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.     

A third order numerical scheme for the two-dimensional sine-Gordon equation

A rational approximant of third order, which is applied to a three-time level recurrence relation, is used to transform the two-dimensional sine-Gordon (SG) equation into a second-order initial-value problem. The resulting nonlinear finite-difference scheme, which is analyzed for stability, is solved by an appropriate predictor–corrector (P–C) scheme, in which the predictor is an explicit one of second order. This scheme is accelerated by using a modification (MPC) in which the already evaluated values are used for the corrector. The behavior of the proposed P–C/MPC schemes is tested numerically on the line and ring solitons known from the bibliography, regarding SG equation and conclusions for both the mentioned schemes regarding the undamped and the damped problem are derived.     

A fourth order numerical scheme for the one-dimensional sine-Gordon equation

A numerical scheme arising from the use of a fourth order rational approximants to the matrix-exponential term in a three-time level recurrence relation is proposed for the numerical solution of the one-dimensional sine-Gordon (SG) equation already known from the bibliography. The method for its implementation uses a predictor–corrector scheme in which the corrector is accelerated by using the already evaluated corrected values modified predictor–corrector scheme. For the implementation of the corrector, in order to avoid extended matrix evaluations, an auxiliary vector was successfully introduced. Both the predictor and the corrector schemes are analysed for stability. The predictor–corrector/modified predictor–corrector (P-C/MPC) schemes are tested on single and soliton doublets as well as on the collision of breathers and a comparison of the numerical results with the corresponding ones in the bibliography is made. Finally, conclusions for the behaviour of the introduced MPC over the standard P-C scheme are derived.     

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.     

An energy-preserving algorithm for nonlinear Hamiltonian wave equations with Neumann boundary conditions

In this paper, a novel energy-preserving numerical scheme for nonlinear Hamiltonian wave equations with Neumann boundary conditions is proposed and analyzed based on the blend of spatial discretization by finite element method (FEM) and time discretization by Average Vector Field (AVF) approach. We first use the finite element discretization in space, which leads to a system of Hamiltonian ODEs whose Hamiltonian can be thought of as the semi-discrete energy of the original continuous system. The stability of the semi-discrete finite element scheme is analyzed. We then apply the AVF approach to the Hamiltonian ODEs to yield a new and efficient fully discrete scheme, which can preserve exactly (machine precision) the semi-discrete energy. The blend of FEM and AVF approach derives a new and efficient numerical scheme for nonlinear Hamiltonian wave equations. The numerical results on a single-soliton problem and a sine-Gordon equation are presented to demonstrate the remarkable energy-preserving property of the proposed numerical scheme.     

New exact solutions of the (n+1)-dimensional sine-Gordon equation using double elliptic equation method

In this paper, the (n+1)-dimensional sine-Gordon equation is studied using double elliptic equation method. With the aid of Maple, more exact solutions expressed by Jacobi elliptic functions are obtained. When the modulus m of Jacobi elliptic function is driven to the limit 1 and 0, some exact solutions expressed by hyperbolic function solutions and trigonometric functions can also be obtained, respectively.     

On the simulation of the energy transmission in the forbidden band-gap of a spatially discrete double sine-Gordon system

In this work, we present a numerical method to consistently approximate solutions of a spatially discrete, double sine-Gordon chain which considers the presence of external damping. In addition to the finite-difference scheme employed to approximate the solution of the difference-differential equations of the model under investigation, our method provides positivity-preserving schemes to approximate the local and the total energy of the system, in such a way that the discrete rate of change of the total energy with respect to time provides a consistent approximation of the corresponding continuous rate of change. Simulations are performed, first of all, to assess the validity of the computational technique against known qualitative solutions of coupled sine-Gordon and coupled double sine-Gordon chains. Secondly, the method is used in the investigation of the phenomenon of nonlinear transmission of energy in double sine-Gordon systems; the qualitative effects of the damping coefficient on the occurrence of the nonlinear process of supratransmission are briefly determined in this work, 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.     

非线性演化方程的新Jacobi椭圆函数解

基于sinh-Gordon方程的椭圆函数解,构造新的试探解来扩展sinh-Gordon方程展开法.利用该方法研究了KdV-mKdV方程,双sine-Gordon方程和BBM方程,获得了这些方程的新Jacobi椭圆函数解.该方法也能用来求解其他数学物理中的非线性演化方程.     

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 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.     

On the numerical solution of space–time fractional diffusion models

A flexible numerical scheme for the discretization of the space–time fractional diffusion equation is presented. The model solution is discretized in time with a pseudo-spectral expansion of Mittag–Leffler functions. For the space discretization, the proposed scheme can accommodate either low-order finite-difference and finite-element discretizations or high-order pseudo-spectral discretizations. A number of examples of numerical solutions of the space–time fractional diffusion equation are presented with various combinations of the time and space derivatives. The proposed numerical scheme is shown to be both efficient and flexible.     

New modification of the HPM for numerical solutions of the sine-Gordon and coupled sine-Gordon equations

We present an efficient modification of the He's homotopy perturbation method that will facilitate the calculations. We apply this modification to solve sine-Gordon and coupled sine-Gordon equations, numerically. The new modification introduces a promising tool for many non-linear problems. The present method performs extremely well in terms of accuracy, efficiency, simplicity and reliability.     

Stability analysis of six-point finite difference schemes for the constant coefficient convective-diffusion equation

A comprehensive and systematic study is presented to derive stability properties of various two-level, six-point finite difference schemes (in particular, difference schemes of Padé type) for the approximation to the constant coefficient convective-diffusion equation. First, the modified equivalent partial differential equation (MEPDE) for a general six-point difference scheme is derive. The MEPDE provides direct information on the order of accuracy of a difference scheme. The von Neumann and matrix methods are then employed to deduce the necessary and sufficient conditions for the numerical stability for the six-point difference schemes. An unified technique is developed to find the stability regions for the difference schemes. Some new second and third order six-point difference schemes for the approximation of the constant coefficient convective-diffusion equation are presented.     

Application of the dual reciprocity boundary integral equation technique to solve the nonlinear Klein-Gordon equation

This paper aims to obtain approximate solutions of the Nonlinear Klein-Gordon (NLKG) equation by employing the Boundary Integral Equation (BIE) method and the Dual Reciprocity Boundary Element Method (DRBEM). This method is improved by using a predictor-corrector scheme to the nonlinearity which appears in the problem. We employ the time stepping scheme to approximate the time derivative, and the Linear Radial Basis Functions (LRBFs), are used in the Dual Reciprocity (DR) technique. To confirm the accuracy of the new approach, the numerical results of a Double-Soliton and a problem with inhomogeneous terms are compared with analytical solutions and for the examples possessing single and periodic waves, two conserved quantities associated to the (NLKG) equation, the energy and the momentum are investigated.     

Semiclassical spectral confinement for the sine-Gordon equation

The inverse scattering method for solving the sine-Gordon equation in laboratory coordinates requires the analysis of the Faddeev–Takhtajan eigenvalue problem. This problem is not self-adjoint and the eigenvalues may lie anywhere in the complex plane, so it is of interest to determine conditions on the initial data that restrict where the eigenvalues can be. We establish bounds on the eigenvalues for a broad class of zero-charge initial data that are applicable in the semiclassical or zero-dispersion limit. It is shown that no point off the coordinate axes or turning point curve can be an eigenvalue if the dispersion parameter is sufficiently small.     

A generalized finite-difference time-domain quantum method for the N-body interacting Hamiltonian

The Quantum Finite-Difference Time-Domain (FDTD-Q) method is a numerical method for solving the time evolution of the Schrödinger equation. It can be applied to systems of interacting particles, allowing for realistic simulations of quantum mechanics of various experimental systems. One of the drawbacks of the method is that divergences in the numerical evolution occur rather easily in the presence of interactions, which necessitates a large number of evolution steps or imaginary time evolution. We present a generalized (GFDTD-Q) method for solving the time-dependent Schrödinger equation including interactions between the particles. The new scheme provides a more relaxed condition for stability when the finite difference approximations for spatial derivatives are employed, as compared with the original FDTD-Q scheme. We demonstrate our scheme by simulating the time evolution of a two-particle interaction Hamiltonian. Our results show that the generalized method allows for stable time evolutions, in contrast to the original FDTD-Q scheme which produces a divergent solution.     

A higher-order alternating group explicit scheme for the diffusion equation

In this paper, a higher-order alternating group explicit scheme for the diffusion equation is developed. The scheme has fourth-order truncation error approximately. The numerical simulations show that the new scheme can provide more accurate solutions. A discussion on the numerical stability of the scheme is also included.     

Numerical Approximation of the Euler-Poisson-Boltzmann Model in the Quasineutral Limit

This paper analyzes various schemes for the Euler-Poisson-Boltzmann (EPB) model of plasma physics. This model consists of the pressureless gas dynamics equations coupled with the Poisson equation and where the Boltzmann relation relates the potential to the electron density. If the quasi-neutral assumption is made, the Poisson equation is replaced by the constraint of zero local charge and the model reduces to the Isothermal Compressible Euler (ICE) model. We compare a numerical strategy based on the EPB model to a strategy using a reformulation (called REPB formulation). The REPB scheme captures the quasi-neutral limit more accurately.     

空间太阳能电站太阳能接收器二维展开过程的保结构分析

针对传统数值方法求解微分-代数方程过程中经常遇到的违约问题,本文以空间太阳能电站太阳能接收器的简化二维模型为例,采用辛算法模拟了简化模型的展开过程,研究了辛算法在求解过程中约束违约问题.首先,基于Hamilton变分原理,将描述简化二维模型展开过程的Euler-Lagrange方程导入Hamilton体系,建立其Hamilton正则方程;随后,采用s级PRK离散方法离散正则方程,得到其辛格式;最后,采用辛PRK格式模拟太阳能接收器的二维展开过程.模拟结果显示:本文构造的辛PRK格式能够很好地满足系统的位移约束.