Numerical solution of shallow water magnetohydrodynamic equations with non-flat bottom topography

The governing equations of shallow water magnetohydrodynamics describe the dynamics of a thin layer of nearly incompressible and electrically conducting fluids for which the evolution is nearly two-dimensional with magnetic equilibrium in the third direction. A high-resolution central-upwind scheme is applied to solve the model equations considering non-flat bottom topography. The suggested method is an upwind biased non-oscillatory finite volume scheme which doées not require a Riemann solver at each time step. To satisfy the divergence-free constraint, the projection method is used. Several case studies are carried out. For validation, a gas kinetic flux vector splitting scheme is also applied to the same model.     

Modified shallow water equations which admit the propagation of discontinuous waves over a dry bed

A method for modeling the propagation of discontinuous waves over a dry bed using the first approximation of shallow water theory is proposed. The method is based on a modified conservation law of total momentum that takes into account the concentrated momentum losses due to the formation of local turbulent vortex structures in the fluid surface layer at a discontinuous-wave front. A quantitative estimate of these losses is obtained by deriving the shallow water equations from the Navier-Stokes equations with allowance for viscosity, which has a rapidly increasing effect in the turbulent flow regions described by discontinuous waves. The stability of the discontinuous waves admitted by the modified system of conservation laws of shallow water theory is examined. As an example, a comparative analysis is performed of the solutions of the dam-break problem obtained for the classical and modified shallow water models. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 6, pp. 22–43, November–December, 2007     

A Newton multigrid method for steady-state shallow water equations with topography and dry areas

A Newton multigrid method is developed for one-dimensional (1D) and two-dimensional (2D) steady-state shallow water equations (SWEs) with topography and dry areas. The nonlinear system arising from the well-balanced finite volume discretization of the steady-state SWEs is solved by the Newton method as the outer iteration and a geometric multigrid method with the block symmetric Gauss-Seidel smoother as the inner iteration. The proposed Newton multigrid method makes use of the local residual to regularize the Jacobian matrix of the Newton iteration, and can handle the steady-state problem with wet/dry transition. Several numerical experiments are conducted to demonstrate the efficiency, robustness, and well-balanced property of the proposed method. The relation between the convergence behavior of the Newton multigrid method and the distribution of the eigenvalues of the iteration matrix is detailedly discussed.     

Numerical viscosity reduction in the resolution of the shallow water equations with turbulent term

A first‐order finite volume model for the resolution of the 2D shallow water equations with turbulent term is presented. An upwind discretization of the equations that include the turbulent term is carried out. A method to reduce the excess of numerical viscosity (or diffusion) produced by the upwinding of the flux term is proposed. Two different discretizations of the turbulent term are compared, and results for uniform distributions of the viscosity are presented. Finally, two discretizations of the time derivative which are more efficient than Euler's are proposed and compared. Copyright © 2008 John Wiley & Sons, Ltd.     

Sensitivity of the 1D shallow water equations with source terms: Solution method for discontinuous flows

A finite volume‐based numerical technique is presented concerning the sensitivity of the solution of the one‐dimensional Shallow Water Equations with scalar transport. An approximate Riemann solver is proposed for direct sensitivity calculation even in the presence of discontinuous solutions. The Shallow Water Sensitivity Equations are first derived as well as the expressions of the sensitivity source terms, initial and boundary conditions. The numerical technique is then detailed and application examples are provided to assess the method's efficiency in estimating the sensitivity to different parameters (friction coefficient and initial and boundary conditions). The application of the dam‐break problem to a trapezoidal channel is also provided. The comparison with the analytical solution and the classical empirical approach illustrates the usefulness of the direct sensitivity calculation. Copyright © 2010 John Wiley & Sons, Ltd.     

Numerical integration of the shallow water equations over a sloping shelf

A finite-difference method is described for the numerical integration of the one-dimensional shallow water equations over a sloping shelf that allows for a continuously moving shoreline. An application of the technique is made to the propagation of non-breaking waves towards the shoreline. The results of the computation are compared with an evaluation based upon an exact analytical treatment of the non-linear equations. Excellent agreement is found for both tsunami and tidal scale oscillations.     

Discontinuous solutions of the shallow water equations on a rotatable attracting sphere

The shallow water equations on a rotatable attracting sphere represent a system of hyperbolic equations on a compact manifold. These equations are derived in a spherical coordinate system from the integral laws of mass and total momentum conservation with account for the Coriolis and centrifugal forces. An analysis of the stability of discontinuous solutions with discontinuous waves and contact discontinuities is made using the closing law of total energy conservation, which represents a convex extension of the basic conservation-law system. The classes of stationary, one-dimensional (latitude-dependent only) exact solutions with contact discontinuities and discontinuous waves are constructed. Within the framework of the one-dimensional equations the test problem of wave flows resulting from the simultaneous break of two dams confining a fluid at rest in the vicinities of the poles is numerically modeled.     

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.     

Some families of exact solutions of the equations of two-dimensional shallow water theory

Results are presented of a study of the exact solutions of the equations of two-dimensional unsteady and steady shallow water theory, based on the group properties of these equations. The first part presents the group properties of the equations in question; the second part presents the invariant solutions of these equations.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 10, No. 6, pp. 62–71, November–December, 1969.The author wishes to thank L. V. Ovsyannikov and N. Kh. Ibragimov for valuable guidance in carrying out this study.     

Backlund transformation and exact solutions for whitham-broer-kaup equations in shallow water

I.IntroductionTodescribethepropagationofshallowwaterwave,manywell-knowncompletelyintegrablemodelsareintroduced,suchasKdVequatioll,Boussinesqequation,K-Pequation,WBKequation,etc.UnderBoussinesqapproximation,Whitham,BroerandKaupl"2'3]obtainednon-lilleurWBKequationwhereu=u(x,l)isthefieldofhorizontalvelocity;v=v(x,t)istheheightthatdeviatefromequilibriumpositionofliquid;a,gareconstantsthatrepresentdift'erentdispersivepower.TheEqs.(1.I),(l.2)areverygoodmodelstodescl.ibedispersivewave.Ifa=0,P…     

Performance of finite volume solutions to the shallow water equations with shock‐capturing schemes

Numerical methods have become well established as tools for solving problems in hydraulic engineering. In recent years the finite volume method (FVM) with shock capturing capabilities has come to the fore because of its suitability for modelling a variety of types of flow; subcritical and supercritical; steady and unsteady; continuous and discontinuous and its ability to handle complex topography easily. This paper is an assessment and comparison of the performance of finite volume solutions to the shallow water equations with the Riemann solvers; the Osher, HLL, HLLC, flux difference splitting (Roe) and flux vector splitting. In this paper implementation of the FVM including the Riemann solvers, slope limiters and methods used for achieving second order accuracy are described explicitly step by step. The performance of the numerical methods has been investigated by applying them to a number of examples from the literature, providing both comparison of the schemes with each other and with published results. The assessment of each method is based on five criteria; ease of implementation, accuracy, applicability, numerical stability and simulation time. Finally, results, discussion, conclusions and recommendations for further work are presented. Copyright © 2002 John Wiley & Sons, Ltd.     

A triangular spectral/hp discontinuous Galerkin method for modelling 2D shallow water equations

We present a spectral/hp element discontinuous Galerkin model for simulating shallow water flows on unstructured triangular meshes. The model uses an orthogonal modal expansion basis of arbitrary order for the spatial discretization and a third‐order Runge–Kutta scheme to advance in time. The local elements are coupled together by numerical fluxes, evaluated using the HLLC Riemann solver. We apply the model to test cases involving smooth flows and demonstrate the exponentially fast convergence with regard to polynomial order. We also illustrate that even for results of ‘engineering accuracy’ the computational efficiency increases with increasing order of the model and time of integration. The model is found to be robust in the presence of shocks where Gibbs oscillations can be suppressed by slope limiting. Copyright 2004 John Wiley & Sons, Ltd.     

Symmetries and exact solutions of the shallow water equations for a two-dimensional shear flow

This paper considers nonlinear equations describing the propagation of long waves in two-dimensional shear flow of a heavy ideal incompressible fluid with a free boundary. A nine-dimensional group of transformations admitted by the equations of motion is found by symmetry methods. Two-dimensional subgroups are used to find simpler integrodifferential submodels which define classes of exact solutions, some of which are integrated. New steady-state and unsteady rotationally symmetric solutions with a nontrivial velocity distribution along the depth are obtained. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 5, pp. 41–54, September–October, 2008.     

Numerical treatment of a discontinuous top surface in 2D shallow water mixed flow modeling

This paper presents a numerical strategy based on shallow water equations (SWE) coupled with the 2D Preissmann slot model to handle a ceiling step discontinuity in finite volume schemes for mixed flow modeling. In practice, a typical situation would be a closed structure, such as a bridge or culvert, which induces a sudden vertical flow constriction and may even run partly or totally full in high flow conditions. In such case, both the inlet and outlet of the structure involve a discontinuity in the top elevation. This special singularity is topologically represented by inserting a fictitious cell between 2 adjacent computational cells characterized by sharply different ceiling elevation. The 2D SWE are solved by means of a well‐balanced quasi‐conservative Godunov‐type numerical scheme based on the Slope Limiter Centered (SLIC) scheme. The flow variables at each boundary of the fictitious cell are reconstructed by adopting the cross‐sectional shape of the adjoining cell. Accordingly, the dynamic effect of the structure deck on the flow is suitably modeled, and the C‐property for a stationary solution is rigorously satisfied, even when the closed structure is partially full. The capability of the numerical scheme is verified by comparison with both novel analytical solutions of 1D Riemann problems with a ceiling step discontinuity and experimental data of steady and unsteady mixed flows available in literature. Finally, a real‐scale application to a multiple arch bridge is presented. The results show that the method is robust and effective in predicting the 2D features induced by a crossing structure on the flow dynamics.     

Practical evaluation of five partly discontinuous finite element pairs for the non‐conservative shallow water equations

This paper provides a comparison of five finite element pairs for the shallow water equations. We consider continuous, discontinuous and partially discontinuous finite element formulations that are supposed to provide second‐order spatial accuracy. All of them rely on the same weak formulation, using Riemann solver to evaluate interface integrals. We define several asymptotic limit cases of the shallow water equations within their space of parameters. The idea is to develop a comparison of these numerical schemes in several relevant regimes of the subcritical shallow water flow. Finally, a new pair, using non‐conforming linear elements for both velocities and elevation (P?P), is presented, giving optimal rates of convergence in all test cases. P?P₁ and P?P₁ mixed formulations lack convergence for inviscid flows. P?P₂ pair is more expensive but provides accurate results for all benchmarks. P?P provides an efficient option, except for inviscid Coriolis‐dominated flows, where a small lack of convergence is observed. Copyright © 2009 John Wiley & Sons, Ltd.     

On discontinuous periodic solution and discontinuous solitary wave of two-dimension shallow water equation

In this paper we discuss discontinuous periodic solution and discontinuous solitary wave of the shallow water model of geophysical fluid dynamics. When we consider the properties of trajectory near non-equilibrium point, i.e. singular point, we find that if we introduce the concept of generalized solution (piecewise smoothing continuous solution), then the system will produce discontinues periodic solution and the condition of discontinuous periodic solution can he obtained. When the system is degenerated, we find that the discontinuous solitary wave is existent in the system. In this paper we consider a series of problems and obtain analytic expression of discontinuous solution. This result is compared with squall line in the atmosphere, and both of them have many things in common.     

Relaxation schemes for the shallow water equations

We present a class of first and second order in space and time relaxation schemes for the shallow water (SW) equations. A new approach of incorporating the geometrical source term in the relaxation model is also presented. The schemes are based on classical relaxation models combined with Runge–Kutta time stepping mechanisms. Numerical results are presented for several benchmark test problems with or without the source term present. Copyright © 2003 John Wiley & Sons, Ltd.     

Numerical solutions of differential equations with fractional L-derivative

Fractional differential equations are solved with L-fractional derivatives, using numerical procedures. Two characteristic fractional differential equations are numerically solved. The first equation describes the motion of a thin rigid plate immersed in a Newtonian fluid connected by a massless spring to a fixed point, and the other one the diffusion of gas in a fluid.     

A correction for balancing discontinuous bed slopes in two‐dimensional smoothed particle hydrodynamics shallow water modeling

In this paper, a smoothed particle hydrodynamics (SPH) numerical model for the shallow water equations (SWEs) with bed slope source term balancing is presented. The solution of the SWEs using SPH is attractive being a conservative, mesh‐free, automatically adaptive method without special treatment for wet‐dry interfaces. Recently, the capability of the SPH–SWEs numerical scheme with shock capturing and general boundary conditions has been used for predicting practical flooding problems. The balance between the bed slope source term and fluxes in shallow water models is desirable for reliable simulations of flooding over bathymetries where discontinuities are present and has received some attention in the framework of Finite Volume Eulerian models. The imbalance because of the source term resulting from the calculation of the the water depth is eradicated by means of a corrected mass, which is able to remove the error introduced by a bottom discontinuity. Two different discretizations of the momentum equation are presented herein: the first one is based on the variational formulation of the SWEs in order to obtain a fully conservative formulation, whereas the second one is obtained using a non‐conservative form of the free‐surface elevation gradient. In both formulations, a variable smoothing length is considered. Results are presented demonstrating the corrections preserve still water in the vicinity of either 1D or 2D bed discontinuities and provide close agreement with 1D analytical solutions for rapidly varying flows over step changes in the bed. The method is finally applied to 2D dam break flow over a square obstacle where the balanced formulation improves the agreement with experimental measurements of the free surface. Copyright © 2012 John Wiley & Sons, Ltd.