Seismic-generated unsteady motions in shallow basins and channels. Part II: Numerical modelling

In this paper, unsteady motions generated by seismic-type excitation are simulated by a 2D depth-averaged mathematical model based on the classic shallow water approximation. A suitable time-dependent forcing term is added in the governing equations, and these are solved by a MUSCL-type shock-capturing finite volume scheme with a splitting treatment of the source term. The HLL approximate Riemann solver is used to estimate the numerical fluxes. The accuracy of the numerical scheme is assessed by comparison with novel exact solutions of test cases concerning sinusoidally-generated sloshing in a prismatic tank, a rectangular open channel, and a parabolic basin. A sensitivity analysis is performed on the influence of the relevant dimensionless parameters. Moreover, numerical results are validated against experimental data available in literature concerning shallow water sloshing in a swaying tank. Finally, real‐scale applications to a reservoir created by a dam and an urban water-supply storage tank are presented. The results show that the model provides accurate solutions of the shallow water equations with a seismic-type source term and can be effectively adopted to predict the main flow features of the unsteady motion induced by horizontal seismic acceleration when the long wave assumption is valid.     

Controlling unpredictability in the randomly driven Hénon–Heiles system

Noisy scattering dynamics in the randomly driven Hénon–Heiles system is investigated in the range of initial energies where the motion is unbounded. In this paper we study, with the help of the exit basins and the escape time distributions, how an external perturbation, be it dissipation or periodic forcing with a random phase, can enhance or mitigate the unpredictability of a system that exhibit chaotic scattering. In fact, if basin boundaries have the Wada property, predictability becomes very complicated, since the basin boundaries start to intermingle, what means that there are points of different basins close to each other. The main responsible of this unpredictability is the external forcing with random phase, while the dissipation can recompose the basin boundaries and turn the system more predictable. Therefore, we do the necessary simulations to find out the values of dissipation and external forcing for which the exit basins present the Wada property. Through these numerical simulations, we show that the presence of the Wada basins have a specific relation with the damping, the forcing amplitude and the energy value. Our approach consists on investigating the dynamics of the system in order to gain knowledge able to control the unpredictability due to the Wada basins.     

Two-dimensional numerical modelling of shallow water flows over multilayer movable beds

The two-dimensional modelling of shallow water flows over multi-sediment erodible beds is presented. A novel approach is developed for the treatment of multiple sediment types in morphodynamics. The governing equations include the two-dimensional shallow water equations for hydrodynamics, an Exner-type equation for morphodynamics, a two-dimensional transport equation for the suspended sediments, and a set of empirical equations for entrainment and deposition. Multilayer sedimentary beds are formed of different erodible soils with sediment properties and new exchange conditions between the bed layers are developed for the model. The coupled equations yield a hyperbolic system of balance laws with source terms. As a numerical solver for the system, we implement a fast finite volume characteristics method. The numerical fluxes are reconstructed using the method of characteristics which employs projection techniques. The proposed finite volume solver is simple to implement, satisfies the conservation property and can be used for two-dimensional sediment transport problems in non-homogeneous isotropic beds without need of complicated three-dimensional equations. To assess the performance of the proposed models, we present numerical results for a wide variety of shallow water flows over sedimentary layers. Comparisons to experimental data for dam-break problems over movable beds are also included in this study.     

Properties and exact solutions of the equations of motion of shallow water in a spinning paraboloid

A transformation is found and, using this, the non-linear system of equations describing the spatial oscillations of a thin layer of liquid in a spinning circular parabolic basin is reduced to the conventional equations of the model of shallow water over a level fixed bottom. This transformation is obtained by analyzing the properties of the symmetry of the equations of motion of spinning shallow water. The existence of non-trivial symmetries in the case of the model considered enabled group multiplication of the solutions to be carried out. Using the known steady-state rotationally symmetric solution, a class of time-periodic solutions is obtained that describes the non-linear oscillations of the liquid in a circular paraboloid with closed or quasiclosed (ergodic) trajectories of the motion of the liquid particles.     

Well-balanced unstaggered central schemes for one and two-dimensional shallow water equation systems

We propose a new well-balanced unstaggered central finite volume scheme for hyperbolic balance laws with geometrical source terms. In particular we construct a new one and two-dimensional finite volume method for the numerical solution of shallow water equations on flat/variable bottom topographies. The proposed scheme evolves a non-oscillatory numerical solution on a single grid, avoids the time consuming process of solving Riemann problems arising at the cell interfaces, and is second-order accurate both in space and time. Furthermore, the numerical scheme follows a well-balanced discretization that first discretizes the geometrical source term according to the discretization of the flux terms, and then mimics the surface gradient method and discretizes the water height according to the discretization of the water level. The resulting scheme exactly satisfies the C-property at the discrete level. The proposed scheme is then applied and classical one and two-dimensional shallow water equation problems with flat or variable bottom topographies are successfully solved. The obtained numerical results are in good agreement with corresponding ones appearing in the recent literature, thus confirming the potential and efficiency of the proposed method.     

DISCONTINUOUS FORCING OF PERIODIC SOLUTIONS IN C1 VECTOR FIELDS WITH APPLICATIONS TO POPULATION MODELS

Averaging methods are used to compare solutions of two-dimensional systems of ordinary differential equations with constant or periodic forcing. The asymptotic separation of solutions of the periodically forced equations from the solutions of the constantly forced equations is proportional to the L¹ norm of the periodic forcing terms. This result is applied to population models of Kolmogorov-type with climax fitness functions where forcing represents stocking or harvesting of a population. The asymptotic behavior of such systems may be controlled, to some extent, by varying the period and/or amplitude of the forcing functions.     

Homogenization Method in the Problem of Long Wave Propagation from a Localized Source in a Basin over an Uneven Bottom

In the framework of the linearized shallow water equations, the homogenization method for wave type equations with rapidly oscillating coefficients that generally cannot be represented as periodic functions of the fast variables is applied to the Cauchy problem for the wave equation describing the evolution of the free surface elevation for long waves propagating in a basin over an uneven bottom. Under certain conditions on the function describing the basin depth, we prove that the solution of the homogenized equation asymptotically approximates the solution of the original equation. Model homogenized wave equations are constructed for several examples of one-dimensional sections of the real ocean bottom profile, and their numerical and asymptotic solutions are compared with numerical solutions of the original equations.     

Bifurcations of basins of attraction from the view point of prime ends

In dynamical systems examples are common in which two or more attractors coexist, and in such cases the basin boundary is nonempty and the basins often have fractal basin boundaries. The purpose of this paper is to describe the structure and properties of unbounded basins and their boundaries for two-dimensional diffeomorphisms. Frequently, if not always, there is a periodic saddle on the boundary that is accessible from the basin. Carathéodory and many others developed an approach in which an open set (in our case a basin) is compactified using so-called prime end theory. Under the prime end compactification of the basin, boundary points of the basin (prime ends) can be characterized as either type 1, 2, 3, or 4. In all well-known examples, most points are of type 1. Many two-dimensional basins have a basin cell, that is, a trapping region whose boundary consists of pieces of the stable and unstable manifolds of a well chosen periodic orbit. Then the basin consists of a central body (the basin cell) and a finite number of channels attached to it, and the basin boundary is fractal. We present a result that says {a basin has a basin cell} if and only if {every prime end that is defined by a chain of unbounded regions (in the basin) is a prime end of type 3 and furthermore all other prime ends are of type 1}. We also prove as a parameter is varied, the basin cell for a basin B is created (or destroyed) if and only if either there is a saddle node bifurcation or the basin B has a prime end that is defined by a chain of unbounded regions and is a prime end of either type 2 or type 4.     

Self-similar solutions and blow-up phenomena for a two-component shallow water system

In this article, we consider a two-component nonlinear shallow water system, which includes the famous 2-component Camassa-Holm and Degasperis-Procesi equations as special cases. The local well-posedess for this equations is established. Some sufficient conditions for blow-up of the solutions in finite time are given. Moreover, by separation method, the self-similar solutions for the nonlinear shallow water equations are obtained, and which local or global behavior can be determined by the corresponding Emden equation.     

A Conservative Formulation and a Numerical Algorithm for the Double-Gyre Nonlinear Shallow-Water Model

We present a conservative formulation and a numerical algorithm for the reduced-gravity shallow-water equations on a beta plane, subjected to a constant wind forcing that leads to the formation of double-gyre circulation in a closed ocean basin. The novelty of the paper is that we reformulate the governing equations into a nonlinear hyperbolic conservation law plus source terms. A second-order fractional-step algorithm is used to solve the reformulated equations. In the first step of the fractional-step algorithm, we solve the homogeneous hyperbolic shallow-water equations by the wave-propagation finite volume method. The resulting intermediate solution is then used as the initial condition for the initial-boundary value problem in the second step. As a result, the proposed method is not sensitive to the choice of viscosity and gives high-resolution results for coarse grids, as long as the Rossby deformation radius is resolved. We discuss the boundary conditions in each step, when no-slip boundary conditions are imposed to the problem. We validate the algorithm by a periodic flow on an $f$-plane with exact solutions. The order-of-accuracy for the proposed algorithm is tested numerically. We illustrate a quasi-steady-state solution of the double-gyre model via the height anomaly and the contour of stream function for the formation of double-gyre circulation in a closed basin. Our calculations are highly consistent with the results reported in the literature. Finally, we present an application, in which the double-gyre model is coupled with the advection equation for modeling transport of a pollutant in a closed ocean basin.     

Quasi-periodic solutions and periodic bursters in quasiperiodically driven oscillators

In this paper, we propose a perturbation method to determine an approximation and conditions of existence of quasi-periodic (QP) solutions and bursting dynamics in a quasi-periodically driven system. The QP forcing consists of two periodic excitations, one with a very slow frequency and the other with a frequency of the same order of the proper frequency of the oscillator. A first averaging is done over the fast dynamics, then the quasi-static solutions of the modulation equations of amplitude and phase are determined and their stability analyzed. We show that a necessary condition for the occurrence of periodic bursters is that the slow excitation is parametric.     

Approximate analytical solution for supercritical flow in rectangular curved channels

In this study, we present an asymptotical mathematical model and an analytical solution for a supercritical flow in curved rectangular open channels. An original approach is proposed for solving the free-surface configuration and features of the flow in the presence of cross shock waves. The two-dimensional steady depth-averaged shallow water equations are transformed into an equivalent one-dimensional (1D) unsteady flow problem and a first order approximation is then obtained using small perturbation theory. Furthermore, the 1D asymptotic model is solved analytically by Laplace integral transformation and the two-dimensional flow field solution is reconstructed according to the translating planes. The free-surface profile along the outer chute wall and downstream channel was compared with the available experimental data, and the results indicated the satisfactory agreement of the maximum flow depth, peak positions, and wavelength. The proposed approach provides accurate predictions of the flow features and it facilitates the safe design of curved channel transitions.     

A new composite scheme for two-layer shallow water flows with shocks

This paper is devoted to solve the system of partial differential equations governing the flow of two superposed immiscible layers of shallow water flows. The system contains source terms due to bottom topography, wind stresses, and nonconservative products describing momentum exchange between the layers. The presence of these terms in the flow model forms a nonconservative system which is only conditionally hyperbolic. In addition, two-layer shallow water flows are often accompanied with moving discontinuities and shocks. Developing stable numerical methods for this class of problems presents a challenge in the field of computational hydraulics. To overcome these difficulties, a new composite scheme is proposed. The scheme consists of a time-splitting operator where in the first step the homogeneous system of the governing equations is solved using an approximate Riemann solver. In the second step a finite volume method is used to update the solution. To remove the non-physical oscillations in the vicinity of shocks a nonlinear filter is applied. The method is well-balanced, non-oscillatory and it is suitable for both low and high values of the density ratio between the two layers. Several standard test examples for two-layer shallow water flows are used to verify high accuracy and good resolution properties for smooth and discontinuous solutions.     

A discontinuous Galerkin method for a new class of Green–Naghdi equations on simplicial unstructured meshes

In this paper, we introduce a discontinuous Finite Element formulation on simplicial unstructured meshes for the study of free surface flows based on the fully nonlinear and weakly dispersive Green-Naghdi equations. Working with a new class of asymptotically equivalent equations, which have a simplified analytical structure, we consider a decoupling strategy: we approximate the solutions of the classical shallow water equations supplemented with a source term globally accounting for the non-hydrostatic effects and we show that this source term can be computed through the resolution of scalar elliptic second-order sub-problems. The assets of the proposed discrete formulation are: (i) the handling of arbitrary unstructured simplicial meshes, (ii) an arbitrary order of approximation in space, (iii) the exact preservation of the motionless steady states, (iv) the preservation of the water height positivity, (v) a simple way to enhance any numerical code based on the nonlinear shallow water equations. The resulting numerical model is validated through several benchmarks involving nonlinear wave transformations and run-up over complex topographies.     

有限体积KFVS方法在二维溃坝中的应用

本文采用了基于KFVS格式的有限体积方法 (FVM)求解了控制水流运动的二维浅水方程 ,建立了二维水坝瞬间溃坝的洪水演进模型 .并应用此模型模拟了二维非对称溃坝和对称溃坝情形下坝左下角有障碍物时的洪水波演进过程 .模拟结果表明该数学模型对二维浅水运动的模拟很有效 .     

Shallow water equations: viscous solutions and inviscid limit

We establish the inviscid limit of the viscous shallow water equations to the Saint-Venant system. For the viscous equations, the viscosity terms are more degenerate when the shallow water is close to the bottom, in comparison with the classical Navier-Stokes equations for barotropic gases; thus, the analysis in our earlier work for the classical Navier-Stokes equations does not apply directly, which require new estimates to deal with the additional degeneracy. We first introduce a notion of entropy solutions to the viscous shallow water equations and develop an approach to establish the global existence of such solutions and their uniform energy-type estimates with respect to the viscosity coefficient. These uniform estimates yield the existence of measure-valued solutions to the Saint-Venant system generated by the viscous solutions. Based on the uniform energy-type estimates and the features of the Saint-Venant system, we further establish that the entropy dissipation measures of the viscous solutions for weak entropy-entropy flux pairs, generated by compactly supported C ² test-functions, are confined in a compact set in H ^?1, which yields that the measure-valued solutions are confined by the Tartar-Murat commutator relation. Then, the reduction theorem established in Chen and Perepelitsa [5] for the measure-valued solutions with unbounded support leads to the convergence of the viscous solutions to a finite-energy entropy solution of the Saint-Venant system with finite-energy initial data, which is relative with respect to the different end-states of the bottom topography of the shallow water at infinity. The analysis also applies to the inviscid limit problem for the Saint-Venant system in the presence of friction.     

Symmetries of shallow water equations on a rotating plane

We consider a system of nonlinear differential equations which describes the spatial motion of an ideal incompressible fluid on a rotating plane in the shallow water approximation and a more general system of the theory of long waves which takes into account the specifics of shear flows. Using the group analysis methods, we calculate the 9-dimensional Lie algebras of infinitesimal operators admissible by the models. We establish an isomorphism of these Lie algebras with a known Lie algebra of operators admissible by the system of equations for the two-dimensional isentropic motions of a polytropic gas with the adiabatic exponent γ = 2. The nontrivial symmetries of the models under consideration enable us to carry out the group generation of the solutions. The class of stationary solutions to the equations of rotating shallow water transforms into a new class of periodic solutions.     

带有集中激励的一类非线性边值问题的奇摄动

本文综合利用近代分析和奇异摄动方法，讨论了一类具有集中激励的非线性向量微分方程组初值总是 的渐近性质，构造了这类问题的一致有效渐近解。并利用这个方法求解几何参数的扁球壳在集中力作用下的非线性稳定性问题，得到了较好的结果。     

非线性演化方程显式精确解的新算法

本文给出了一种求解非线性演化方程的新算法 .将这种算法运用于变形浅水波方程 ,获得了八组显式精确解 ,其中包括新的孤波解和周期解 .借助于 Mathematica软件 ,这种算法能够在 Computer上实现 .     

A class of quasilinear degenerate elliptic problems

We establish the existence of solutions for a class of quasilinear degenerate elliptic equations. The equations in this class satisfy a structure condition which provides ellipticity in the interior of the domain, and degeneracy only on the boundary. Equations of transonic gas dynamics, for example, satisfy this property in the region of subsonic flow and are degenerate across the sonic surface. We prove that the solution is smooth in the interior of the domain but may exhibit singular behavior at the degenerate boundary. The maximal rate of blow-up at the degenerate boundary is bounded by the “degree of degeneracy” in the principal coefficients of the quasilinear elliptic operator. Our methods and results apply to the problems recently studied by several authors which include the unsteady transonic small disturbance equation, the pressure-gradient equations of the compressible Euler equations, and the singular quasilinear anisotropic elliptic problems, and extend to the class of equations which satisfy the structure condition, such as the shallow water equation, compressible isentropic two-dimensional Euler equations, and general two-dimensional nonlinear wave equations. Our study provides a general framework to analyze degenerate elliptic problems arising in the self-similar reduction of a broad class of two-dimensional Cauchy problems.