关于色散方程的具有高稳定性的显式差分格式

本文对色散方程u_t=au_(xxx)构造了显式差分格式J_4,其截断误差和稳定条件分别为O(τ+ h~2)和|r|≤4.0884,稳定性比[1]的结果|r|≤0.7016和[2]的结果|r|≤1.1851有很大改进,而且格式的形式也比[2]的格式简单得多.     

色散方程的四点显式差分格式

本文对色散方程u_t=au>_xxx构造了一类高稳定性的、在中间层涉及四个网格点的三层显式差分格式,其局部截断误差为O(τ+h),其稳定条件为|R|=|α|τ/h3≤0.25至|R|≤10,它们较大地改善了同类格式的稳定条件|R|≤0.25[1].     

关于色散方程u_t=au_(xxx)的两个显式差分格式

§1.前言 本文对色散方程u_t=au_(xxx)(a为常数,可正可负)构造了两个三层显式差分格式,其截断误差为O(τ十h~2)(τ=△t,h=△x),稳定条件为|r|≤0.7016,r=aτ/h~3.这个条件比[1]中显格式的最好条件|r|≤0.3849为宽,文末用数值例子验证了此点.     

色散方程ut=auxxx的一类具高稳定性的三层显式格式D3

本文提出中层点数为六点的一类三层显式格式,其截断误差为O(τh h~2),最佳稳定性条件为|R|≤4.67377。     

五维热传导方程的一族两层显格式

构造了五维热传导方程的一族两层显格式,证明了当截断误差阶为O（τ＋h2）时,其稳定性条件为网比r=hτ2≤21,优于同类的其它显格式,当截断误差阶为O（τ2＋h2）时,可以得到一个简洁而实用的二阶精度的两层显格式.     

关于色散方程u_t=au_(xxx)的一类绝对稳定的半显式格式

1.引言在[1]-[6]中讨论了色散方程u_t=au_(xxx)(a为常数,可正可负)的差分解法,但是, 显式格式的稳定性条件较苛刻,其中以[5]中提出的 H_3类显式格式最好,稳定条件为|R|=|a|τ/h~3≤1.1851;而隐式格式虽然绝对稳定且具有高精度,但每前进一步需要解一个具有五对角线的线性方程组,计算量较大. 本文针对显式格式与隐式格式存在的问题,提出一类三层绝对稳定半显式格式,其截     

色散方程两组三阶恒稳显格式

采用组合差商法对色散方程ut=auxxx(a为常数)的初边值问题,构造了两组互为对称带参数的三层显式差分格式.它们空间宽度为4,其局部截断误差为O(τ+h3),绝对稳定.而且计算时无方向性的约束,即不管a的符号如何,每一组格式均可以计算.最后给出了数值例子,数值结果表明了理论分析的正确性.     

一类具高稳定性的三层显式格式H_3

本文对色散方程u_t=au_(xxx)构造了新的三层显式格式H_3,它的稳定性条件为R=|a|τ/h~3≤1.1851,比[1]的结果R≤0.7018有较大改进.在中间层取点数不超过6的三层显式格式类中,尚未找到稳定性更好的格式.     

解色散方程Ut=auxxx绝对稳定的两层半显式格式

本文对色散方程u_i=au_(xxx)的初边值问题,构造两层半显式差分格式S_2~R,S_2~L,E_2~R,E_2~L,其截断误差分别为O(τ h~2 τ/h~2)和O(τ h τ/h~2),这些格式当参数β≥2/3时为绝对稳定的且可显式地计算。     

色散方程u_t=au_(xxx)的一类具高稳定性的三层显式格式

本文对色散方程u_1=au_(xxx)提出一类三层显式格式,它的稳定性条件为|r|=|a|△t/(△x)~3≤2.382484,比[1,2]中的|r|≤0.3849和[3]中的|r|≤0.701659以及[4]中的|r|≤1.1851有较大改进.     

带扩散项四阶抛物方程一类新的交替分组显格式

首先给出逼近带扩散项四阶抛物方程初边值问题一类非对称差分格式,利用该组非对称格式构造了一类新的交替分组显格式算法,并给出了截断误差分析和绝对稳定性结论,最后给出数值实验.     

解Schrdinger方程的绝对稳定半显式与显式差分格式

其中τ,h分别为t,x方向的步长,u_j~k为u(jh,τk)的差分逼近.尽管它们是绝对稳定的,但需解方程组.许多方便的显格式均为绝对不稳定的,如Enler格式.因此,自然要问,是否存在稳定的显格式?这个问题有理论价值,而且实用.比起隐格式,显格     

色散方程两层绝对稳定隐格式

对色散方程ut=auxxx的初边值问题,构造了两组带参数绝对稳定两层四点去心隐式差分格式,其截断误差为0（τ＋h^2）．若适当选取参数,格式的精确度可高达0（τ＋h^3）．若特殊的令某个节点前的系数为0,则得到二阶的半显格式．最后的数例验证了理论分析的正确性．这是两组灵活、实用的差分格式．     

Operator-difference schemes for a class of systems of evolution equations

For a special system of evolution equations of first order, discrete time approximations for the approximate solution of the Cauchy problem are considered. Such problems arise after the spatial approximation in the Schrödinger equation and the subsequent separation of the imaginary and real parts and in nonstationary problems of acoustics and electrodynamics. Unconditionally stable two time level operator-difference weighted schemes are constructed. The second class of difference schemes is based on the formal passage to explicit operator-difference schemes for evolution equations of second order when explicit-implicit approximation is used for isolated equations of the system. The regularization of such schemes in order to obtain unconditionally stable operator difference schemes is discussed. Splitting schemes involving the solution of simplest problems at each time step are constructed.     

高阶Schrdinger方程的一族恒稳差分格式

1 引言 高阶Schrdinger方程在量子力学、非线性光学及流体力学中有着广泛的应用,其最简单的模型方程为,即 (其中,i=(-1)~(1/2),m为正整数) (1) 文[2-3]研究了方程(1)的辛算法及差分解法。文[2]先将方程(1)写成Hamilton形     

解三维抛物型方程的一个高精度显式差分格式

提出了求解三维抛物型方程的一个高精度显式差分格式.首先,推导了一个特殊节点处一阶偏导数(■u)/(■/t)的一个差分近似表达式,利用待定系数法构造了一个显式差分格式,通过选取适当的参数使格式的截断误差在空间层上达到了四阶精度和在时间层上达到了三阶精度.然后,利用Fourier分析法证明了当r1/6时,差分格式是稳定的.最后,通过数值试验比较了差分格式的解与精确解的区别,结果说明了差分格式的有效性.     

High-order accurate monotone compact running scheme for multidimensional hyperbolic equations

Monotone absolutely stable conservative difference schemes intended for solving quasilinear multidimensional hyperbolic equations are described. For sufficiently smooth solutions, the schemes are fourth-order accurate in each spatial direction and can be used in a wide range of local Courant numbers. The order of accuracy in time varies from the third for the smooth parts of the solution to the first near discontinuities. This is achieved by choosing special weighting coefficients that depend locally on the solution. The presented schemes are numerically efficient thanks to the simple two-diagonal (or block two-diagonal) structure of the matrix to be inverted. First the schemes are applied to system of nonlinear multidimensional conservation laws. The choice of optimal weighting coefficients for the schemes of variable order of accuracy in time and flux splitting is discussed in detail. The capabilities of the schemes are demonstrated by computing well-known two-dimensional Riemann problems for gasdynamic equations with a complex shock wave structure.     

Explicit finite difference schemes with extended stability for advection equations

In this study an explicit central difference approximation of the generalized leap-frog type is applied to the one- and two-dimensional advection equations. The stability of the considered numerical schemes is investigated and the scheme with the largest stable time step is found. For the linear and nonlinear advection equations numerical experiments with different schemes from the considered class are performed in order to evaluate the practical stability of the designed schemes.     

An inverse problem of finding a source parameter in a semilinear parabolic equation

An inverse problem concerning diffusion equation with source control parameter is considered. Several finite-difference schemes are presented for identifying the control parameter. These schemes are based on the classical forward time centred space (FTCS) explicit formula, and the 5-point FTCS explicit method and the classical backward time centred space (BTCS) implicit scheme, and the Crank–Nicolson implicit method. The classical FTCS explicit formula and the 5-point FTCS explicit technique are economical to use, are second-order accurate, but have bounded range of stability. The classical BTCS implicit scheme and the Crank–Nicolson implicit method are unconditionally stable, but these schemes use more central processor (CPU) times than the explicit finite difference mehods. The basis of analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach, developed from the 1974 work of Warming and Hyett. This allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference schemes. The results of a numerical experiment are presented, and the accuracy and CPU time needed for this inverse problem are discussed.