Multi-symplectic Runge-Kutta methods for Landau-Ginzburg-Higgs equation

Nonlinear wave equations have been extensively investigated in the last sev- eral decades. The Landau-Ginzburg-Higgs equation, a typical nonlinear wave equation, is studied in this paper based on the multi-symplectic theory in the Hamilton space. The multi-symplectic Runge-Kutta method is reviewed, and a semi-implicit scheme with certain discrete conservation laws is constructed to solve the first-order partial differential equations （PDEs） derived from the Landau-Ginzburg-Higgs equation. The numerical re- sults for the soliton solution of the Landau-Ginzburg-Higgs equation are reported, showing that the multi-symplectic Runge-Kutta method is an efficient algorithm with excellent long-time numerical behaviors.     

非线性弹性杆中纵波传播过程的数值模拟

基于Hamilton空间体系的多辛理论研究了包含材料非线性效应和几何弥散效应的非线性弹性杆中纵波传播问题。导出了其Bridges意义下的多辛形式及其多种守恒律,并构造了等价于Box多辛格式的隐式多辛格式对不考虑材料非线性效应和几何弥散效应、只考虑材料非线性效应、只考虑几何弥散效应、同时考虑材料非线性效应和几何弥散效应四种情况下不同截面参数的圆杆中的纵波传播过程进行数值模拟,模拟结果不仅全面地反映了非线性效应和几何弥散效应对纵波传播的影响,而且也反映了多辛方法的两大优点:精确的保持多种守恒律和良好的长时间数值行为。     

THE MULTI-SYMPLECTIC ALGORITHM FOR "GOOD" BOUSSINESQ EQUATION

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｒｅｃｅｎｔｙｅａｒｓａｒｅｍａｒｋａｂｌｅｄｅｖｅｌｏｐｍｅｎｔｈａｓｔａｋｅｎｐｌａｃｅｉｎｔｈｅｓｔｕｄｙｏｆｎｏｎｌｉｎｅａｒｅｖｏｌｕｔｉｏｎａｒｙｐａｒｔｉａｌｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓ.Ａｎｅｘａｍｐｌｅｉｓｔｈｅ“Ｇｏｏｄ”Ｂｏｕｓｓｉｎｅｓｑ (Ｇ .Ｂ .)ｅｑｕａｔｉｏｎｕｔｔ =-ｕｘｘｘｘ+ｕｘｘ+ (ｕ2 ) ｘｘ ( 1 )ｗｈｉｃｈｄｅｓｃｒｉｂｅｓｓｈａｌｌｏｗｗａｔｅｒｗａｖｅｓｐｒｏｐａｇａｔｉｎｇｉｎｂｏｔｈｄｉｒｅｃｔｉｏｎｓ.Ｔｈｅａ…     

New explicit multi-symplectic scheme for nonlinear wave equation简

Based on splitting multi-symplectic structures, a new multi-symplectic scheme is proposed and applied to a nonlinear wave equation. The explicit multi-symplectic scheme of the nonlinear wave equation is obtained, and the corresponding multi-symplectic conservation property is proved. The backward error analysis shows that the explicit multi-symplectic scheme has good accuracy. The sine-Gordon equation and the Klein-Gordon equation are simulated by an explicit multi-symplectic scheme. The numerical results show that the new explicit multi-symplectic scheme can well simulate the solitary wave behaviors of the nonlinear wave equation and approximately preserve the relative energy error of the equation.     

Analyzing dynamic response of non-homogeneous string fixed at both ends

To survey the local conservation properties of complex dynamic problems, a structure-preserving numerical method, named as generalized multi-symplectic method, is proposed to analyze the dynamic response of a non-homogeneous string fixed at both ends. Firstly, based on the multi-symplectic idea, a generalized multi-symplectic form derived from the vibration equation of a non-homogeneous string is presented. Secondly, several conservation laws are deduced from the generalized multi-symplectic form to illustrate the local properties of the system. Thirdly, a centered box difference scheme satisfying the discrete local momentum conservation law exactly, named as a generalized multi-symplectic scheme, is constructed to analyze the dynamic response of the non-homogeneous string fixed at both ends. Finally, numerical experiments on the generalized multi-symplectic scheme are reported. The results illustrate the high accuracy, the good local conservation properties as well as the excellent long-time numerical behavior of the generalized multi-symplectic scheme well.     

THE MULTI-SYMPLECTIC ALGORITHM FOR “GOOD” BOUSSINESQ EQUATION

The multi-symplectic formulations of the “Good” Boussinesq equation were considered. For the multi-symplectic formulation, a new fifteen-point difference scheme which is equivalent to the multi-symplectic Preissman integrator was derived. The numerical experiments show that the multi-symplectic scheme have excellent long-time numerical behavior. Foundation items: the Foundation for Key Laboratory of Scientific/Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing, Chinese Academy of Sciences; the Natural Science Foundation of Huaqiao University. Biography: ZENG Wen-ping (1940-), Professor (E-mail: qmz@1sec.cc.ac.cn)     

A new numerical model for simulations of wave transformation,breaking and long‐shore currents in complex coastal regions

In this paper, we propose a model based on a new contravariant integral form of the fully nonlinear Boussinesq equations in order to simulate wave transformation phenomena, wave breaking, and nearshore currents in computational domains representing the complex morphology of real coastal regions. The aforementioned contravariant integral form, in which Christoffel symbols are absent, is characterized by the fact that the continuity equation does not include any dispersive term. A procedure developed in order to correct errors related to the difficulties of numerically satisfying the metric identities in the numerical integration of fully nonlinear Boussinesq equation on generalized boundary‐conforming grids is presented. The Boussinesq equation system is numerically solved by a hybrid finite volume–finite difference scheme. The proposed high‐order upwind weighted essentially non‐oscillatory finite volume scheme involves an exact Riemann solver and is based on a genuinely two‐dimensional reconstruction procedure, which uses a convex combination of biquadratic polynomials. The wave breaking is represented by discontinuities of the weak solution of the integral form of the nonlinear shallow water equations. The capacity of the proposed model to correctly represent wave propagation, wave breaking, and wave‐induced currents is verified against test cases present in the literature. The results obtained are compared with experimental measures, analytical solutions, or alternative numerical solutions. Copyright © 2015 John Wiley & Sons, Ltd.     

一类偏微分方程的Hamilton正则表示

主要给出一系列关于力学中的偏微分方程的无穷维Ｈａｍｉｌｔｏｎ正则表示．其中包括变系数线性偏微分方程,ＫｄＶ方程,ＭＫｄＶ方程,ＫＰ方程,Ｂｏｕｓｉｎｅｓｑ方程等的无穷维Ｈａｍｉｌｔｏｎ正则表示．     

基于数据驱动的流场控制方程的稀疏识别

利用有限数据建立系统的非线性动力学模型是具有挑战性的重要课题. 数据驱动的稀疏识别方法是近年来发展的从数据识别动力系统控制方程的有效方法. 本文基于数据驱动稀疏识别方法对不同流场的控制方程进行了识别. 采用非线性动力学偏微分方程函数识别(partial differential equations functional identification of nonlinear dynamics, PDE-FIND)方法和最小绝对收缩和选择算子(least absolute shrinkage and selection operator, LASSO)方法对二维圆柱绕流、顶盖驱动方腔流、Rayleigh-Bénard (RB)对流和三维槽道湍流的控制方程进行了识别. 在稀疏识别过程中, 采用直接数值模拟得到的流场数据来计算过完备候选库中的每一项, 候选库中变量最高保留到二次, 变量导数最高保留到二阶, 非线性项最高保留到四阶. 结果发现PDE-FIND方法和LASSO方法对于不含有非线性项的控制方程, 如涡量输运方程、热输运方程和连续性方程, 都能准确识别. 对于含有强非线性项的控制方程, 如Navier-Stokes方程的识别, PDE-FIND方法正确地识别出了控制方程及流场的Rayleigh数和Reynolds数, 而LASSO方法识别结果不正确, 这是因为候选库中的项之间存在分组效应, LASSO方法通常只取分组中的一项. 本文还发现选择流动结构丰富的区域的数据进行控制方程的稀疏识别可以提高识别的准确性.      

Multi-symplectic method for the generalized (2+1)-dimensional KdV-mKdV equation

In the present paper,a general solution involving three arbitrary functions for the generalized(2+1)dimensional KdV-mKdV equation,which is derived fromthe generalized(1+1)-dimensional KdV-mKdV equation,is first introduced by means of the Wiess,Tabor,Carnevale(WTC) truncation method.And then multisymplectic formulations with several conservation lawstaken into account are presented for the generalized(2+1)dimensional KdV-mKdV equation based on the multisymplectic theory of Bridges.Subsequently,in order tosimulate the periodic wave solutions in terms of rationalfunctions of the Jacobi elliptic functions derived from thegeneral solution,a semi-implicit multi-symplectic schemeis constructed that is equivalent to the Preissmann scheme.From the results of the numerical experiments,we can conclude that the multi-symplectic schemes can accurately simulate the periodic wave solutions of the generalized(2+1)dimensional KdV-mKdV equation while preserve approximately the conservation laws.     

The integral bifurcation method combined with factoring technique for investigating exact solutions and their dynamical properties of a generalized Gardner equation

It is well known that it is difficult to obtain exact solutions of some partial differential equations with highly nonlinear terms or high order terms because these kinds of equations are not integrable in usual conditions. In this paper, by using the integral bifurcation method and factoring technique, we studied a generalized Gardner equation which contains both highly nonlinear terms and high order terms, some exact traveling wave solutions such as non-smooth peakon solutions, smooth periodic solutions and hyperbolic function solutions to the considered equation are obtained. Moreover, we demonstrate the profiles of these exact traveling wave solutions and discuss their dynamic properties through numerical simulations.     

Free Convection Boundary Layer Flow Past a Vertical Surface in a Porous Medium with Temperature-Dependent Properties

Numerical solutions for the free convection heat transfer in a viscous fluid at a permeable surface embedded in a saturated porous medium, in the presence of viscous dissipation with temperature-dependent variable fluid properties, are obtained. The governing equations for the problem are derived using the Darcy model and the Boussinesq approximation (with nonlinear density temperature variation in the buoyancy force term). The coupled non-linearities arising from the temperature-dependent density, viscosity, thermal conductivity, and viscous dissipation are included. The partial differential equations of the model are reduced to ordinary differential equations by a similarity transformation and the resulting coupled, nonlinear ordinary differential equations are solved numerically by a second order finite difference scheme for several sets of values of the parameters. Also, asymptotic results are obtained for large values of | f _w|. Moreover, the numerical results for the velocity, the temperature, and the wall-temperature gradient are presented through graphs and tables, and are discussed. It is observed that by increasing the fluid variable viscosity parameter, one could reduce the velocity and thermal boundary layer thickness. However, quite the opposite is true with the non-linear density temperature variation parameter.     

A Smooth Center Manifold Theorem which Applies to Some Ill-Posed Partial Differential Equations with Unbounded Nonlinearities

We prove the existence of a smooth center manifold for several partial differential equations, including ill posed equations with unbounded nonlinearities. We also prove smooth dependence on parameters with respect to some perturbations, including unbounded ones. More concretely, we prove an abstract theorem and present applications to several concrete equations: ill posed Boussinesq, equation and system and nonlinear Laplace equations in cylindrical domains. We also consider the effect of some geometric structures.     

The Generalized Nonlinear Galerkin Method Applied to Hydrodynamics of the Open Sea

In this paper we present new numerical algorithms based on a generalized nonlinear Galerkin method in order to solve coastal and oceanic circulation problems. The equations system is based on the primitive equations of the ocean under Boussinesq and hydrostatic approximations. These equations are transformed using, at the same time, the classical σ transformation and an original homogenization of the boundary conditions. We use a well adapted special basis to apply the usual Galerkin method and the nonlinear Galerkin method. This basis is built on a modelization of the energetic transfers through the different scales of flow. Two approaches are proposed to solve the continuity equation: the (nonlinear) Galerkin method and the method of the characteristics. We present the advantages and drawbacks of both methods.     

Effects of thermal radiation on MHD viscous fluid flow and heat transfer over nonlinear shrinking porous sheet

This paper investigates the effects of thermal radiation on the magnetohy-drodynamic (MHD) flow and heat transfer over a nonlinear shrinking porous sheet. The surface velocity of the shrinking sheet and the transverse magnetic field are assumed to vary as a power function of the distance from the origin. The temperature dependent viscosity and the thermal conductivity are also assumed to vary as an inverse function and a linear function of the temperature, respectively. A generalized similarity transformarion is used to reduce the governing partial differential equations to their nonlinear coupled ordinary differential equations, and is solved numerically by using a finite difference scheme. The numerical results concern with the velocity and temperature profiles as well as the local skin-friction coefficient and the rate of the heat transfer at the porous sheet for different values of several physical parameters of interest.     

A high order spectral volume solution to the Burgers' equation using the Hopf–Cole transformation

A limiter free high order spectral volume (SV) formulation is proposed in this paper to solve the Burgers' equation. This approach uses the Hopf–Cole transformation, which maps the Burgers' equation to a linear diffusion equation. This diffusion equation is solved in an SV setting. The local discontinuous Galerkin (LDG) and the LDG2 viscous flux discretization methods were employed. An inverse transformation was used to obtain the numerical solution to the Burgers' equation. This procedure has two advantages: (i) the shock can be captured, without the use of a limiter; and (ii) the effects of SV partitioning becomes almost redundant as the transformed equation is not hyperbolic. Numerical studies were performed to verify. These studies also demonstrated (i) high order accuracy of the scheme even for very low viscosity; (ii) superiority of the LDG2 scheme, when compared with the LDG scheme. In general, the numerical results are very promising and indicate that this procedure can be applied for obtaining high order numerical solutions to other nonlinear partial differential equations (for instance, the Korteweg–de Vries equations) which generate discontinuous solutions. Copyright © 2011 John Wiley & Sons, Ltd.     

Stochastic averaging of energy harvesting systems

A stochastic averaging method is proposed for nonlinear energy harvesters subjected to external white Gaussian noise and parametric excitations. The Fokker–Planck–Kolmogorov equation of the coupled electromechanical system of energy harvesting is a three variables nonlinear parabolic partial differential equation whose exact stationary solutions are generally hard to find. In order to overcome difficulties in solving higher dimensional nonlinear partial differential equations, a transformation scheme is applied to decouple the electromechanical equations. The averaged Itô equations are derived via the standard stochastic averaging method, then the FPK equations of the decoupled system are obtained. The exact stationary solution of the averaged FPK equation is used to determine the probability densities of the displacement, the velocity, the amplitude, the joint probability densities of the displacement and velocity, and the power of the stationary response. The effects of the system parameters on the output power are examined. The approximate analytical outcomes are qualitatively and quantitatively supported by the Monte Carlo simulations.     

Asymptotic Solution for a Kind of Boundary Layer Problem

This paper studies the partial differential equation with a small coefficient in the highest-order item. This kind of equation is also named as boundary layer problem. The Burgers equation and modified Burgers equation are analyzed in this approach. First, these equations are transferred into the strong nonlinear ones, and then the corresponding strong nonlinear equations are solved based on the perturbation method. The results from the asymptotic method are comparable with those obtained from numerical computation. An erratum to this article is available at .     

基于Boussinesq方程的波浪模型

从欧拉方程出发，提供了另一种推导完全非线性Boussinesq方程的方法，并对方程的 线性色散关系和线性变浅率进行了改进. 改进后方程的线性色散关系达到了一阶Stokes波 色散关系的Pad\'{e}[4,4]近似，在相对水深达1.0的强色散波浪时仍保持较高的准确性，并且方程的非线性和线性 变浅率都得到了不同程度的改善. 方程的水平一维形式用预估-校正的有限差分格式求解， 建立了一个适合较强非线性波浪的Boussinesq波浪数值模型. 作为验证，模拟了波浪在潜 堤上的传播变形，计算结果和实验数据的比较发现两者符合良好.     

NONLINEAR RESPONSES OF A FLUID-CONVEYING PIPE EMBEDDED IN NONLINEAR ELASTIC FOUNDATIONS

The nonlinear responses of planar motions of a fluid-conveying pipe embedded in nonlinear elastic foundations are investigated via the differential quadrature method discretization （DQMD） of the governing partial differential equation. For the analytical model, the effect of the nonlinear elastic foundation is modeled by a nonlinear restraining force. By using an iterative algorithm, a set of ordinary differential dynamical equations derived from the equation of motion of the system are solved numerically and then the bifurcations are analyzed. The numerical results, in which the existence of chaos is demonstrated, are presented in the form of phase portraits of the oscillations. The intermittency transition to chaos has been found to arise.