Numerical solution of the ‘classical’ Boussinesq system

We consider the ‘classical’ Boussinesq system of water wave theory, which belongs to the class of Boussinesq systems modelling two-way propagation of long waves of small amplitude on the surface of water in a horizontal channel. (We also consider its completely symmetric analog.) We discretize the initial-boundary-value problem for these systems, corresponding to homogeneous Dirichlet boundary conditions on the velocity variable at the endpoints of a finite interval, using fully discrete Galerkin-finite element methods of high accuracy. We use the numerical schemes as exploratory tools to study the propagation and interactions of solitary-wave solutions of these systems, as well as other properties of their solutions.     

C3: A finite volume-finite difference hybrid model for tsunami propagation and runup

A numerical model that couples Finite Difference and Finite Volume schemes has been developed for tsunami propagation and runup study. An explicit leap-frog scheme and a first order upwind scheme has been considered in the Finite Difference module, while in the Finite Volume scheme a Godunov Type method based on the f-waves approach has been used. The Riemann solver included in the model corresponds to an approximate augmented solver for the Shallow Water Equations (SWE) in the presence of variable bottom surface. With this hybrid model some of the problems inherent to the Godunov type schemes are avoided in the offshore region, while in the coastal area the use of a conservative method ensures the correct computation of the runup and wave breaking. The model has been tested and validated using different problems with a known analytical solution and also with laboratory experiments, considering both non breaking and breaking waves. The results are very satisfactory, showing that the hybrid approach is a useful technique for practical usages.     

Numerical study of the generation and propagation of tsunami waves caused by a running bottom rift

Within shallow water equations, the numerical simulation problem of the generation and propagation of shallow waves caused by a running bottom rupture is considered. The rupture is a trench with a width increasing with time. As a result of fluid drainage into the trench, waves are formed. The basic problem parameters influencing the characteristics of the formed waves are found.     

Numerical results for linear Fredholm integral equations of the first kind over surfaces in three dimensions

Linear Fredholm integral equations of the first kind over surfaces are less familiar than those of the second kind, although they arise in many applications like computer tomography, heat conduction and inverse scattering. This article emphasizes their numerical treatment, since discretization usually leads to ill-conditioned linear systems. Strictly speaking, the matrix is nearly singular and ordinary numerical methods fail. However, there exists a numerical regularization method – the Tikhonov method – to deal with this ill-conditioning and to obtain accurate numerical results.     

A posteriori error analysis of an augmented fully-mixed formulation for the stationary Boussinesq model

In this paper we undertake an a posteriori error analysis along with its adaptive computation of a new augmented fully-mixed finite element method that we have recently proposed to numerically simulate heat driven flows in the Boussinesq approximation setting. Our approach incorporates as additional unknowns a modified pseudostress tensor field and an auxiliary vector field in the fluid and heat equations, respectively, which possibilitates the elimination of the pressure. This unknown, however, can be easily recovered by a postprocessing formula. In turn, redundant Galerkin terms are included into the weak formulation to ensure well-posedness. In this way, the resulting variational formulation is a four-field augmented scheme, whose Galerkin discretization allows a Raviart–Thomas approximation for the auxiliary unknowns and a Lagrange approximation for the velocity and the temperature. In the present work, we propose a reliable and efficient, fully-local and computable, residual-based a posteriori error estimator in two and three dimensions for the aforementioned method. Standard arguments based on duality techniques, stable Helmholtz decompositions, and well-known results from previous works, are the main underlying tools used in our methodology. Several numerical experiments illustrate the properties of the estimator and further validate the expected behavior of the associated adaptive algorithm.     

Adaptive domain decomposition for the bound method: Application to the incompressible Navier-Stokes and Energy equations in three space dimensions

Large-scale numerical simulations of the flow and associated transport phenomena governed by the Navier-Stokes and Energy equations are routinely calculated in engineering practice. Nevertheless, the uncertainty due to spatial discretization limits the confidence of practitioners in numerical solutions. An approach to provide information about the accuracy of the quantity of interest is proposed here-in. The novel a posteriori error estimation technique - the bound method - is based on relaxing Lagrange multipliers that enforces continuity between sub-domains. The method provides fast, efficient, asymptotic but reliable lower and upper bounds to the output of underlying partial differential equations (PDEs). Herein, we highlight the method when applied to outputs of the steady incompressible Navier-Stokes and Energy equations. The bound method in this paper follows the directly equilibrated hybrid-flux approach for the flux calculation between sub-domains and uses the Crouzeix-Raviart () approximation spaces. To improve the effectiveness of the bound method, an adaptive sub-domain refinement strategy leading to sharper bounds is adopted. A convective heat transfer problem in a series of electronic chip devices is investigated. The novelty of this paper is to present bounds using adaptive domain decomposition for outputs associated to a complex three-dimensional field solution of the Navier-Stokes and Energy equations.     

Efficient numerical algorithms for the solution of “good” Boussinesq equation in water wave propagation

This paper proposes three fast and high accuracy numerical methods for solving a nonlinear partial differential equation (PDE) describing water waves and called the Boussinesq (Bq) equation. We numerically solve the Bq equation with fourth-order time-stepping schemes in combination with discrete Fourier transform. We discretize the original PDE with discrete Fourier transform in space and obtain a system of ordinary differential equations (ODEs) which will be solved with fourth-order time-stepping methods. After transforming the equation to a system of ODEs, the linear operator is not diagonal, but we can implement the methods such as diagonal case which reduces the CPU time. Comparing numerical solutions with analytical solutions demonstrates that those methods are accurate and readily implemented. Also we investigate the conservation of mass for Bq equation.     

On the strong stability of finite difference schemes for hyperbolic systems in two space dimensions

We study the stability of some finite difference schemes for symmetric hyperbolic systems in two space dimensions. For the so-called upwind scheme and the Lax–Wendroff scheme with a stabilizer, we show that stability is equivalent to strong stability, meaning that both schemes are either unstable or $\ell ^2$ -decreasing. These results improve on a series of partial results on strong stability. We also show that, for the Lax–Wendroff scheme without stabilizer, strong stability may not occur no matter how small the CFL parameters are chosen. This partially invalidates some of Turkel’s conjectures in Turkel (16(2):109–129, 1977).     

Well-balanced RKDG2 solutions to the shallow water equations over irregular domains with wetting and drying

This paper presents a new one-dimensional (1D) second-order Runge–Kutta discontinuous Galerkin (RKDG2) scheme for shallow flow simulations involving wetting and drying over complex domain topography. The shallow water equations that adopt water level (instead of water depth) as a flow variable are solved by an RKDG2 scheme to give piecewise linear approximate solutions, which are locally defined by an average coefficient and a slope coefficient. A wetting and drying technique proposed originally for a finite volume MUSCL scheme is revised and implemented in the RKDG2 solver. Extra numerical enhancements are proposed to amend the local coefficients associated with water level and bed elevation in order to maintain the well-balanced property of the RKDG2 scheme for applications with wetting and drying. Friction source terms are included and evaluated using splitting implicit discretization, implemented with a physical stopping condition to ensure stability. Several steady and unsteady benchmark tests with/without friction effects are considered to demonstrate the performance of the present model.     

A complex variable boundary element method for elliptic partial differential equations in a multiple-connected region

A simple boundary element method based on the Cauchy integral formulae is proposed for the numerical solution of a class of boundary value problems involving a system of elliptic partial differential equations in a multiple-connected region of infinite extent. It can be easily and efficiently implemented on the computer.     

Application of a fractional step method for the numerical solution of the shallow water waves in a rotating rectangular basin

A numerical method is applied to the problem of an incompressible fluid in a slowly rotating rectangular basin for the simulation of wave propagation in shallow water. The present work is a complete study of the wave motion through evaluation of the wave height and the velocity components. The results are found by the application of a fractional step method and illustrated graphically. The technique is applied by splitting the shallow water equations and successive integration in every direction along the characteristics using the Riemann invariants associated with cubic spline interpolation. It has the advantage of reducing the multidimensional matrix inversion problem into an equivalent one-dimensional problem. Numerical results are represented in three dimensions for the velocity components at different times. The distribution of temperature and concentration are also calculated and plotted.     

The PDF shape control of the state variable for a class of stochastic systems

The shape control of probability density function (PDF) is an important subject in stochastic systems. The PDF-shaping control study has ranged from linear systems to non-linear systems. In this paper we present a PDF-shaping control technique which is useful for a class of non-linear stochastic systems. Controlling the PDF shape requires designing a controller to make the state PDF follow the goal PDF; it is actually to determine the gains of the controller. As we know, the stationary PDF of the state variable is equivalent to the solution of the Fokker–Planck–Kolmogorov (FPK) equation arising from the stochastic system driven with Gaussian white noise. After designing the controller, we derive the solution with some parameters to the corresponding FPK equation, and then solve out the parameters in the solution by the linear-least-squares method, therefore obtaining the gains of the controller. Finally, simulation experiments have been carried out to verify the effectiveness of the approach.     

A numerically efficient and stable algorithm for animating water waves

Published online: 24 January 2002     

Study of the LTI relations between the outputs of two coupled Wiener systems and its application to the generation of initial estimates for Wiener-Hammerstein systems

This paper consists of two parts. In the first, more theoretic part, two Wiener systems driven by the same Gaussian noise excitation are considered. For each of these systems, the best linear approximation (BLA) of the output (in mean square sense) is calculated, and the residuals, defined as the difference between the actual output and the linearly simulated output is considered for both outputs. The paper is focused on the study of the linear relations that exist between these residuals. Explicit expressions are given as a function of the dynamic blocks of both systems, generalizing earlier results obtained by Brillinger [Brillinger, D. R. (1977). The identification of a particular nonlinear time series system. Biometrika, 64(3), 509-515] and Billings and Fakhouri [Billings, S. A., & Fakhouri, S. Y. (1982). Identification of systems containing linear dynamic and static nonlinear elements. Automatica, 18(1), 15-26]. Compared to these earlier results, a much wider class of static nonlinear blocks is allowed, and the efficiency of the estimate of the linear approximation between the residuals is considerably improved. In the second, more practical, part of the paper, this new theoretical result is used to generate initial estimates for the transfer function of the dynamic blocks of a Wiener-Hammerstein system. This method is illustrated on experimental data.     

Application of the recursion method in the numerical solution of multidiagonal nonlinear systems

The recursion method of solving banded matrix equations is presented. End conditions are shown to be automatically incorporated in the recursion formulae thus simplifying computer programming. Application of the method to the numerical solution of the nonlinear liquid jet equations by finite differences is given as an example.     

Haze reduction from the visible bands of LANDSAT TM and ETM+ images over a shallow water reef environment

A method for haze reduction in the visible bands of Landsat TM and ETM+ images over a shallow water marine environment is presented in this paper. This method uses the near infrared (NIR) band to estimate the spatial distribution of haze intensity in each visible band through a linear regression model established over deep water areas. As a first order approximation, the signal received at the sensor is assumed to be the arithmetic sum of radiance contributed by haze and the radiance leaving the water surface. Reduction of haze is then carried out by a simple subtraction procedure. Images acquired over the Southern Tip of Palawan, Philippines are used for the experiments. Results show that the method works well for compensating signals contaminated by optically thin haze. Overcorrection occurs when haze is optically thick and geometrically complex. When images are acquired under hazy conditions the method can be applied to drastically improve image interpretability and may also be considered as a necessary pre-processing step for subsequent analyses and information extraction.     

On sparse lu factorization procedures for the solution of parabolic differential equations in three space dimensions

The numerical solution of partial differential equations in 3 dimensions by finite difference methods leads to the problem of solving large order sparse structured linear systems.

In this paper, a factorization procedure in algorithmic form is derived yielding direct and iterative methods of solution of some interesting boundary value problems in physics and engineering.     

Two parallel algorithms for the convex hull problem in a two dimensional space

Two parallel algorithms for determining the convex hull of a set of data points in two dimensional space are presented. Both are suitable for MIMD parallel systems. The first is based on the strategy of divide-and-conquer, in which some simplest convex-hulls are generated first and then the final convex hull of all points is achieved by the processes of merging 2 sub-convex hulls. The second algorithm is by the process of picking up the points that are necessarily in the convex hull and discarding the points that are definitely not in the convex hull. Experimental results on a MIMD parallel system of 4 processors are analysed and presented.     

Infrastructures and tools for multiagent systems for the new generation of distributed systems

In order for multiagent systems to be included in real domains (media and Internet, logistics, e-commerce, and health care), infrastructures and tools for multiagent systems should provide efficiency, scalability, security, management, monitoring, and other features related to building real applications. Thus, infrastructures and tools that support multiagent systems are needed, especially those that promote the adoption of agent-based systems by designers and programmers in both academia and industry. This special issue is a selection of contributions whose preliminary versions were presented at the ITMAS 2010 workshop, which was held in conjunction with the International Conference on Autonomous Agents and Multi-agent Systems.     

Adaptive open loop control for a class of distributed parameter systems

The real time implementation of closed loop control laws obtained by the minimization of a quadratic criterion with finite time intervals is a difficult problem, if we consider computing time and the eventual determination of the state variables. It would seem useful therefore to define, for complex systems such as distributed parameter systems, a control type which unites the advantages of open loop control, with its relative simplicity of implementation, and those of closed loop control, with its ability to reduce perturbations. It is for this reason that we use the principle of adaptive open loop control. In this paper, we describe the principles of such a method and results obtained from one example studied by numerical computation or hybrid simulation in real time.