Numerical simulation of two-dimensional sine-Gordon solitons by differential quadrature method

During the past few decades, the idea of using differential quadrature methods for numerical solutions of partial differential equations (PDEs) has received much attention throughout the scientific community. In this article, we proposed a numerical technique based on polynomial differential quadrature method (PDQM) to find the numerical solutions of two-dimensional sine-Gordon equation with Neumann boundary conditions. The PDQM reduced the problem into a system of second-order linear differential equations. Then, the obtained system is changed into a system of ordinary differential equations and lastly, RK4 method is used to solve the obtained system. Numerical results are obtained for various cases involving line and ring solitons. The numerical results are found to be in good agreement with the exact solutions and the numerical solutions that exist in literature. It is shown that the technique is easy to apply for multidimensional problems.     

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.     

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.     

Differential quadrature solution of nonlinear Klein–Gordon and sine-Gordon equations

Differential quadrature method (DQM) is proposed to solve the one-dimensional quadratic and cubic Klein–Gordon equations, and two-dimensional sine-Gordon equation. We apply DQM in space direction and also blockwise in time direction. Initial and derivative boundary conditions are also approximated by DQM. DQM provides one to obtain numerical results with very good accuracy using considerably small number of grid points. Numerical solutions are obtained by using Gauss–Chebyshev–Lobatto (GCL) grid points in space intervals, and GCL grid points in each equally divided time blocks.     

A modified explicit numerical scheme for the two-dimensional sine-Gordon equation

Abstract

A fourth-order rational approximant to the matrix-exponential term in a three-time-level recurrence relation is used to transform the two-dimensional sine-Gordon equation into a second-order initial-value problem. The resulting nonlinear system is solved using an appropriate predictor–corrector (P-C) scheme in which the predictor is an explicit one of second order. The procedure of the corrector is accelerated by using a modification (MPC) in which the already evaluated values are used for the corrector. Both the nonlinear method and the predictor–corrector are analysed for local truncation error and stability. The MPC scheme has been tested on line and circular ring solitons known from the literature, and numerical experiments have proved that there is an improvement in accuracy over the standard predictor–corrector implementation.     

Polynomial Time-Marching for Three-Dimensional Wave Equations

As a continuation of the efficient and accurate polynomial interpolation time-marching technique in one-dimensional and two-dimensional cases, this paper proposes a method to extend it to three-dimensional problems. By stacking all two-dimensional (x, y) slice matrices along Z direction, three-dimensional derivative matrices are constructed so that the polynomial interpolation time-marching can be performed in the same way as one-dimensional and two-dimensional cases. Homogeneous Dirichlet and Neumann boundary conditions are also incorporated into the matrix operators in a similar way as in one- and two-dimensional problems. A simple numerical example of scalar wave propagation in a closed-cube has validated this extended method.     

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.     

Stratified smooth two-phase flow using the immersed interface method

The immersed interface method is used to derive a numerical method for solving fully developed, stratified smooth two-phase flow in pipes. This sharp interface technique makes the representation of the interface independent of the grid structure, and it allows for using an arbitrary shaped interface. The two-dimensional steady-state axial momentum equation is discretized and solved using a finite difference scheme on a composite, overlapping grid with local grid refinement near the interface and near the pipe wall. A low Reynolds number k-ε turbulence model is adopted to account for the effect of turbulence. A level set function is used to represent the interface. Numerical results are presented for laminar and turbulent flows. The numerical method compares well with analytical solution for laminar flow, and it shows acceptable agreement with experimental data for turbulent flow. A few examples are given to demonstrate the capability of the method to solve flow problems with a complex shaped interface.     

An efficient Chebyshev-Tau spectral method for Ginzburg-Landau-Schrödinger equations

We propose an efficient time-splitting Chebyshev-Tau spectral method for the Ginzburg-Landau-Schrödinger equation with zero/nonzero far-field boundary conditions. The key technique that we apply is splitting the Ginzburg-Landau-Schrödinger equation in time into two parts, a nonlinear equation and a linear equation. The nonlinear equation is solved exactly; while the linear equation in one dimension is solved with Chebyshev-Tau spectral discretization in space and Crank-Nicolson method in time. The associated discretized system can be solved very efficiently since they can be decoupled into two systems, one for the odd coefficients, the other for the even coefficients. The associated matrices have a quasi-tridiagonal structure which allows a direction solution to be obtained. The computation cost of the method in one dimension is O(Nlog(N)) compared with that of the non-optimized one, which is O(N²). By applying the alternating direction implicit (ADI) technique, we extend this efficient method to solve the Ginzburg-Landau-Schrödinger equation both in two dimensions and in three dimensions, respectively. Numerical accuracy tests of the method in one dimension, two dimensions and three dimensions are presented. Application of the method to study the semi-classical limits of Ginzburg-Landau-Schrödinger equation in one dimension and the two-dimensional quantized vortex dynamics in the Ginzburg-Landau-Schrödinger equation are also presented.     

Numerical Verification of Solutions of Nekrasov’s Integral Equation

This paper describes numerical verification of solutions of Nekrasov’s integral equation which is a mathematical model of two-dimensional water waves. This nonlinear and periodic integral equation includes a logarithmic singular kernel which is typically found in some two-dimensional potential problems. We propose the verification method using some properties of the singular integral for trigonometric polynomials and Schauder’s fixed point theorem in the periodic Sobolev space. A numerical example shows effectiveness of the present method.     

The approximate Dirichlet Domain Decomposition method. Part I: An algebraic approach

We present a new approach to the construction of Domain Decomposition (DD) preconditioners for the conjugate gradient method applied to the solution of symmetric and positive definite finite element equations. The DD technique is based on a non-overlapping decomposition of the domain Ω intop subdomains connected later with thep processors of a MIMD computer. The DD preconditioner derived contains three block matrices which must be specified for the specific problem considered. One of the matrices is used for the transformation of the nodal finite element basis into the approximate discrete harmonic basis. The other two matrices are block preconditioners for the Dirichlet problems arising on the subdomains and for a modified Schur complement defined over all nodes on the coupling boundaries between the subdomains. The relative spectral condition number is estimated. Relations to the additive Schwarz method are discussed. In the second part of this paper, we will apply the results of this paper to two-dimensional, symmetric, second-order, elliptic boundary value problems and present numerical results performed on a transputer-network.     

A numerical study of singularly perturbed generalized Burgers-Huxley equation using three-step Taylor-Galerkin method

In the present work, a numerical study has been carried out for the singularly perturbed generalized Burgers-Huxley equation using a three-step Taylor-Galerkin finite element method. A Burgers-Huxley equation represents the traveling wave phenomena. In singular perturbed problems, a very small positive parameter, ?, called the singular perturbation parameter is multiplied with the highest order derivative term. As this parameter tends towards zero, the problem exhibits boundary layers. The traditional methods fail to capture the boundary layers when ? becomes very small. In this paper a three-step Taylor-Galerkin finite element method is used to capture the boundary layers. The method is third-order accurate and has inbuilt upwinding. Stability analysis has been carried out and the numerical results show that the method is efficient in capturing the boundary layers.     

A new preconditioner for generalized saddle point matrices with highly singular(1,1) blocks

In this paper, based on the preconditioners presented by Cao [A note on spectrum analysis of augmentation block preconditioned generalized saddle point matrices, Journal of Computational and Applied Mathematics 238(15) (2013), pp. 109–115], we introduce and study a new augmentation block preconditioners for generalized saddle point matrices whose coefficient matrices have singular (1,1) blocks. Moreover, theoretical analysis gives the eigenvalue distribution, forms of the eigenvectors and its minimal polynomial. Finally, numerical examples show that the eigenvalue distribution with presented preconditioner has the same spectral clustering with preconditioners in the literature when choosing the optimal parameters and the preconditioner in this paper and in the literature improve the convergence of BICGSTAB and GMRES iteration efficiently when they are applied to the preconditioned BICGSTAB and GMRES to solve the Stokes equation and two-dimensional time-harmonic Maxwell equations by choosing different parameters.     

Using implicit equations of parametric curves and surfaces without computing them: Polynomial algebra by values

The availability of the implicit equation of a plane curve or of a 3D surface can be very useful in order to solve many geometric problems involving the considered curve or surface: for example, when dealing with the point position problem or answering intersection questions. On the other hand, it is well known that in most cases, even for moderate degrees, the implicit equation is either difficult to compute or, if computed, the high degree and the big size of the coefficients makes extremely difficult its use in practice.We will show that, for several problems involving plane curves, 3D surfaces and some of their constructions (for example, offsets), it is possible to use the implicit equation (or, more precisely, its properties) without needing to explicitly determine it. We replace the computation of the implicit equation with the evaluation of the considered parameterizations in a set of points. We then translate the geometric problem in hand, into one or several generalized eigenvalue problems on matrix pencils (depending again on several evaluations of the considered parameterizations).This is the so-called “polynomial algebra by values” approach where the huge polynomial equations coming from Elimination Theory (e.g., using resultants) are replaced by big structured and sparse numerical matrices. For these matrices there are well-known numerical techniques allowing to provide the results we need to answer the geometric questions on the considered curves and surfaces.     

The new numerical method for solving the system of two-dimensional Burgers’ equations

In this paper, the system of two-dimensional Burgers’ equations are solved by local discontinuous Galerkin (LDG) finite element method. The new method is based on the two-dimensional Hopf–Cole transformations, which transform the system of two-dimensional Burgers’ equations into a linear heat equation. Then the linear heat equation is solved by the LDG finite element method. The numerical solution of the heat equation is used to derive the numerical solutions of Burgers’ equations directly. Such a LDG method can also be used to find the numerical solution of the two-dimensional Burgers’ equation by rewriting Burgers’ equation as a system of the two-dimensional Burgers’ equations. Three numerical examples are used to demonstrate the efficiency and accuracy of the method.     

Bee colony optimization for the p-center problem

Bee colony optimization (BCO) is a relatively new meta-heuristic designed to deal with hard combinatorial optimization problems. It is biologically inspired method that explores collective intelligence applied by the honey bees during nectar collecting process. In this paper we apply BCO to the p-center problem in the case of symmetric distance matrix. On the contrary to the constructive variant of the BCO algorithm used in recent literature, we propose variant of BCO based on the improvement concept (BCOi). The BCOi has not been significantly used in the relevant BCO literature so far. In this paper it is proved that BCOi can be a very useful concept for solving difficult combinatorial problems. The numerical experiments performed on well-known benchmark problems show that the BCOi is competitive with other methods and it can generate high-quality solutions within negligible CPU times.     

Energy control of distributed parameter systems via speed-gradient method: case study of string and sine-Gordon benchmark models

Energy control problems are analysed for infinite dimensional systems. Benchmark linear wave equation and nonlinear sine-Gordon equation are chosen for exposition. The relatively simple case of distributed yet uniform over the space control is considered. The speed-gradient method for energy control of Hamiltonian systems proposed by A. Fradkov in 1996, has already successfully been applied to numerous nonlinear and adaptive control problems is presently developed and justified for the above partial differential equations (PDEs). An infinite dimensional version of the Krasovskii–LaSalle principle is validated for the resulting closed-loop systems. By applying this principle, the closed-loop trajectories are shown to either approach the desired energy level set or converge to a system equilibrium. The numerical study of the underlying closed-loop systems reveals reasonably fast transient processes and the feasibility of a desired energy level if initialised with a lower energy level.     

A new kind of discretization scheme for solving a two-dimensional time-independent Schrödinger equation

In this paper we present a new kind of discretization scheme for solving a two-dimensional time-independent Schrödinger equation. The scheme uses a symmetrical multi-point difference formula to represent the partial differentials of the two-dimensional variables, which can improve the accuracy of the numerical solutions to the order of Δx2Nq+2 when a (2Nq+1)-point formula is used for any positive integer Nq with Δx=Δy, while Nq=1 equivalent to the traditional scheme. On the other hand, the new scheme keeps the same form of the traditional matrix equation so that the standard algebraic eigenvalue algorithm with a real, symmetric, large sparse matrix is still applicable. Therefore, for the same dimension, only a little more CPU time than the traditional one should be used for diagonalizing the matrix. The numerical examples of the two-dimensional harmonic oscillator and the two-dimensional Henon-Heiles potential demonstrate that by using the new method, the error in the numerical solutions can be reduced steadily and extensively through the increase of Nq, which is more efficient than the traditional methods through the decrease of the step size.     

A partially structure-preserving algorithm for the permanents of adjacency matrices of fullerenes

A partially structure-preserving method for sparse symmetric matrices is proposed. Computational results on the permanents of adjacency matrices arising from molecular chemistry are presented. The largest adjacency matrix of fullerenes computed before is that of C₆₀ with a cost of several hours on supercomputers, while only about 6 min on an Intel Pentium PC (1.8 GHz) with our method. Further numerical computations are given for larger fullerenes and other adjacency matrices with n=60,80. This shows that our method is promising for problems from molecular chemistry.     

On mixed methods for fourth-order problems

A mixed finite element method for the problem v + σ²Δ²v = x with different types of boundary conditions is described. The method converges, and it is well suited for the analysis of various evolution problems. The computation of the discrete solution is made by applying a sequence of iterative methods for block matrices: correspondingly to each iteration either a couple of Poisson problems or a couple of problems for the identity operator are solved, according to the value of the parameter σ. Some numerical results for two model examples are presented.