Cauchy problem for viscous rotating shallow water equations

We consider the Cauchy problem for a viscous compressible rotating shallow water system with a third-order surface-tension term involved, derived recently in the modeling of motions for shallow water with free surface in a rotating sub-domain Marche (2007) [19]. The global existence of the solution in the space of Besov type is shown for initial data close to a constant equilibrium state away from the vacuum. Unlike the previous analysis about the compressible fluid model without Coriolis forces, see for instance Danchin (2000) [10], Haspot (2009) [16], the rotating effect causes a coupling between two parts of Hodge's decomposition of the velocity vector field, and additional regularity is required in order to carry out the Friedrichs' regularization and compactness arguments.     

Symmetries of the Maxwell-Bloch equations with the rotating wave approximation

In this paper a symplectic realization for the Maxwell-Bloch equations with the rotating wave approximation is given, which also leads to a Lagrangian formulation. We show how Lie point symmetries generate a third constant of motion for the dynamical system considered.     

A parallel time integrator for solving the linearized shallow water equations on the rotating sphere

With the stagnation of processor core performance, further reductions in the time to solution for geophysical fluid problems are becoming increasingly difficult with standard time integrators. Parallel‐in‐time exposes and exploits additional parallelism in the time dimension, which is inherently sequential in traditional methods. The rational approximation of exponential integrators (REXI) method allows taking arbitrarily long time steps based on a sum over a number of decoupled complex PDEs that can be solved independently massively parallel. Hence, REXI is assumed to be well suited for modern massively parallel super computers, which are currently trending. To date, the study and development of the REXI approach have been limited to linearized problems on the periodic two‐dimensional plane. This work extends the REXI time stepping method to the linear shallow‐water equations on the rotating sphere, thus moving the method one step closer to solving fully nonlinear fluid problems of geophysical interest on the sphere. The rotating sphere poses particular challenges for finding an efficient solver due to the zonal dependence of the Coriolis term. Here, we present an efficient REXI solver based on spherical harmonics, showing the results of a geostrophic balance test, a comparison with alternative time stepping methods, an analysis of dispersion relations indicating superior properties of REXI, and finally, a performance comparison on the Cheyenne supercomputer. Our results indicate that REXI not only can take larger time steps but also can be used to gain higher accuracy and significantly reduced time to solution compared with currently existing time stepping methods.     

Symmetries of plane partitions

We introduce a new symmetry operation, called complementation, on plane partitions whose three-dimensional diagram is contained in a given box. This operation was suggested by work of Mills, Robbins, and Rumsey. There then arise a total of ten inequivalent problems concerned with the enumeration of plane partitions with a given symmetry. Four of these ten problems had been previously considered. We survey what is known about the ten problems and give a solution to one of them. The proof is based on the theory of Schur functions, in particular the Littlewood-Richardson rule. Of the ten problems, seven are now solved while the remaining three have conjectured simple solutions.     

On a numerical flux for the shallow water equations

The finite volume scheme of Vijayasundaram and Osher-Solomon type for shallow water equations are proposed. The numerical results with discontinuous initial condition and the comparison with Lax-Friedrichs numerical flux are presented for homogeneous case. The extension of the method for the inhomogeneous case is described.     

A characteristic-Galerkin approximation to a system of shallow water equations

Summary. Characteristic methods are known to handle advective flow better than traditional Galerkin methods and allow large time steps to be taken when compared to standard time-stepping methods. In this paper, we investigate a characteristic-Galerkin approximation to the 2-dimensional system of shallow water equations. We derive bounds for elevation and velocity, showing these to be optimal for velocity in . Received October 15, 1998 / Revised version received March 13, 1999 / Published online April 20, 2000     

Well-posedness for the fifth-order shallow water equations

The Cauchy problems for some kind of fifth-order shallow water equations

     

On the dynamics of Navier-Stokes equations for a shallow water model

In this paper, we study a free boundary problem of one-dimensional compressible Navier-Stokes equations with a density-dependent viscosity, which include, in particular, a shallow water model. Under some suitable assumptions on the initial data, we obtain the global existence, uniqueness and the large time behavior of weak solutions. In particular, it is shown that a stationary wave pattern connecting a gas to the vacuum continuously is asymptotically stable for small initial general perturbations.     

Evolution of weak discontinuities in shallow water equations

In this paper, we determine the critical time, when a weak discontinuity in the shallow water equations culminates into a bore. Invariance group properties of the governing system of partial differential equations (PDEs), admitting Lie group of point transformations with commuting infinitesimal operators, are presented. Some appropriate canonical variables are characterized that transform equations at hand to an equivalent form, which admits non-constant solutions. The propagation of weak discontinuities is studied in the medium characterized by the particular solution of the governing system.     

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.     

Structure preserving model order reduction of shallow water equations

In this paper, we present two different approaches for constructing reduced‐order models (ROMs) for the two‐dimensional shallow water equation (SWE). The first one is based on the noncanonical Hamiltonian/Poisson form of the SWE. After integration in time by the fully implicit average vector field method, ROMs are constructed with proper orthogonal decomposition(POD)/discrete empirical interpolation method that preserves the Hamiltonian structure. In the second approach, the SWE as a partial differential equation with quadratic nonlinearity is integrated in time by the linearly implicit Kahan's method, and ROMs are constructed with the tensorial POD that preserves the linear‐quadratic structure of the SWE. We show that in both approaches, the invariants of the SWE such as the energy, enstrophy, mass and circulation are preserved over a long period of time, leading to stable solutions. We conclude by demonstrating the accuracy and the computational efficiency of the reduced solutions by a numerical test problem.     

Generic singularities in solutions of the shallow water equations

Doklady Mathematics -     

Convergence analysis of a space-time approximation to a two-dimensional system of shallow water equations

A characteristic-Galerkin approximation of a new two-dimensional Shallow Water viscous parabolic model is presented and a semi-implicit-Lagrangian finite element scheme is used. A convergence result is proved and numerical experiments on academic tests are shown.     

Flatness based trajectory planning for the shallow water equations

A flatness based feed-forward control design approach for an open channel flow modelled by the shallow water equations is discussed. The control input consists of the adjustable height of opening of a delimiting sluice gate. In order to compute the trajectory of the control input, the problem is reduced to an initial value problem w.r.t. the spatial coordinate. The solution to this problem can be obtained with the method of characteristics. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Absorbing boundary conditions for the linearized shallow water equations

A mixed problem imitating the Cauchy problem for the linearized shallow water equations is considered. This problem is also a mixed problem with perfectly absorbing conditions (cp. [1], [3]). An exact formula for the conditions has been given.     

On curl-preserving finite volume discretizations for shallow water equations

The preservation of intrinsic or inherent constraints, like divergence-conditions, has gained increasing interest in numerical simulations of various physical evolution equations. In Torrilhon and Fey, SIAM J. Numer. Anal. (42/4) 2004, a general framework is presented how to incorporate the preservation of a discrete constraint into upwind finite volume methods. This paper applies this framework to the wave equation system and the system of shallow water equations. For the wave equation a curl-preservation for the momentum variable is present and easily identified. The preservation in case of the shallow water system is more involved due to the presence of convection. It leads to the vorticity evolution as generalized curl-constraint. The mechanisms of vorticity generation are discussed.For the numerical discretization special curl-preserving flux distributions are discussed and their incorporation into a finite volume scheme described. This leads to numerical discretizations which are exactly curl-preserving for a specific class of discrete curl-operators.The numerical experiments for the wave equation show a significant improvement of the new method against classical schemes. The extension of the curl-free numerical discretization to the shallow water case is possible after isolating the pressure flux. Simulation examples demonstrate the influence of the modification. The vortex structure is more clearly resolved.     

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.