Dimensional Shallow Water Equations Research Articles

Bilinear Form, Bilinear Bäcklund Transformations, Breather and Periodic-Wave Solutions for a (2+1)-Dimensional Shallow Water Equation with the Time-Dependent Coefficients

Bilinear Form, Bilinear Bäcklund Transformations, Breather and Periodic-Wave Solutions for a (2+1)-Dimensional Shallow Water Equation with the Time-Dependent Coefficients

Analytic solutions for (2+1)-dimensional shallow water equations with flat bottom through Lie symmetry approach

Analytic solutions for (2+1)-dimensional shallow water equations with flat bottom through Lie symmetry approach

Diverse novel analytical and semi-analytical wave solutions of the generalized (2+1)-dimensional shallow water waves model

This article studies the generalized (2 + 1)-dimensional shallow water equation by applying two recent analytical schemes (the extended simplest equation method and the modified Kudryashov method) for constructing abundant novel solitary wave solutions. These solutions describe the bidirectional propagating water wave surface. Some obtained solutions are sketched in two- and three-dimensional and contour plots for demonstrating the dynamical behavior of these waves along shallow water. The accuracy of the obtained solutions and employed analytical schemes is investigated using the evaluated solutions to calculate the initial condition, and then the well-known variational iterational (VI) method is applied. The VI method is one of the most accurate semi-analytical solutions, and it can be applied for high derivative order. The used schemes’ performance shows their effectiveness and power and their ability to handle many nonlinear evolution equations.

Abundant exact analytical solutions and novel interaction phenomena of the generalized (3+1)-dimensional shallow water equation

This paper reveals abundant exact analytical solutions to the generalized (3+1)-D shallow water equation. The generalized bilinear method is used in the solution process and the obtained solutions include the high-order lump-type solutions, the three-wave solutions, the breather solutions. The interaction between the high-order lump-type solutions and the soliton solutions is also elucidated. These solutions have greatly enriched the generalized (3+1)-D shallow water equation in open literature.

Interaction of the elementary waves for shallow water equations with discontinuous topography

The Riemann problem of one dimensional shallow water equations with discontinuous topography has been constructed recently. The elementary waves include shock waves, rarefaction waves, and the stationary wave. The stationary wave appears when the water depth changes, especially when there exists a bottom step. In this paper, we are mainly concerned with the interaction between a stationary wave with either a shock wave or a rarefaction wave. By using the characteristic analysis methods, the evolution of waves is described during the interaction process. The solution in large time scale is also presented in each case. The results may contribute to research on more complicated wave interaction problems.

High-order entropy stable discontinuous Galerkin methods for the shallow water equations: Curved triangular meshes and GPU acceleration

We present a high-order entropy stable discontinuous Galerkin (ESDG) method for the two dimensional shallow water equations (SWE) on curved triangular meshes. The presented scheme preserves a semi-discrete entropy inequality and remains well-balanced for continuous bathymetry profiles. We provide numerical experiments which confirm the high-order accuracy and theoretical properties of the scheme, and compare the presented scheme to an entropy stable scheme based on simplicial summation-by-parts (SBP) finite difference operators. Finally, we report the computational performance of an implementation on Graphics Processing Units (GPUs) and provide comparisons to existing GPU-accelerated implementations of high-order DG methods on quadrilateral meshes.

A Numerical Model for 1D Bed Morphology Calculations

The present work proposes a new morphodynamic model for flow simulation in open channels. The governing one dimensional (1D) shallow water equations along with the Exner sediment continuity equation are solved here using coupled approach. A TVD based MacCormack predictor corrector method is used here for solution of the coupled governing equations. The developed numerical model is used to replicate several test cases of known experimental and approximate solutions. These cases include bed aggradation due to sediment overloading, bed degradation due to sediment shut-off, dam break flow over erodible bed and movable bump due to subcritical and transcritical flows. The results obtained are found to be excellent and free from oscillations near sharp gradient. The numerical model is also found to be capable of capturing the wave-fronts of both water surface and bed sediment. Quantitative analysis of the results in terms of two statistical parameters establishes the potential of the present model in more complicated practical problems.

Two dimensional bed deformation model in turbulent streams

ABSTRACTA coupled model is developed to simulate two dimensional water surface profile, suspended sediment load and bed deformation in unsteady open channels. The hydrodynamical component employs the two dimensional shallow water equations to obtain the hydraulic variables. These, in turn, are used in the morphdynamical component to determine the bed deformation. For the turbulence variables; two turbulence models are supervened to the governing equations. Triangular meshes were developed to discretize the domain of open channel. In order to discretize the governing equations, the explicit finite volume method is used by the total variation diminishing (TVD) schemes. The performance of the developed model is compared to that of the Flow3D software. The comparison results are in good agreement.

A (2+1)-dimensional shallow water equation and its explicit lump solutions

We introduce a new (2+1)-dimensional equation by modifying the potential form of the Calogero–Bogoyavlenskii–Schiff (CBS) equation. By applying the Hirota bilinear method, we construct explicit lump solutions to this new equation and establish necessary and sufficient conditions that guarantee that the solutions are analytic and rationally localized in all directions in space. We also depict the evolution of the profiles of some selected lump solutions with three-dimensional and contour plots. It is immediately observed that the lump solutions generated are solitary wave type solutions as is the case with the KP equation.

One Dimensional Shallow Water Equation Streamflow Modeling using WASH123D Model

Reservoir inflow forecasting assists dam operator in reservoir operation by providing advance information on lake level. This paper discusses on the application of the physical-based numerical model to simulate one-dimensional channel network using WASH123D Model. The model was developed to simulate streamflow at two locations namely Sg Kejar and Sg Tiang, located in the Temengor catchment. The WASH123D model performed channel routing using shallow water equation. The model input data includes rainfall from 5 rainfall stations, river cross sections and simulated runoff data using SCS Method. Due to unavailable observed data, results comparisons were performed using streamflow results obtained using InfoWorks RS Platform. The peak flow from simulation results at Sg Kejar & Sg Tiang Station is 152.6m3/s and 36.6m3/s. The analysis shows good agreement for both simulations with Nash-Sutcliffe Efficiency of 0.68 for Sg Kejar and 0.99 for Sg Tiang. It is suggested that model recalibration shall be made once there is enough water level data to enable more accurate representation of spatial heterogeneity in the catchment processes.

Solving shallow water equations with crisp and uncertain initial conditions

Purpose This paper aims to deal with the application of variational iteration method and homotopy perturbation method (HPM) for solving one dimensional shallow water equations with crisp and fuzzy uncertain initial conditions. Design/methodology/approach Firstly, the study solved shallow water equations using variational iteration method and HPM with constant basin depth and crisp initial conditions. Further, the study considered uncertain initial conditions in terms of fuzzy numbers, which leads the governing equations to fuzzy shallow water equations. Then using cut and parametric concepts the study converts fuzzy shallow water equations to crisp form. Then, HPM has been used to solve the fuzzy shallow water equations. Findings Results obtained by both methods HPM and variational iteration method are compared graphically in crisp case. Solution of fuzzy shallow water equations by HPM are presented in the form triangular fuzzy number plots. Originality/value Shallow water equations with crisp and fuzzy initial conditions have been solved.

An entropy stable discontinuous Galerkin method for the shallow water equations on curvilinear meshes with wet/dry fronts accelerated by GPUs

We extend the entropy stable high order nodal discontinuous Galerkin spectral element approximation for the non-linear two dimensional shallow water equations presented by Wintermeyer et al. (2017) [46] with a shock capturing technique and a positivity preservation capability to handle dry areas. The scheme preserves the entropy inequality, is well-balanced and works on unstructured, possibly curved, quadrilateral meshes. For the shock capturing, we introduce an artificial viscosity to the equations and prove that the numerical scheme remains entropy stable. We add a positivity preserving limiter to guarantee non-negative water heights as long as the mean water height is non-negative. We prove that non-negative mean water heights are guaranteed under a certain additional time step restriction for the entropy stable numerical interface flux. We implement the method on GPU architectures using the abstract language OCCA, a unified approach to multi-threading languages. We show that the entropy stable scheme is well suited to GPUs as the necessary extra calculations do not negatively impact the runtime up to reasonably high polynomial degrees (around N=7). We provide numerical examples that challenge the shock capturing and positivity properties of our scheme to verify our theoretical findings.

Lump and Rogue Wave Solutions of a Reduced (3 + 1)-Dimensional Shallow Water Equation

Lump and Rogue Wave Solutions of a Reduced (3 + 1)-Dimensional Shallow Water Equation

Численные модели для расчета гидрофизических процессов в системах открытых водотоков и водоемов

The present work is devoted to carry out numerical modeling for the combine mathematical model testing. In this model a possible junction of different dimension mathematical models is foresee. It is researched the possibility of this model using for the hydro-physical (thermal, water-salt) description unsteady processes in the water bodies under the influence of nature factors and different actions of the water management. The research used the programs involving in the structure of mathematical model complex developed in HI SB RAS and based on one dimensional shallow water equations (Sent-Venan’ equations) and mean for solution of water resources efficient management problems. The represent numerical results (a sea-water intrusion into fresh water the Ob-Taz outfall sections, formation of thermally-stratified unsteady flows at standing water body) are showed the possibility for using of the complex mathematical model, as well as its efficacy and economy for the study of hydro-physical processes in the open channels and water bodies systems contained the waterways with very different morphology and geometrical sizes.

SHALLOW WATER EQUATION SOLUTION IN 2D USING FINITE DIFFERENCE METHOD WITH EXPLICIT SCHEME

Abstract. Modeling the dynamics of seawater typically uses a shallow water model. The shallow water model is derived from the mass conservation equation and the momentum set into shallow water equations. A two-dimensional shallow water equation alongside the model that is integrated with depth is described in numerical form. This equation can be solved by finite different methods either explicitly or implicitly. In this modeling, the two dimensional shallow water equations are described in discrete form using explicit schemes.Keyword: shallow water equation, finite difference and schema explisit.REFERENSI 1. Bunya, S., Westerink, J. J. dan Yoshimura. 2005. Discontinuous Boundary Implementation for the Shallow Water Equations. Int. J. Numer. Meth. Fluids. 47: 1451-1468.2. Kampf Jochen. 2009. Ocean Modelling For Beginners. Springer Heidelberg Dordrecht. London New York.3. Rezolla, L 2011. Numerical Methods for the Solution of Partial Diferential Equations. Trieste. International Schoolfor Advanced Studies.4. Natakussumah, K. D., Kusuma, S. B. M., Darmawan, H., Adityawan, B. M. Dan Farid, M. 2007. Pemodelan Hubungan Hujan dan Aliran Permukaan pada Suatu DAS dengan Metode Beda Hingga. ITB Sain dan Tek. 39: 97-123.5. Casulli, V. dan Walters, A. R. 2000. An unstructured grid, three-dimensional model based on the shallow water equations. Int. J. Numer. Meth. Fluids. 32: 331-348.6. Triatmodjo, B. 2002. Metode Numerik Beta Offset. Yogyakarta.

Generalized finite difference method for two-dimensional shallow water equations

A novel meshless numerical scheme, based on the generalized finite difference method (GFDM), is proposed to accurately analyze the two–dimensional shallow water equations (SWEs). The SWEs are a hyperbolic system of first-order nonlinear partial differential equations and can be used to describe various problems in hydraulic and ocean engineering, so it is of great importance to develop an efficient and accurate numerical model to analyze the SWEs. According to split-coefficient matrix methods, the SWEs can be transformed to a characteristic form, which can easily present information of characteristic in the correct directions. The GFDM and the second-order Runge-Kutta method are adopted for spatial and temporal discretization of the characteristic form of the SWEs, respectively. The GFDM is one of the newly-developed domain-type meshless methods, so the time-consuming tasks of mesh generation and numerical quadrature can be truly avoided. To use the moving-least squares method of the GFDM, the spatial derivatives at every node can be expressed as linear combinations of nearby function values with different weighting coefficients. In order to properly cooperate with the split-coefficient matrix methods and the characteristic of the SWEs, a new way to determine the shape of star in the GFDM is proposed in this paper to capture the wave transmission. Numerical results and comparisons from several examples are provided to verify the merits of the proposed meshless scheme. Besides, the numerical results are compared with other solutions to validate the accuracy and the consistency of the proposed meshless numerical scheme.

An entropy stable nodal discontinuous Galerkin method for the two dimensional shallow water equations on unstructured curvilinear meshes with discontinuous bathymetry

We design an arbitrary high-order accurate nodal discontinuous Galerkin spectral element approximation for the non-linear two dimensional shallow water equations with non-constant, possibly discontinuous, bathymetry on unstructured, possibly curved, quadrilateral meshes. The scheme is derived from an equivalent flux differencing formulation of the split form of the equations. We prove that this discretization exactly preserves the local mass and momentum. Furthermore, combined with a special numerical interface flux function, the method exactly preserves the mathematical entropy, which is the total energy for the shallow water equations. By adding a specific form of interface dissipation to the baseline entropy conserving scheme we create a provably entropy stable scheme. That is, the numerical scheme discretely satisfies the second law of thermodynamics. Finally, with a particular discretization of the bathymetry source term we prove that the numerical approximation is well-balanced. We provide numerical examples that verify the theoretical findings and furthermore provide an application of the scheme for a partial break of a curved dam test problem.

2D Versus 1D Models for Shallow Water Equations

In this paper we present a general framework to construct 1D width averaged models when the flow is constrained -e.g. by topography- to be almost 1D. We start from two dimensional shallow water equations, perform an asymptotic expansion of the fluid elevation and velocity field in the spirit of wave diffusive equations and establish a set of 1D equations made of a mass, momentum and energy equations which are close to the one usually used in hydraulic engineering. We show that in some special cases, like the U-shaped river bed, that our set of equations reduces to the classical 1d shallow water equations. Out of these configurations, there is an O (1) deviation of our model from the classical one.

A comparison of two entropy stable discontinuous Galerkin spectral element approximations for the shallow water equations with non-constant topography

In this work, we compare and contrast two provably entropy stable and high-order accurate nodal discontinuous Galerkin spectral element methods applied to the one dimensional shallow water equations for problems with non-constant bottom topography. Of particular importance for numerical approximations of the shallow water equations is the well-balanced property. The well-balanced property is an attribute that a numerical approximation can preserve a steady-state solution of constant water height in the presence of a bottom topography. Numerical tests are performed to explore similarities and differences in the two high-order schemes.

A well balanced and entropy conservative discontinuous Galerkin spectral element method for the shallow water equations

In this work, we design an arbitrary high order accurate nodal discontinuous Galerkin spectral element type method for the one dimensional shallow water equations. The novel method uses a skew-symmetric formulation of the continuous problem. We prove that this discretisation exactly preserves the local mass and momentum. Furthermore, we show that combined with a special numerical interface flux function, the method exactly preserves the entropy, which is also the total energy for the shallow water equations. Finally, we prove that the surface fluxes, the skew-symmetric volume integrals, and the source term are well balanced. Numerical tests are performed to demonstrate the theoretical findings.