广义非线性Sin-Gordon方程的整体解及数值计算

本文考察了一类广义非线性Sin-Gordon方程的周期初值问题，利用非线性Galerkin方法，证明了其整体解的存在性和唯一性，并给出了其有界吸引集的存在性．构造了全离散的Fourier拟谱显格式，利用有界延拓法证明了其格式的收敛性与稳定性，并给出了误差估计、算法分析及计算复杂度，最后，通过数值例子，检验了理论结果的可信性．为对此模型的数值分析提供了理论基础和一个有效的算法．     

MKdV方程的多辛Fourier拟谱方法

基于谱微分矩阵方法,给出MKdV方程的多辛Fourier拟谱格式及其相应多辛离散守恒律,证明了它等价于通常的Fourier拟谱格式．数值结果表明,格式对于长时间计算具有稳定性与高精度．     

非定常不可压Navier-Stokes方程两重网格方法的稳定性和L2误差估计

讨论了二维非定常不可压Navier-Stokes方程的两重网格方法．此方法包括在粗网格上求解一个非线性问题,在细网格上求解一个Stokes问题．采用一种新的全离散(时间离散用Crank-Nicolson格式,空间离散用混合有限元方法)格式数值求解N-S方程．证明了该全离散格式的稳定性．给出了L2误差估计．对比标准有限元方法,在保持同样精度的前提下,TGM能节省大量的计算量．     

粘弹性方程全离散化有限体积元格式及数值模拟

本文利用有限体积元方法研究二维粘弹性方程, 给出一种时间二阶精度的全离散化有限体积元格式, 并给出这种全离散化有限体积元解的误差估计, 最后用数值例子验证数值结果与理论结果是相吻合的. 通过与有限元方法和有限差分方法相比较, 进一步说明了全离散化有限体积元格式是求解二维粘弹性方程数值解的最有效方法之一.     

Cahn-Hilliard方程的拟谱逼近

该文讨论用Ｌｅｇｅｎｄｒｅ拟谱方法数值求解非线性Ｃａｈｎ Ｈｉｌｌｉａｒｄ方程的Ｄｉｒｉｃｈｌｅｔ问题．建立了其半离散和全离散逼近格式，它们保持原问题能量耗散的性质．证明了离散解的存在唯一性，并给出了最佳误差估计．数值实验也证实了我们的结果．     

一类非线性Schrodinger方程的差分／Legendre谱元法

考察一类带幂次非线性项的Schrodinger方程的Dirichlet初边值问题，提出了一个有效的计算格式，其中时间方向上应用了一种守恒的二阶差分隐格式，空间方向上采用Legendre谱元法．对于时间半离散格式，证职了该格式具有能量守恒性质，并给出了L^2误差估计，对于全离散格式，应用不动点原理证明了数值解的存在唯一性，并给出了L^2误差估计．最后，通过数值试验验证了结果的可信性．     

一类非线性双曲型方程扩展混合有限元方法的误差估计

该文针对一类非线性双曲型方程提出了扩展混合有限元方法.首先,建立了半离散扩展混合元格式,获得了半离散扩展混合元解的L∞(L2)先验误差估计.然后,利用有限差分法对时间项进行离散,建立了全离散扩展混合元格式,并给出了全离散格式下的先验误差估计.最后,通过数值算例验证了理论结果.     

非饱和土壤水流方程的CN广义差分法

首先给出二维非饱和土壤水流方程时间二阶精度的Crank-Nicolson(CN)时间半离散化格式,然后直接从CN时间半离散化格式出发,建立具有时间二阶精度的全离散化CN广义差分格式,并给出误差分析,最后用数值例子验证全离散化CN广义差分格式的优越性.这种方法能提高时间离散的精度,极大地减少时间方向的迭代步,从而减少实际计算中截断误差的积累,提高计算精度和计算效率.而且该方法可以绕开对空间变量的半离散化广义差分格式的讨论,使得理论研究更简便.     

二维阻尼非线性sine-Gordon方程的共形多辛Fourier拟谱格式

为二维阻尼非线性sine-Gordon方程构造了一个新的共形多辛Fourier拟谱格式.基于原系统的共形多辛哈密尔顿形式,首先在时间和空间方向上分别用辛中点和Fourier拟谱方法进行离散,得到一个全离散格式.随后证明了构造的格式保持离散的共形多辛守恒律.最后数值实验验证了格式的有效性.     

二维非饱和水流的全离散广义差分格式分析及其数值模拟

建立二维非饱和水流问题的全离散广义差分格式,讨论了全离散广义差分解的存在唯一性,并给出最优误差估计的证明．最后给出数值算例,验证方法的有效性.     

广义Korteweg-de Vries-Burgers方程组的谱方法及其误差估计

引言 近十多年来,偏微分方程的谱方法发展迅速,主要是由于在谱方法计算中应用了快速Pourier变换(FFT),减少了计算量,使之具有实用价值.谱方法的另一优点是具有“无穷阶”的快敛速,即,若原微分方程的解是无穷可微的,则合适的(半离散)谱方法逼近的收敛性比N~(-1)的任何幂次都快(这里N是所取基函数的个数).郭本瑜提供了证明KdV-Burgers方程谱方法格式产格误差估计的技巧,并在[6]中推广到二维涡度方程,证     

Uniform convergence of the legendre spectral method for the Zakharov equations

The Legendre spectral and pseudospectral approximations are proposed for the standard Zakharov equations with initial boundary conditions. Optimal H¹ error estimate of the method is given for both semidiscrete and fully discrete schemes. The uniform convergence for the parameter ε relative to the acoustic speed is proved. Moreover, the multidomain Legendre spectral scheme is also constructed, which can be implemented in parallel. Finally, numerical results in single domain and multidomain verify the high accuracy of the Legendre spectral method. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

Geometric numerical schemes for the KdV equation

Geometric discretizations that preserve certain Hamiltonian structures at the discrete level has been proven to enhance the accuracy of numerical schemes. In particular, numerous symplectic and multi-symplectic schemes have been proposed to solve numerically the celebrated Korteweg-de Vries equation. In this work, we show that geometrical schemes are as much robust and accurate as Fourier-type pseudospectral methods for computing the long-time KdV dynamics, and thus more suitable to model complex nonlinear wave phenomena.     

Numerical approximations for the nonlinear time fractional reaction–diffusion equation

In this article, we propose an implicit pseudospectral scheme for nonlinear time fractional reaction–diffusion equations with Neumann boundary conditions, which is based upon Gauss–Lobatto–Legendre–Birkhoff pseudospectral method in space and finite difference method in time. A priori estimate of numerical solution is given firstly. Then the existence of numerical solution is proved by Brouwer fixed point theorem and the uniqueness is obtained. It is proved rigorously that the fully discrete scheme is unconditionally stable and convergent. Furthermore, we develop a modified scheme by adding correction terms for the problem with nonsmooth solutions. Numerical examples are given to verify the theoretical analysis.     

Conformal structure-preserving schemes for damped-driven stochastic nonlinear Schrödinger equation with multiplicative noise

Since many physical phenomena are often influenced by dispersive medium, energy compensation and random perturbation, exploring the dynamic behaviors of the damped-driven stochastic system has becoming a hot topic in mathematical physics in recent years. In this paper, inspired by the stochastic conformal structure, we investigate the geometric numerical integrators for the damped-driven stochastic nonlinear Schrödinger equation with multiplicative noise. To preserve the conformal structures of the system, by using symplectic Euler method, implicit midpoint method and Fourier pseudospectral method, we propose three kinds of stochastic conformal schemes satisfying corresponding discrete stochastic multiconformal-symplectic conservation laws and discrete global/local charge conservation laws. Numerical experiments illustrate the structure-preserving properties of the proposed schemes, as well as favorable results over traditional nonconformal schemes, which are consistent with our theoretical analysis.     

Construction of a Mindlin pseudospectral plate element and evaluating efficiency of the element

In this paper, a Mindlin pseudospectral plate element is constructed to perform static, dynamic, and wave propagation analyses of plate-like structures. Chebyshev polynomials are used as basis functions and Chebyshev–Gauss–Lobatto points are used as grid points. Two integration schemes, i.e., Gauss–Legendre quadrature (GLEQ) and Chebyshev points quadrature (CPQ), are employed independently to form the elemental stiffness matrix of the present element. A lumped elemental mass matrix is generated by only using CPQ due to the discrete orthogonality of Chebyshev polynomials and overlapping of the quadrature points with the grid points. This results in a remarkable reduction of numerical operations in solving the equation of motion for being able to use explicit time integration schemes. Numerical calculations are carried out to investigate the influence of the above two numerical integration schemes in the elemental stiffness formation on the accuracy of static and dynamic response analyses. By comparing with the results of ABAQUS, this study shows that CPQ performs slightly better than GLEQ in various plates with different thicknesses, especially in thick plates. Finally, a one dimensional (1D) and a 2D wave propagation problems are used to demonstrate the efficiency of the present Mindlin pseudospectral plate element.     

A STRUCTURE-PRESERVING DISCRETIZATION OFNONLINEAR SCHRDINGER EQUATION

1.IntroductionManyimportantdifferentialequationsofevolutiontypeinphysicsandmechanicshavespecificgeometricstructure.Forinstance,theHamiltoniansystemsinclassicalmechanics,theSchrodingerequationinquantum,theKorteweg-deVriesandKleinGordonequationsofnonli...     

A LEGENDRE PSEUDOSPECTRAL METHOD FOR SOLVINGNONLINEAR KLEIN-GORDON EQUATION

1.IntroductionAsweknow,theKlein-Gordonequationisanimportantmathematicalmodelinquantummechanics.Itisoftheformwherefi=(--1,1)",x=(xl,x25'jx.),bisarealnumber.AssumethatU000=UI(x)=0onOnandAsin[1],itcanbeshownthatifUOEHI(n)nH(n),UIEL'(fl)andfEL'(o,T;L'(n)),then(1.1)hasuniquesolutionUELoo(o,T;Hi(n)nH(fl)).oftheotherhand,somefinitedifferenceschemeswereproposedin12,31withstrictproo]ofgeneralizedstabilityandconvergence.TheirnumericalsolutionskeepthediscretEconservations.Oneofspecialcases(…     

JACOBI PSEUDOSPECTRAL METHOD FOR FOURTH ORDER PROBLEMS

In this paper, we investigate Jacobi pseudospectral method for fourth order problems. We establish some basic results on the Jacobi-Gauss-type interpolations in non-uniformly weighted Sobolev spaces, which serve as important tools in analysis of numerical quadratures, and numerical methods of differential and integral equations. Then we propose Jacobi pseudospectral schemes for several singular problems and multiple-dimensional problems of fourth order. Numerical results demonstrate the spectral accuracy of these schemes, and coincide well with theoretical analysis.     

Efficient splitting schemes for magneto-hydrodynamic equations

We present in this paper several efficient numerical schemes for the magneto-hydrodynamic (MHD) equations. These semi-discretized (in time) schemes are based on the standard and rotational pressure-correction schemes for the Navier-Stokes equations and do not involve a projection step for the magnetic field. We show that these schemes are unconditionally energy stable, present an effective algorithm for their fully discrete versions and carry out demonstrative numerical experiments.