M.K.D.V方程高阶全离散Galerkin有限元格式

本文用对角隐式Runge-Kutta方法(D.I.R.K),对M.K.D.V.方程在时间方向离散,采用增加扰动项的办法,得到了L~2模意义下时间方向具有三阶精度的格式。数值实例表明,其精度比无拢动项及C-N格式好。还证明了收敛性和稳定性,用Newton迭代法求解非线性方程组,并证明选取适当的初始值,Newton迭代仅需一步完成。     

M.K.d.V.方程的数值方法

自从Zabusky和Kruskal用数值方法发现Soliton以来,Korteweg-de Vries(K.d.V.)方程的数值解法引起了广泛的兴趣,出现了大量的实际计算和某些理论分析结果。例如[1—8],郭本瑜最近的报告回顾了这方面的工作。比较起来,另一类具有Soliton解的广义K.d.V.方程的数值解法研究得还不多,严格的理论分析结果尚未     

广义Rosenau-KdV方程的高精度守恒差分格式

对广义Rosenau-KdV方程提出一种在时间层和空间层上分别具有二阶和四阶精度的三层线性差分格式,所建格式是离散质量守恒和离散能量守恒的,利用离散能量法证明了差分格式的可解性、收敛性和稳定性.数值实验验证了该格式的精度和守恒性.     

带广义边界条件的四阶抛物型方程的混合间断时空有限元法

构造具有广义边界条件的四阶线性抛物型方程的混合间断时空有限元格式,利用混合有限元方法将高阶方程降阶,利用空间连续而时间允许间断的时空有限元方法离散方程,证明了离散解的存在唯一性,稳定性和收敛性,并给出数值算例验证了方法的有效性.     

广义混合非线性Schrdinger方程的拟谱方法

本文讨论了广义混合非线性Schrdinger 方程的周期初值问题,构造了守恒的半离散Fourier 拟谱格式,对其近似解进行了先验估计,并证明了格式的收敛性.证明了该方程存在孤立子解,并给出其孤立子解的精确表达式.研究了线性化方程的稳定性问题,即在初值有扰动的情况下,该方程只有振荡解和鞍点.最后,通过数值例子验证了格式的可信性,数值计算表明,本格式时间方向可取大步长且是长时间稳定的,我们还计算了孤立子解,并绘出了在初值有扰动的情况下,相空间的轨线图.     

广义混合非线性Schrödinger方程的拟谱方法

本文讨论了广义混合非线性Schrodinger方程的周期初值问题,构造了守恒的半离散Fourier拟谱格式,对其近似解进行了先验估计,并证明了格式的收敛性.证明了该方程存在孤立子解,并给出其孤立子解的精确表达式.研究了线性化方程的稳定性问题,即在初值有扰动的情况下,该方程只有振荡解和鞍点.最后,通过数值例子验证了格式的可信性,数值计算表明,本格式时间方向可取大步长且是长时间稳定的,我们还计算了孤立子解,并绘出了在初值有扰动的情况下,相空间的轨线图.     

广义混合非线性Schr？dinger方程的拟谱方法

本文讨论了广义混合非线性Schrodinger方程的周期初值问题,构造了守恒的半离散Fourier拟谱格式,对其近似解进行了先验估计,并证明了格式的收敛性.证明了该方程存在孤立子解,并给出其孤立子解的精确表达式.研究了线性化方程的稳定性问题,即在初值有扰动的情况下,该方程只有振荡解和鞍点.最后,通过数值例子验证了格式的可信性,数值计算表明,本格式时间方向可取大步长且是长时间稳定的,我们还计算了孤立子解,并绘出了在初值有扰动的情况下,相空间的轨线图.     

广义混合非线性Schr(o)dinger方程的拟谱方法

本文讨论了广义混合非线性Schrodinger方程的周期初值问题,构造了守恒的半离散Fourier拟谱格式,对其近似解进行了先验估计,并证明了格式的收敛性.证明了该方程存在孤立子解,并给出其孤立子解的精确表达式.研究了线性化方程的稳定性问题,即在初值有扰动的情况下,该方程只有振荡解和鞍点.最后,通过数值例子验证了格式的可信性,数值计算表明,本格式时间方向可取大步长且是长时间稳定的,我们还计算了孤立子解,并绘出了在初值有扰动的情况下,相空间的轨线图.     

广义对称正则长波方程的守恒差分格式

文章考虑了具有齐次边界条件的广义对称正则长波方程的有限差分格式.提出了一个守恒并且线性非耦合的三层有限差分格式,由于格式在计算中只需要解三对角线性方程组,从而避免了其中的迭代计算.文中先讨论了一个离散守恒量,然后我们利用离散泛函分析方法证明了格式的收敛性和稳定性,从理论上得到了收敛阶为O(h~2+τ~2).通过数值试验表明,所提的方法是可靠有效的.     

一维边界阻尼波动方程指数稳定的半离散有限差分一致逼近格式

通过在时间方向引入一个平均算子,对一维边界阻尼波动方程构造了一个等距网格上的半离散有限差分格式.利用离散乘子法,证明了对偶系统半离散格式的一致可观测不等式,进而证明了原系统半离散格式的一致指数稳定性.数值实验验证了理论结果.     

Numerical methods for Hamilton Jacobi functional differential equations

Initial and initial boundary value problems for first order partial functional differential equations are considered. Explicit difference schemes of the Euler type and implicit difference methods are investigated. The following theoretical aspects of the methods are presented. Sufficient conditions for the convergence of approximate solutions are given and comparisons of the methods are presented. It is proved that assumptions on the regularity of given functions are the same for both the methods. It is shown that conditions on the mesh for explicit difference schemes are more restrictive than suitable assumptions for implicit methods. There are implicit difference schemes which are convergent and corresponding explicit difference methods are not convergent. Error estimates for both the methods are construted.     

The completely conservative difference schemes for the nonlinear Landau-Fokker-Planck equation

Conservativity and complete conservativity of finite difference schemes are considered in connection with the nonlinear kinetic Landau-Fokker-Planck equation. The characteristic feature of this equation is the presence of several conservation laws. Finite difference schemes, preserving density and energy are constructed for the equation in one- and two-dimensional velocity spaces. Some general methods of constructing such schemes are formulated. The constructed difference schemes allow us to carry out the numerical solution of the relaxation problem in a large time interval without error accumulation. An illustrative example is given.     

Implicit Solution of a Two-Dimensional Parabolic Inverse Problem with Temperature Overspecification

Three different implicit finite difference schemes for solving the two-dimensional parabolic inverse problem with temperature overspecification are considered. These schemes are developed for indentifying the control parameter which produces, at any given time, a desired temperature distribution at a given point in the spatial domain. The numerical methods discussed, are based on the second-order (5,1) Backward Time Centered Space (BTCS) implicit formula, and the second-order (5,5) Crank-Nicolson implicit finite difference formula and the fourth-order (9,9) implicit scheme. These finite difference schemes are unconditionally stable. The (9,9) implicit formula takes a huge amount of CPU time, but its fourth-order accuracy is significant. The results of a numerical experiment are presented, and the accuracy and central processor (CPU) times needed for each of the methods are discussed and compared. The implicit finite difference schemes use more central processor times than the explicit finite difference techniques, but they are stable for every diffusion number.     

Convergence of Difference Methods for Inverse Problems of a One-Dimensional Hyperbolic System of First Order

In this paper, the difference methods for solving the inverse problem of a one-dimensional hyperbolic system of first order are discussed. Some difference schemes are constructed and the convergence of these schemes is proved.     

Factorized SM-stable two-level schemes

Additional requirements for unconditionally stable schemes were formulated by analyzing higher order accurate difference schemes in time as applied to boundary value problems for second-order parabolic equations. These requirements concern the inheritance of the basic properties of the differential problem and lead to the concept of an SM-stable difference scheme. An earlier distinguished class of SM-stable schemes consists of the schemes based on various Padé approximations. The computer implementation of such higher order accurate schemes deserves special consideration because certain matrix polynomials must be inverted at each new time level. Factorized SM-stable difference schemes are constructed that can be interpreted as diagonally implicit Runge-Kutta methods.     

Local refinement techniques for elliptic problems on cell-centered grids; II. Optimal order two-grid iterative methods

Two preconditioning techniques for solving difference equations arising in finite difference approximation of elliptic problems on cell-centered grids are studied. It is proven that the BEPS and the FAC preconditioners are spectrally equivalent to the corresponding finite difference schemes, including a nonsymmetric one, which is of higher-order accuracy. Numerical experiments that demonstrate the fast convergence of the preconditioned iterative methods (CG and GCG-LS in the nonsymmetric case) are presented.     

Implicit difference methods for Hamilton-Jacobi functional differential equations

Classical solutions of initial boundary value problems are approximated by solutions of associated implicit difference functional equations. A stability result is proved by using a comparison technique with nonlinear estimates of the Perron type for given functions. The Newton method is used to numerically solve nonlinear equations generated by implicit difference schemes. It is shown that there are implicit difference schemes which are convergent whereas the corresponding explicit difference methods are not. The results obtained can be applied to differential integral problems and differential equations with deviated variables.     

Two-level finite difference scheme of improved accuracy order for time-dependent problems of mathematical physics

In the theory of finite difference schemes, the most complete results concerning the accuracy of approximate solutions are obtained for two- and three-level finite difference schemes that converge with the first and second order with respect to time. When the Cauchy problem is numerically solved for a system of ordinary differential equations, higher order methods are often used. Using a model problem for a parabolic equation as an example, general requirements for the selection of the finite difference approximation with respect to time are discussed. In addition to the unconditional stability requirements, extra performance criteria for finite difference schemes are presented and the concept of SM stability is introduced. Issues concerning the computational implementation of schemes having higher approximation orders are discussed. From the general point of view, various classes of finite difference schemes for time-dependent problems of mathematical physics are analyzed.     

Parameterized families of finite difference schemes for the wave equation

This article is devoted to an analysis of simple families of finite difference schemes for the wave equation. These families are dependent on several free parameters, and methods for obtaining stability bounds as a function of these parameters are discussed in detail. Access to explicit stability bounds such as those derived here may, it is hoped, lead to optimization techniques for so‐called spectral‐like methods, which are difference schemes dependent on many free parameters (and for which maximizing the order of accuracy may not be the defining criterion). Though the focus is on schemes for the wave equation in one dimension, the analysis techniques are extended to two dimensions; implicit schemes such as ADI methods are examined in detail. Numerical results are presented. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 463–480, 2004.     

Some Stability Inequalities for Compact Finite Difference Schemes

For finite difference schemes of compact form on nonuniform grids approximating m-th order two-point boundary value problems stability inequalities are proved which use a norm analogous to the Spijker-norm in the case of multistep methods. The results are applied to a number of finite difference schemes for which they establish a higher order of convergence than naively expected.