Entropy‑Conservative Discontinuous Galerkin Methods for the Shallow Water Equations with Uncertainty

In this paper,we develop an entropy-conservative discontinuous Galerkin(DG)method for the shallow water(SW)equation with random inputs.One of the most popular methods for uncertainty quantifcation is the generalized Polynomial Chaos(gPC)approach which we consider in the following manuscript.We apply the stochastic Galerkin(SG)method to the stochastic SW equations.Using the SG approach in the stochastic hyperbolic SW system yields a purely deterministic system that is not necessarily hyperbolic anymore.The lack of the hyperbolicity leads to ill-posedness and stability issues in numerical simulations.By transforming the system using Roe variables,the hyperbolicity can be ensured and an entropy-entropy fux pair is known from a recent investigation by Gerster and Herty(Commun.Comput.Phys.27(3):639–671,2020).We use this pair and determine a corresponding entropy fux potential.Then,we construct entropy conservative numerical twopoint fuxes for this augmented system.By applying these new numerical fuxes in a nodal DG spectral element method(DGSEM)with fux diferencing ansatz,we obtain a provable entropy conservative(dissipative)scheme.In numerical experiments,we validate our theoretical fndings.

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.

Two-dimensional shallow water equations with porosity and their numerical scheme on unstructured grids

In this study, porosity was introduced into two-dimensional shallow water equations to reflect the effects of obstructions, leading to the modification of the expressions for the flux and source terms. An extra porosity source term appears in the momentum equation. The numerical model of the shallow water equations with porosity is presented with the finite volume method on unstructured grids and the modified Roe-type approximate Riemann solver. The source terms of the bed slope and porosity are both decomposed in the characteristic direction so that the numerical scheme can exactly satisfy the conservative property. The present model was tested with a dam break with discontinuous porosity and a flash flood in the Toce River Valley. The results show that the model can simulate the influence of obstructions, and the numerical scheme can maintain the flux balance at the interface with high efficiency and resolution.

An Explicit High Resolution Scheme for Nonlinear Shallow Water Equations

The present study develops a numerical model of the two-dimensional fully nonlinear shallow water equations （NSWE） for the wave run-up on a beach. The finite volume method （FVM） is used to solve the equations, and a second-order explicit scheme is developed to improve the computation efficiency. The numerical fluxes are obtained by the two dimensional Roe＇ s flux function to overcome the errors caused by the use of one dimensional fluxes in dimension splitting methods. The high-resolution Godunov-type TVD upwind scheme is employed and a second-order accuracy is achieved based on monotonic upstream schemes for conservation laws （MUSCL） variable extrapolation; a nonlinear limiter is applied to prevent unwanted spurious oscillation. A simple but efficient technique is adopted to deal with the moving shoreline boundary. The verification of the solution technique is carried out by comparing the model output with documented results and it shows that the solution technique is robust.

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.

GLOBAL WELL-POSEDNESS FOR A FIFTH-ORDER SHALLOW WATER EQUATION ON THE CIRCLE

The periodic initial value problem of a fifth-order shallow water equation t u 2 x t u ＋ 3 x u 5 x u ＋ 3u x u 2 x u 2 x u u 3 x u = 0 is shown to be globally well-posed in Sobolev spaces˙ H s （T） for s 〉 2/3 by I-method. For this equation lacks scaling invariance, we first reconsider the local result and pay special attention to the relationship between the lifespan of the local solution and the initial data, and then prove the almost conservation law, and finally obtain the global well-posedness by an iteration process.

MIXED FINITE ELEMENT METHODS FOR THE SHALLOW WATER EQUATIONS INCLUDING CURRENT AND SILT SEDIMENTATION (Ⅱ)——THE DISCRETE-TIME CASE ALONG CHARACTERISTICS

The mixed finite element(MFE) methods for a shallow water equation system consisting of water dynamics equations,silt transport equation,and the equation of bottom topography change were derived.A fully discrete MFE scheme for the discrete_time along characteristics is presented and error estimates are established.The existence and convergence of MFE solution of the discrete current velocity,elevation of the bottom topography,thickness of fluid column,and mass rate of sediment is demonstrated.

MIXED FINITE ELEMENT METHODS FOR THE SHALLOW WATER EQUATIONS INCLUDING CURRENT AND SILT SEDIMENTA-TION (Ⅰ)-THE CONTINUOUS-TIME CASE

An initial-boundary value problem for shallow equation system consisting of water dynamics equations,silt transport equation, the equation of bottom topography change,and of some boundary and initial conditions is studied, the existence of its generalized solution and semidiscrete mixed finite element(MFE) solution was discussed, and the error estimates of the semidiscrete MFE solution was derived.The error estimates are optimal.

High-resolution central difference scheme for the shallow water equations

A two-dimensional nonoscillatory central difference scheme was extended to the shallow water equations. A high-resolution numerical method for solving the shallow water equations was presented. In order to prevent oscillation, the nonlinear limiter is employed to approximate the discrete slopes. The main advantage of the presented method is simplicity comparable with the upwind schemes. This method does not require Riemann solvers or some form of flux difference splitting methods. Furthermore, the discrete derivatives of flux can be approximated by the component-wise approach and thus the computation of Jacobian can be avoided. The method retains high resolution and high accuracy similar to the upwind results. It is applied to simulating several tests, including circular dam-break problem, shock focusing problem and partial dam-break problem. The results are in good agreement with the numerical results obtained by other methods. The simulated results also demonstrate that the presented method is stable and efficient.

THE CAUCHY PROBLEM FOR THE TWO LAYER VISCOUS SHALLOW WATER EQUATIONS

In this paper,the Cauchy problem for the two layer viscous shallow water equations is investigated with third-order surface-tension terms and a low regularity assumption on the initial data.The global existence and uniqueness of the strong solution in a hybrid Besov space are proved by using the Littlewood-Paley decomposition and Friedrichs'regularization method.

A Regional Spectral Nested Shallow Water Equation Model

A method to expand meteorological elements in terms of finite double Fourier series in a limited-region and a spectral nested shallow water equation model based upon the method with conformal map projection in rectangular coordinates, have been proposed, and computational stability and efficiency of time integration have been discussed.

A second-order accurate fluid-in-cell (FLIC) method for the 2D shallow water equations with topography

The fluid-in-cell （FLIC） approach of Gentry et al. （1966） is extended to second-order accuracy in space and applied to solve the 2D shallow water equations with topography. The FLIC method can be interpreted in a finite volume sense, it therefore conserves both water mass and momentum. Like the original FLIC method the second-order FLIC method presented here is able to handle wetting-drying fronts without any special treatment. Moreover, the resulting method is shock capturing and well-balanced, satisfying both the C- and extended C-properties exactly. Published by Elsevier Ltd on behalf of The Chinese Society of Theoretical and Applied Mechanics.

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.

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.

Residual symmetry, CRE integrability and interaction solutions of two higher-dimensional shallow water wave equations

Two(3+1)-dimensional shallow water wave equations are studied by using residual symmetry and the consistent Riccati expansion(CRE) method. Through localization of residual symmetries, symmetry reduction solutions of the two equations are obtained. The CRE method is applied to the two equations to obtain new B?cklund transformations from which a type of interesting interaction solution between solitons and periodic waves is generated.

NUMERICAL SIMULATION FOR 2D SHALLOW WATER EQUATIONS BY USING GODUNOV-TYPE SCHEME WITH UNSTRUCTURED MESH

In order to establish a well-balanced scheme, 2D shallow water equations were transformed and solved by using the Finite Volume Method （FVM） with unstructured mesh. The numerical flux from the interface between cells was computed with an exact Riemann solver, and the improved dry Riemann solver was applied to deal with the wet/dry problems. The model was verified through computing some typical examples and the tidal bore on the Qiantang River. The results show that the scheme is robust and accurate, and could be applied extensively to engineering problems.

AN UNSTRUCTURED FINITE-VOLUME ALGORITHM FOR NONLINEAR TWO-DIMENSIOAL SHALLOW WATER EQUATION

An unstructured finite-volume numerical algorithm was presented for solution of the two-dimensional shallow water equations, based on triangular or arbitrary quadrilateral meshes. The Roe type approximate Riemann solver was used to the system. A second-order TVD scheme with the van Leer limiter was used in the space discretization and a two-step Runge-Kutta approach was used in the time discretization. An upwind, as opposed to a pointwise, treatment of the slope source terms was adopted and the semi-implicit treatment was used for the friction source terms. Verification for two-dimension dam-break problems are carried out by comparing the present results with others and very good agreement is shown.

A CHARACTERISTIC-BASED FINITE VOLUME SCHEME FOR SHALLOW WATER EQUATIONS

We propose a new characteristic-based finite volume scheme combined with the method of Central Weighted Essentially Non-Oscillatory （CWENO） reconstruction and characteristics, to solve shallow water equations. We apply the scheme to simulate dam-break problems. A number of challenging test cases are considered, such as large depth differences even wet/dry bed. The numerical solutions well agree with the analytical solutions. The results demonstrate the desired accuracy, high-resolution and robustness of the presented scheme.

A HIGH RESOLUTION FINITE VOLUME METHOD FOR SOLVING SHALLOW WATER EQUATIONS

A high resolution finite volume numerical method for solving the shallow water equations is developed in this paper. In order to extend finite difference TVD scheme to finite volume method, a new geometry and topology of control bodies is defined by considering the corresponding relationships between nodes and elements. This solver is implemented on arbitrary quadrilateral meshes and their satellite elements, and based on a second order hybrid type of TVD scheme in space discretization and a two step Runge Kutta method in time discretization. Then it is used to deal with two typical dam break problems and very satisfactory results are obtained comparied with other numerical solutions. It can be considered as an efficient implement for the computation of shallow water problems, especially concerning those having discontinuities, subcritical and supercritical flows and complex geometries.

The Cauchy Problem for the Fifth Order Shallow Water Equation

The local well-posedness of the Cauchy problem for the fifth order shallow water equation δtu＋αδx^5u＋βδx^3u＋rδxu＋μuδru=0,x,t∈R, is established for low regularity data in Sobolev spaces H^s（s≥-3/8） by the Fourier restriction norm method. Moreover, the global well-posedness for L^2 data follows from the local well-posedness and the conserved quantity. For data in H^s（s〉0）, the global well-posedness is also proved, where the main idea is to use the generalized bilinear estimates associated with the Fourier restriction norm method to prove that the existence time of the solution only depends on the L^2 norm of initial data.