Numerical simulation of two-dimensional sine-Gordon solitons via a split cosine scheme

This paper proposes a split cosine scheme for simulating solitary solutions of the sine-Gordon equation in two dimensions, as it arises, for instance, in rectangular large-area Josephson junctions. The dispersive nonlinear partial differential equation allows for soliton-type solutions, a ubiquitous phenomenon in a large variety of physical problems. The semidiscretization approach first leads to a system of second-order nonlinear ordinary differential equations. The system is then approximated by a nonlinear recurrence relation which involves a cosine function. The numerical solution of the system is obtained via a further application of a sequential splitting in a linearly implicit manner that avoids solving the nonlinear scheme at each time step and allows an efficient implementation of the simulation in a locally one-dimensional fashion. The new method has potential applications in further multi-dimensional nonlinear wave simulations. Rigorous analysis is given for the numerical stability. Numerical demonstrations for colliding circular solitons are given.     

Numerical simulation of two-dimensional sine-Gordon solitons via a local weak meshless technique based on the radial point interpolation method (RPIM)

In this paper the meshless local radial point interpolation method (LRPIM) is adopted to simulate the two-dimensional nonlinear sine-Gordon (S-G) equation. The meshless LRPIM is one of the “truly meshless” methods since it does not require any background integration cells. In this case, all integrations are carried out locally over small quadrature domains of regular shapes, such as circles or squares in two dimensions and spheres or cubes in three dimensions. A technique is proposed to construct shape functions using radial basis functions. These shape functions which are constructed by point interpolation method using the radial basis functions have delta function property. The time derivatives are approximated by the time-stepping method. In order to eliminate the nonlinearity, a simple predictor-corrector scheme is performed. Numerical results are obtained for various cases involving line and ring solitons. Also the conservation of energy in undamped sine-Gordon equation is investigated.     

Solving 2D-wave problems by the iterative differential quadrature method

In this paper, the numerical stability of an iterative method based on differential quadrature (DQ) rules when applied to solve a two-dimensional (2D) wave problem is discussed. The physical model of a vibrating membrane, with different initial conditions, is considered. The stability analysis is performed by the matrix method generalized for a 2D space-time domain. This method was presented few years ago by the same author as an analytical support to check the stability of the iterative differential quadrature method in 1D space-time domains. The stability analysis confirms here the conditionally stable nature of the method. The accuracy of the solution is discussed too.     

Numerical simulation of the N-dimensional sine-Gordon equation via operational matrices

In this paper, we develop a numerical method for the N-dimensional sine-Gordon equation using differentiation matrices, in the theoretical frame of matrix differential equations.Our method avoids calculating exponential matrices, is very intuitive and easy to express, and can be implemented without toil in any number of spatial dimensions. Although there is currently a vast literature on the numerical treatment of the one-dimensional sine-Gordon equation, the references for the two-dimensional case are much sparser, and virtually nonexistent for higher dimensions.We apply it to a battery of two-dimensional problems taken from the literature, showing that it largely outperforms the previously existing algorithms; while for three-dimensional problems, the results seem very promising.     

Solution of magnetohydrodynamic flow in a rectangular duct by differential quadrature method

The polynomial based differential quadrature and the Fourier expansion based differential quadrature method are applied to solve magnetohydrodynamic (MHD) flow equations in a rectangular duct in the presence of a transverse external oblique magnetic field. Numerical solution for velocity and induced magnetic field is obtained for the steady-state, fully developed, incompressible flow of a conducting fluid inside of the duct. Equal and unequal grid point discretizations are both used in the domain and it is found that the polynomial based differential quadrature method with a reasonable number of unequally spaced grid points gives accurate numerical solution of the MHD flow problem. Some graphs are presented showing the behaviours of the velocity and the induced magnetic field for several values of Hartmann number, number of grid points and the direction of the applied magnetic field.     

Buckling and post-buckling analysis of extensible beam-columns by using the differential quadrature method

Buckling and post-buckling analysis of extensible beam-columns is performed numerically in this paper. It was experienced earlier that in some cases the numerical integration would not produce the convergent post-buckling solutions, especially under high loads. Therefore, a new differential quadrature (DQ) based iterative numerical integration method is proposed to solve post-buckling differential equations of extensible beam-columns. Six cases, including five classical Euler buckling cases, are analyzed. Critical loads and convergent post-buckling solutions under different applied loads are obtained. The results are compared with the existing multiple scales solutions. It is found that under high applied loads, the small rotation assumption in obtaining multiple scales solutions is no longer valid. The proposed iterative DQ based numerical integration method can yield reliable and accurate post-buckling solutions even at high applied loads for the cases investigated.     

Solution of initial value problems by the differential quadrature method with Hermite bases

The differential quadrature method (DQM) is used to solve the first-order initial value problem. The initial condition is given at the beginning of the interval. The derivative of a space-independent variable at a sampling grid point within the interval can be defined as a weighted linear sum of the given initial conditions and the function values at the sampling grid points within the defined interval. Hermite polynomials have advantages compared with Lagrange and Chebyshev polynomials, and so, unlike other work, they are chosen as weight functions in the DQM. The proposed method is applied to a numerical example and it is shown that the accuracy of the quadrature solution obtained using the proposed sampling grid points is better than solutions obtained with the commonly used Chebyshev–Gauss–Lobatto sampling grid points.     

Numerical study of the one-dimensional coupled nonlinear sine-Gordon equations by a novel geometric meshless method

Engineering with Computers - In the paper, we derive a geometric meshless method for coupled nonlinear sine-Gordon (CNSG) equations. Approximate solutions of the CNSG equations are supposed to be...     

用微分求积法分析输液管道的非线性动力学行为

将微分求积法(Differential Quadrature Method,简称DQM)应用于输液管道的非线性动力学分析,采用此法研究了受非线性约束输液管道的分岔现象和混沌运动问题.从悬臂输液管道模型出发,利用微分求积法形成管道的动力学方程.以分岔图、相平面图、时间历程图和Poincare映射等分析手段考察了系统参数(管内流速)变化对管道振动形态的影响.结果表明,在所研究的系统中存在出现倍周期分岔现象和混沌运动的参数区域,这与前人的研究成果具有一致性.这为一类结构的非线性动力响应问题提供了一种新的研究思路.     

Numerical simulation of two-dimensional plasma flow in channels

Axisymmetric plasma flow in a channel between two coaxial electrodes is considered. The problem is described by magnetohydrodynamics (MHD) equations taking into account the Hall effect. Two-dimensional time-dependent equations are solved numerically. As a rule solutions stabilize and become steady-state. This paper contains the mathematical statement of the problem and the method of its numerical solution. This kind of technique may be applied to a large class of two-dimensional plasma flows across the magnetic field. Physical results obtained in computation have been published in the papers [1]–[12].     

Numerical solution of fuzzy differential equations by predictor-corrector method

In this paper three numerical methods to solve “The fuzzy ordinary differential equations” are discussed. These methods are Adams-Bashforth, Adams-Moulton and predictor-corrector. Predictor-corrector is obtained by combining Adams-Bashforth and Adams-Moulton methods. Convergence and stability of the proposed methods are also proved in detail. In addition, these methods are illustrated by solving two fuzzy Cauchy problems.     

Stability and accuracy of differential quadrature method in solving dynamic problems

In this paper, the differential quadrature method is used to solve dynamic problems governed by second-order ordinary differential equations in time. The Legendre, Radau, Chebyshev, Chebyshev–Gauss–Lobatto and uniformly spaced sampling grid points are considered. Besides, two approaches using the conventional and modified differential quadrature rules to impose the initial conditions are also investigated. The stability and accuracy properties are studied by evaluating the spectral radii and truncation errors of the resultant numerical amplification matrices. It is found that higher-order accurate solutions can be obtained at the end of a time step if the Gauss and Radau sampling grid points are used. However, the conventional approach to impose the initial conditions in general only gives conditionally stable time step integration algorithms. Unconditionally stable algorithms can be obtained if the modified differential quadrature rule is used. Unfortunately, the commonly used Chebyshev–Gauss–Lobatto sampling grid points would not generate unconditionally stable algorithms.     

Numerical simulation of two-dimensional turbulence in a plane channel

Two-dimensional flow of an incompressible viscous fluid in a plane channel is studied under fixed flux and supercritical Reynolds number, Re = 10⁴. For numerical simulation, and original algorithm, possessing good stability and accuracy properties, is used. Calculation of the flow over a large interval of time leads to a statistically steady-state condition of the flow and to stabilization of the averaged characteristics, e.g., profiles of averaged velocity, averaged gradient of pressure, energy, etc. The computations show that qualitatively proper characteristics of “two-dimensional” turbulence can be obtained by numerical simulation of the Navier-Stokes equations.     

A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions

The nonlinear sine-Gordon equation arises in various problems in science and engineering. In this paper, we propose a numerical scheme to solve the two-dimensional damped/undamped sine-Gordon equation. The proposed scheme is based on using collocation points and approximating the solution employing the thin plate splines (TPS) radial basis function (RBF). The new scheme works in a similar fashion as finite difference methods. Numerical results are obtained for various cases involving line and ring solitons.     

Application of differential quadrature method to the delamination buckling of composite plates

The delamination buckling response of a composite panel containing through-the-width delamination is numerically modeled using a solution that is based on the differential quadrature method (DQM). The composite is modeled as a general one-dimensional beam–plate having a through-the -width delamination that can be at any arbitrary location through its thickness; hence, dividing the domain into four regions. The DQM is applied to each region and with the imposition of appropriate boundary conditions, the problem is transformed into a standard eigenvalue problem. Numerical results are presented, illustrating the stability and validity of the method. The results also demonstrate the efficiency of the method in treating this class of engineering problem.     

Numerical simulation of the MHD equations by a kinetic-type method

We introduce a flux-splitting formula for the approximation of the ideal MHD equations in conservative form. The Faraday equation is considered as the average of an abstract kinetic equation, giving a flux-splitting formula. For the other part of the equations, we generalize formally the classical half-Maxwellian flux-splitting of the Euler equations. Numerical results on MHD shock tube problems are displayed.     

Analysis of flexural vibration band gaps in periodic beams using differential quadrature method

To determine the flexural vibration band gaps in periodic beams, the theoretical equations are derived by employing the Bloch–Floquet theorem and then solved by the use of the differential quadrature method. Moreover, a comprehensive parametric study is also conducted to highlight the influences of shear deformation, geometrical parameters and material parameters on the gaps. The results show that the method is efficient and accurate and that the bandwidth can be enlarged by changing the geometrical or material parameters. The existence of vibration gaps lends a new insight into vibration isolation applications in areas such as mechanical and civil engineering.     

Solitary wave simulations of Complex Modified Korteweg-de Vries Equation using differential quadrature method

Complex Modified Korteweg-deVries Equation is solved numerically using differential quadrature method based on cosine expansion. Three test problems, motion of single solitary wave, interaction of solitary waves and wave generation, are simulated. The accuracy of the method is measured via the discrete root mean square error norm L₂, maximum error norm L_∞ for the motion of single solitary wave since it has an analytical solution. A rate of convergency analysis for motion of single solitary wave containing both real and imaginary parts is also given. Lowest three conserved quantities are computed for all test problems. A comparison with some earlier works is given.     

Numerical simulation of forest fire propagation based on modified two-dimensional model

A modified two-dimensional two-phase mathematical model of forest wildfires propagation is considered. The model is based on the averaging of three-dimensional equations of two-phase medium over the height of the forest fuel (FF) layer and it includes the (k?ε)-turbulence model with additional turbulence production and dissipation terms in the forest layer and the Eddy Break-up Model for the combustion rate in the gas phase. The developed model can be used to carry out numerical simulation of the forest fire-front propagation under the conditions of a heterogeneous FF distribution, the presence of obstacles to the fire propagation, and the wind effects. This model can be used for real-time computation of the fire propagation, for expert assessments of emergency situations, and for assessments of the damage caused by forest fires.