A robust central scheme for the shallow water flows with an abrupt topography based on modified hydrostatic reconstructions

We study a second-order central scheme for the shallow water flows with a discontinuous bottom topography based on modified hydrostatic reconstructions (HRs). The first HR scheme was proposed in Audusse et al, which may be missing the effect of the large discontinuous bottom topography. We introduce a modified HR method to cope with this numerical difficulty. The new scheme is well-balanced for still water solutions and can guarantee the positivity of the water depth. Finally, several numerical results of classical problems of the shallow water equations confirmed these properties of the new scheme. Especially, the new scheme yields superior results for the shallow water downhill flow over a step.     

Simulation of flows with strong shocks with an adaptive conservative scheme

In the present work, numerical simulations of unsteady flows with moving shocks are presented. An unsteady mesh adaptation method, based on error equidistribution criteria, is adopted to capture the most important flow features. The modifications to the topology of the grid are locally interpreted in terms of continuous deformation of the finite volumes built around the nodes. The arbitrary Lagrangian–Eulerian formulation of the Euler equations is then applied to compute the flow variable over the new grid without resorting to any explicit interpolation step. The numerical results show an increase in the accuracy of the solution, together with a strong reduction of the computational costs, with respect to computations with a uniform grid using a larger number of nodes.     

A Comparison of positivity preserving techniques without time step restriction for the DG scheme applied to shallow water flows

Positivity preservation in the context of shallow water flows is important as soon as wet regions become dry or a dry region is flooded. However, positivity preserving explicit time stepping of DG semi-discretizations can not be applied in a straigthforward manner to the case of implicit time integration as the enforcement of non-negativity generally demands rather restrictive time step constraints. In this context, we compare two different modifications to the strategy of positivity preservation based on a production-destruction splitting of the DG scheme for the cell means of water height. One strategy uses the so-called Patankar trick while the other one is based on an iterative redistribution of the water level. Both modifications guarantee non-negativity of the water height for any time step size while still preserving conservativity. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A well-balanced van Leer-type numerical scheme for shallow water equations with variable topography

A well-balanced van Leer-type numerical scheme for the shallow water equations with variable topography is presented. The model involves a nonconservative term, which often makes standard schemes difficult to approximate solutions in certain regions. The construction of our scheme is based on exact solutions in computational form of local Riemann problems. Numerical tests are conducted, where comparisons between this van Leer-type scheme and a Godunov-type scheme are provided. Data for the tests are taken in both the subcritical region as well as supercritical region. Especially, tests for resonant cases where the exact solutions contain coinciding waves are also investigated. All numerical tests show that each of these two methods can give a good accuracy, while the van Leer -type scheme gives a better accuracy than the Godunov-type scheme. Furthermore, it is shown that the van Leer-type scheme is also well-balanced in the sense that it can capture exactly stationary contact discontinuity waves.     

Central-upwind scheme for shallow water equations with discontinuous bottom topography

Finite-volume central-upwind schemes for shallow water equations were proposed in [A. Kurganov and G. Petrova, Commun. Math. Sci., 5 (2007), 133–160]. These schemes are capable of maintaining “lake-at-rest” steady states and preserving the positivity of the computed water depth. The well-balanced and positivity preserving features of the central-upwind schemes are achieved, in particular, by using continuous piecewise linear interpolation of the bottom topography function. However, when the bottom function is discontinuous or a model with a moving bottom topography is studied, the continuous piecewise linear approximationmay not be sufficiently accurate and robust.     

A new model of Saint Venant and Savage–Hutter type for gravity driven shallow water flows

We introduce a new model for shallow water flows with non-flat bottom. A prototype is the Saint Venant equation for rivers and coastal areas, which is valid for small slopes. An improved model, due to Savage–Hutter, is valid for small slope variations. We introduce a new model which relaxes all restrictions on the topography. Moreover it satisfies the properties (i) to provide an energy dissipation inequality, (ii) to be an exact hydrostatic solution of Euler equations. The difficulty we overcome here is the normal dependence of the velocity field, that we are able to establish exactly. Applications we have in mind concern, in particular, computational aspects of flows of granular material (for example in debris avalanches) where such models are especially relevant. To cite this article: F. Bouchut et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Regularized shallow water equations and an efficient method for numerical simulation of shallow water flows

Regularized shallow water equations are derived as based on a regularization of the Navier-Stokes equations in the form of quasi-gasdynamic and quasi-hydrodynamic equations. Efficient finite-difference algorithms based on the regularized shallow water equations are proposed for the numerical simulation of shallow water flows. The capabilities of the model are examined by computing a test Riemann problem, the flow over an obstacle, and asymmetric dam break.     

A two-dimensional artificial viscosity technique for modelling discontinuity in shallow water flows

In this study, a two-dimensional cell-centred finite volume scheme is used to simulate discontinuity in shallow water flows. Instead of using a Riemann solver, an artificial viscosity technique is developed to minimise unphysical oscillations. This is constructed from a combination of a Laplacian and a biharmonic operator using a maximum eigenvalue of the Jacobian matrix. In order to achieve high-order accuracy in time, we use the fourth-order Runge–Kutta method. A hybrid formulation is then proposed to reduce computational time, in which the artificial viscosity technique is only performed once per time step. The convective flux of the shallow water equations is still re-evaluated four times, but only by averaging left and right states, thus making the computation much cheaper. A comparison of analytical and laboratory results shows that this method is highly accurate for dealing with discontinuous flows. As such, this artificial viscosity technique could become a promising method for solving the shallow water equations.     

A kinetic scheme for unsteady pressurised flows in closed water pipes

The aim of this paper is to present a kinetic numerical scheme for the computations of transient pressurised flows in closed water pipes. Firstly, we detail the mathematical model written as a conservative hyperbolic partial differentiel system of equations, and then we recall how to obtain the corresponding kinetic formulation. Then we build the kinetic scheme ensuring an upwinding of the source term due to the topography performed in a close manner described by Perthame and Simeoni (2001) [1] and Botchorishvili et al. (2003) [2] using an energetic balance at microscopic level. The validation is lastly performed in the case of a water hammer in an uniform pipe: we compare the numerical results provided by an industrial code used at EDF-CIH (France), which solves the Allievi equation (the commonly used equation for pressurised flows in pipes) by the method of characteristics, with those of the kinetic scheme. It appears that they are in a very good agreement.     

Regularized shallow water equations for numerical simulation of flows with a moving shoreline

A numerical algorithm for simulating free-surface flows based on regularized shallow water equations is adapted to flows involving moving dry-bed areas. Well-balanced versions of the algorithm are constructed. Test computations of flows with dry-bed areas in the cases of water runup onto a plane beach and a constant-slope beach are presented. An example of tsunami simulation is given.     

A new shallow water model with polynomial dependence on depth

In this paper, we study two‐dimensional Euler equations in a domain with small depth. With this aim, we introduce a small non‐dimensional parameter ε related to the depth and we use asymptotic analysis to study what happens when ε becomes small. We obtain a model for ε small that, after coming back to the original domain, gives us a shallow water model that considers the possibility of a non‐constant bottom, and the horizontal velocity has a dependence on z introduced by the vorticity when it is not zero. This represents an interesting novelty with respect to shallow water models found in the literature. We stand out that we do not need to make a priori assumptions about velocity or pressure behaviour to obtain the model. The new model is able to approximate the solutions to Euler equations with dependence on z (reobtaining the same velocities profile), whereas the classic model just obtains the average velocity. Copyright © 2007 John Wiley & Sons, Ltd.     

Transonic shocks for steady Euler flows with cylindrical symmetry

In this paper, we prove the existence of transonic shocks adjacent to a uniform one for the full Euler system for steady compressible fluids with cylindrical symmetry in a cylinder, and consequently show the stability of such uniform transonic shocks. Mathematically we solve a free boundary problem for a quasi-linear elliptic–hyperbolic composite system. This reveals that the boundary conditions and equations interact in a subtle way. The key point is to “separate” in a suitable way the elliptic and hyperbolic parts of the system. The approach developed here can be applied to deal with certain multidimensional problems concerning stability of transonic shocks for the full Euler system.     

Simulation of shallow water flows with shoaling areas and bottom discontinuities

A numerical method based on a second-order accurate Godunov-type scheme is described for solving the shallow water equations on unstructured triangular-quadrilateral meshes. The bottom surface is represented by a piecewise linear approximation with discontinuities, and a new approximate Riemann solver is used to treat the bottom jump. Flows with a dry sloping bottom are computed using a simplified method that admits negative depths and preserves the liquid mass and the equilibrium state. The accuracy and performance of the approach proposed for shallow water flow simulation are illustrated by computing one- and two-dimensional problems.     

Semi-implicit staggered mesh scheme for the multi-layer shallow water system

We present a semi-implicit scheme for a two-dimensional multilayer shallow water system with density stratification, formulated on general staggered meshes. The main result of the present note concerns the control of the mechanical energy at the discrete level, principally based on advective fluxes implying a diffusion term expressed in terms of the gradient pressure. The scheme is also designed to capture the dynamics of low-Froude-number regimes and offers interesting positivity and well-balancing results. A numerical test is proposed to highlight the scheme's efficiency in the one-layer case.     

A new highly nonlinear shallow water wave equation

We derive a quasilinear shallow water equation directly from the governing equations for gravity water waves within a certain regime for large-amplitude waves which has not been studied so far. Furthermore, we demonstrate local well-posedness of the corresponding Cauchy problem and finally discuss some aspects of the blowup behavior of solutions.     

Space–time discontinuous Galerkin finite element method for shallow water flows

A space–time discontinuous Galerkin (DG) finite element method is presented for the shallow water equations over varying bottom topography. The method results in nonlinear equations per element, which are solved locally by establishing the element communication with a numerical HLLC flux. To deal with spurious oscillations around discontinuities, we employ a dissipation operator only around discontinuities using Krivodonova's discontinuity detector. The numerical scheme is verified by comparing numerical and exact solutions, and validated against a laboratory experiment involving flow through a contraction. We conclude that the method is second order accurate in both space and time for linear polynomials.     

Three‐layer flows in the shallow water limit

We formulate and discuss the shallow water limit dynamics of the layered flow with three layers of immiscible fluids of different densities bounded above and below by horizontal walls. We obtain a resulting system of four equations, which may be nonlocal in the non‐Boussinesq case. We provide a systematic way to pass to the Boussinesq limit, and then study those equations, which are first‐order PDEs of mixed type, more carefully. We show that in a symmetric case the solutions remain on an invariant surface and using simple waves we illustrate that this is not the case for nonsymmetric cases. Reduced models consisting of systems of two equations are also proposed and compared to the full system.     

A new averaging scheme for the riemann problem in pure water

The numerical investigation of shock phenomena in gas or liquid media where enthalpy is the preferred thermodynamic variable poses special problems. When an expression for internal energy is available, the usual procedure is to employ a splitting scheme to remove source terms from the Euler equations, then upwind-biased shock capturing algorithms are built around the Riemann problem for the conservative system which remains. However, when the governing equations arc formulated in terms of total enthalpy, treatment of a pressure time derivative as a source term leads to a Riemann problem for a system where one equation is not a conservation law. The present research establishes that successful upwind-biased shock capturing schemes can be based upon the pseudo-conservative system. A new averaging scheme for solving the associated Riemann problem is developed. The method is applied to numerical simulations of shock wave propagation in pure water.     

On unconditionally positivity preserving DG schemes for shallow water flows with shock capturing by adaptive filtering procedures

In this work, we present an unconditionally positivity preserving implicit time integration scheme for the DG method applied to shallow water flows. For locally refined grids with very small elements, the ODE resulting from space discretization is stiff and requires implicit or partially implicit time stepping. However, for simulations including wetting and drying or regions with small water height, severe time step restrictions have to be imposed due to positivity preservation. Nevertheless, as implicit time stepping demands a significant amount of computational time in order to solve a large system of nonlinear equations for each time step we need large time steps to obtain an efficient scheme. In this context, we propose a novel approach to the strategy of positivity preservation. This new technique is based on the so-called Patankar trick and guarantees non-negativity of the water height for any time step size while still preserving conservativity. Due to this modification, the implicit scheme can take full advantage of larger time steps and is therefore able to beat explicit time stepping in terms of CPU time. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)