux Globalization Based Well-Balanced Path-Conservative Central-Upwind Scheme for the Thermal Rotating Shallow Water Equations

We present an extension of the flux globalization based well-balanced pathconservative central-upwind scheme to the one-and two-dimensional thermal rotating shallow water equations.The scheme is well-balanced in the sense that it can exactly preserve a variety of physically relevant steady states.In the one-dimensional case,it can preserve different“lake-at-rest”equilibria,thermo-geostrophic equilibria,as well as general moving-water steady states.In the two-dimensional case,preserving general moving-water steady states is difficult,and to the best of our knowledge,none of existing schemes can achieve this ultimate goal.The proposed scheme can exactly preserve the x-and y-directional jets in the rotational frame as well as certain genuinely two-dimensional equilibria.Furthermore,our approach employs a path-conservative technique for discretizing nonconservative product terms,which are incorporated into the global fluxes.This allows the developed scheme to exactly preserve some of the discontinuous steady states as well.We provide a number of numerical examples to demonstrate the advantages of the proposed scheme over some alternative finitevolume methods.

A well-balanced positivity preserving two-dimensional shallow flow model with wetting and drying fronts over irregular topography

This paper presents an improved well-balanced Godunov-type 2-D finite volume model with structured grids to simulate shallow flows with wetting and drying fronts over an irregular topography. The intercell flux is computed using a central upwind scheme, which is a Riemann-problem-solver-free method for hyperbolic conservation laws. The nonnegative reconstruction method for the water depth is implemented to resolve the stationary or wet/dry fronts. The bed slope source term is discretized using a central difference method to capture the static flow state over the irregular topography. Second-order accuracy in space is achieved by using the slope limited linear reconstruction method. With the proposed method, the model can avoid the partially wetting/drying cell problem and maintain the mass conservation. The proposed model is tested and verified against three theoretical benchmark tests and two experimental dam break flows. Further, the model is applied to predict the maximum water level and the flood arrival time at different gauge points for the Malpasset dam break event. The predictions agree well with the numerical results and the measurement data published in literature, which demonstrates that with the present model, a well-balanced state can be achieved and the water depth can be nonnegative when the Courant number is kept less than 0.25.

High-Order Positivity-Preserving Well-Balanced Discontinuous Galerkin Methods for Euler Equations with Gravitation on Unstructured Meshes

In this paper,we propose a high-order accurate discontinuous Galerkin(DG)method for the compressible Euler equations under gravitationalfields on un-structured meshes.The scheme preserves a general hydrostatic equilibrium state and provably guarantees the positivity of density and pressure at the same time.Compar-ing with the work on the well-balanced scheme for Euler equations with gravitation on rectangular meshes,the extension to triangular meshes is conceptually plausible but highly nontrivial.Wefirst introduce a special way to recover the equilibrium state and then design a group of novel variables at the interface of two adjacent cells,which plays an important role in the well-balanced and positivity-preserving properties.One main challenge is that the well-balanced schemes may not have the weak positivity property.In order to achieve the well-balanced and positivity-preserving properties simultaneously while maintaining high-order accuracy,we carefully design DG spa-tial discretization with well-balanced numericalfluxes and suitable source term ap-proximation.For the ideal gas,we prove that the resulting well-balanced scheme,cou-pled with strong stability preserving time discretizations,satisfies a weak positivity property.A simple existing limiter can be applied to enforce the positivity-preserving property,without losing high-order accuracy and conservation.Extensive one-and two-dimensional numerical examples demonstrate the desired properties of the pro-posed scheme,as well as its high resolution and robustness.

Well-Balanced Central Scheme for the System of MHD Equations with Gravitational Source Term

A well-balanced second order finite volume central scheme for the magnetohydrodynamic(MHD)equations with gravitational source term is developed in this paper.The scheme is an unstaggered central scheme that evolves the numerical solution on a single grid and avoids solving Riemann problems at the cell interfaces using ghost staggered cells.A subtraction technique is used on the conservative variables with the support of a known steady state in order to manifest the well-balanced property of the scheme.The divergence-free constraint of themagnetic field is satisfied after applying the constrained transport method(CTM)for unstaggered central schemes at the end of each time-step by correcting the components of the magnetic field.The robustness of the proposed scheme is verified on a list of numerical test cases from the literature.

A Well-Balanced Runge-Kutta Discontinuous Galerkin Method for Multilayer ShallowWater Equations with Non-Flat Bottom Topography

A well-balanced Runge-Kutta discontinuous Galerkin method is presented for the numerical solution of multilayer shallow water equations with mass exchange and non-flat bottom topography.The governing equations are reformulated as a non-linear system of conservation laws with differential source forces and reaction terms.Coupling between theflow layers is accounted for in the system using a set of ex-change relations.The considered well-balanced Runge-Kutta discontinuous Galerkin method is a locally conservativefinite element method whose approximate solutions are discontinuous across the inter-element boundaries.The well-balanced property is achieved using a special discretization of source terms that depends on the nature of hydrostatic solutions along with the Gauss-Lobatto-Legendre nodes for the quadra-ture used in the approximation of source terms.The method can also be viewed as a high-order version of upwindfinite volume solvers and it offers attractive features for the numerical solution of conservation laws for which standardfinite element methods fail.To deal with the source terms we also implement a high-order splitting operator for the time integration.The accuracy of the proposed Runge-Kutta discontinuous Galerkin method is examined for several examples of multilayer free-surfaceflows over bothflat and non-flat beds.The performance of the method is also demonstrated by comparing the results obtained using the proposed method to those obtained using the incompressible hydrostatic Navier-Stokes equations and a well-established kinetic method.The proposed method is also applied to solve a recirculationflow problem in the Strait of Gibraltar.

A Well-Balanced Weighted Compact Nonlinear Scheme for Pre-Balanced Shallow Water Equations

It is well known that developing well-balanced schemes for the balance laws is useful for reducing numerical errors.In this paper,a well-balanced weighted compact nonlinear scheme(WCNS)is proposed for shallow water equations in prebalanced forms.The scheme is proved to be well-balanced provided that the source term is treated appropriately as the advection term.Some numerical examples in oneand two-dimensions are also presented to demonstrate the well-balanced property,high order accuracy and good shock capturing capability of the proposed scheme.

An Efficient High Order Well-Balanced Finite Difference WENO Scheme for the Blood Flow Model

The blood flow model admits the steady state,in which the flux gradient is non-zero and is exactly balanced by the source term.In this paper,we present a high order well-balanced finite difference weighted essentially non-oscillatory(WENO)scheme,which exactly preserves the steady state.In order to maintain the wellbalanced property,we propose to reformulate the equation and apply a novel source term approximation.Extensive numerical experiments are carried out to verify the performances of the current scheme such as the maintenance of well-balanced property,the ability to capture the perturbations of such steady state and the genuine high order accuracy for smooth solutions.

Capturing Near-Equilibrium Solutions: A Comparison between High-Order Discontinuous Galerkin Methods and Well-Balanced Schemes

Equilibrium or stationary solutions usually proceed through the exact balance between hyperbolic transport terms and source terms. Such equilibrium solutions are affected by truncation errors that prevent any classical numerical scheme from capturing the evolution of small amplitude waves of physical significance. In order to overcome this problem, we compare two commonly adopted strategies: going to very high order and reduce drastically the truncation errors on the equilibrium solution, or design a specific scheme that preserves by construction the equilibrium exactly, the so-called well-balanced approach. We present a modern numerical implementation of these two strategies and compare them in details, using hydrostatic but also dynamical equilibrium solutions of several simple test cases. Finally, we apply our methodology to the simulation of a protoplanetary disc in centrifugal equilibrium around its star and model its interaction with an embedded planet, illustrating in a realistic application the strength of both methods.

AHigh-OrderWell-Balanced Discontinuous Galerkin Method Based on the Hydrostatic Reconstruction for the Ripa Model

In this work,we present a high-order discontinuous Galerkin method for the shallow water equations incorporating horizontal temperature gradients(also known as the Ripa model),which exactly maintains the lake at rest steady state.Herein,we propose original numerical fluxes defined on the basis of the hydrostatic reconstruction idea and a simple source term approximation.This novel approach allows us to achieve the well-balancing of the discontinuous Galerkin method without complication.Moreover,the proposed method retains genuinely high-order accuracy for smooth solutions and it shows good resolution for discontinuous solutions at the same time.Rigorous numerical analysis as well as extensive numerical results all verify the good performances of the proposed method.

On the Use of Monotonicity-Preserving Interpolatory Techniques in Multilevel Schemes for Balance Laws

Cost-effective multilevel techniques for homogeneous hyperbolic conservation laws are very successful in reducing the computational cost associated to high resolution shock capturing numerical schemes.Because they do not involve any special data structure,and do not induce savings in memory requirements,they are easily implemented on existing codes and are recommended for 1D and 2D simulations when intensive testing is required.The multilevel technique can also be applied to balance laws,but in this case,numerical errors may be induced by the technique.We present a series of numerical tests that point out that the use of monotonicity-preserving interpolatory techniques eliminates the numerical errors observed when using the usual 4-point centered Lagrange interpolation,and leads to a more robust multilevel code for balance laws,while maintaining the efficiency rates observed forhyperbolic conservation laws.

High Order Finite Difference WENO Methods for Shallow Water Equations on Curvilinear Meshes

A high order finite difference numerical scheme is developed for the shallow water equations on curvilinear meshes based on an alternative flux formulation of the weighted essentially non-oscillatory(WENO)scheme.The exact C-property is investigated,and comparison with the standard finite difference WENO scheme is made.Theoretical derivation and numerical results show that the proposed finite difference WENO scheme can maintain the exact C-property on both stationarily and dynamically generalized coordinate systems.The Harten-Lax-van Leer type flux is developed on general curvilinear meshes in two dimensions and verified on a number of benchmark problems,indicating smaller errors compared with the Lax-Friedrichs solver.In addition,we propose a positivity-preserving limiter on stationary meshes such that the scheme can preserve the non-negativity of the water height without loss of mass conservation.

浅水波方程组的熵稳定有限体积格式

本文针对非平底地形上的浅水波方程组,提出了一种高精度熵稳定有限体积格式。首先,我们构造了一个具有二阶精度的well-balanced的半离散熵守恒格式,该格式满足给定熵对的熵恒等式,并精确地保持静水稳态。本文的关键点是使通量梯度和源项的离散化相互匹配;其次,二阶熵守恒格式的可负担熵守恒通量对于最终的高阶格式也是至关重要的。然后,以二阶熵守恒格式为基本模块,实现了高阶well-balanced的半离散熵守恒格式。第三,通过在现有的熵守恒格式中添加适当的耗散项,提出了高阶精度的well-balanced的半离散熵稳定格式,该耗散项基于标度熵变量的加权本质非振荡重构,以克服熵守恒的数值振荡。最后,使用Runge-Kutta方法对半离散格式进行时间积分,以实现最终格式。用大量的数值结果说明所提出的格式满足离散熵不等式,具有良好的平衡性,保持了光滑解的真正高阶精度,并且能够很好地捕获稳态下的小扰动。

浅水动边界问题的位移法模拟

为精确模拟浅水波非线性演化过程中的动边界,提出一种基于位移的Hamilton变分原理,并进而导出一种基于位移的浅水方程(Shallow Water Equation based on Displacement,SWE-D).SWE-D以位移为基本未知量,可以精确满足动边界处的零水深要求并精确捕捉动态边界位置,且解具有协调性.在Hamilton变分原理的框架下,分别采用有限元和保辛积分算法对该浅水方程进行空间离散和时间积分,可有效地处理不平水底情况,保证对非线性演化进行长时间仿真的精度.数值算例表明该方法适用于浅水动边界问题的数值模拟.

Coupled 2D Hydrodynamic and Sediment Transport Modeling of Megaflood due to Glacier Dam-break in Altai Mountains,Southern Siberia

One of the largest known megafloods on earth resulted from a glacier dam-break,which occurred during the Late Quaternary in the Altai Mountains in Southern Siberia.Computational modeling is one of the viable approaches to enhancing the understanding of the flood events.The computational domain of this flood is over 9460 km2 and about 3.784 × 106 cells are involved as a 50 m × 50 m mesh is used,which necessitates a computationally efficient model.Here the Open MP(Open Multiprocessing) technique is adopted to parallelize the code of a coupled 2D hydrodynamic and sediment transport model.It is shown that the computational efficiency is enhanced by over 80% due to the parallelization.The floods over both fixed and mobile beds are well reproduced with specified discharge hydrographs at the dam site.Qualitatively,backwater effects during the flood are resolved at the bifurcation between the Chuja and Katun rivers.Quantitatively,the computed maximum stage and thalweg are physically consistent with the field data of the bars and deposits.The effects of sediment transport and morphological evolution on the flood are considerable.Sensitivity analyses indicate that the impact of the peak discharge is significant,whilst those of the Manningroughness,medium sediment size and shape of the inlet discharge hydrograph are marginal.

间断有限元方法在重力场下欧拉方程中的应用

重力场作用下的欧拉方程在满足熵恒定的前提下,方程保持等熵定常状态。通过等价改写欧拉方程源项并对改写的方程源项进行合理离散,以及修正方程的数值流通量设计出well-balanced间断有限元方法。在离散状态下,well-balanced间断有限元方法可以保持方程的等熵定常状态,并且大、小振幅波的传播测试表明在网格较粗前提下该方法能有效捕捉方程等熵定常状态的小扰动。

Moving-Water Equilibria Preserving HLL-Type Schemes for the Shallow Water Equations

We construct new HLL-type moving-water equilibria preserving upwind schemes for the one-dimensional Saint-Venant system of shallow water equations with nonflat bottom topography.The designed first-and secondorder schemes are tested on a number of numerical examples,in which we verify the well-balanced property as well as the ability of the proposed schemes to accurately capture small perturbations of moving-water steady states.