共查询到18条相似文献,搜索用时 140 毫秒
1.
2.
针对Burgers方程,采用余项修正法和欧拉公式,推导了一种新的四层高精度紧致差分隐格式,其截断误差为O(τ~2+τh~2+h~4),即当τ=O(h~2)时,格式空间具有四阶精度;然后通过数值实验验证了格式的精确性和可靠性. 相似文献
3.
4.
《系统科学与数学》2016,(4)
基于双二次元及其梯度空间,建立了抛物型积分微分方程的一种新混合有限元逼近格式.在不需要Ritz-Volterra投影的前提下,直接利用双二次元插值的高精度结果及关于时间变量的导数转移技巧,在半离散格式下,得到了原始变量u和中间变量p=▽u+integral from n=0 to t▽u(s)ds分别关于H~1模和L~2模的O(h~4)阶超逼近结果,相比插值误差估计,提高了二阶精度.与此同时,对向后Euler格式,导出了u和p分别在H~1模与L~2模意义下的O(h~4+τ)阶超逼近;对Crank-Nicolson-Galerkin格式,在L~2模意义下证明了u和p分别具有O(h~4+τ~2)和O(h~3+τ~2)阶的超逼近性质.其中,h,τ分别表示空间剖分参数和时间步长,t代表时间变量. 相似文献
5.
《数学物理学报(A辑)》2017,(5)
该文将研究二维分数阶发展型方程的正式的二阶向后微分公式(BDF)的交替方向隐式(ADI)紧致差分格式.在时间方向上用二阶向后微分公式离散一阶时间导数,积分项用二阶卷积求积公式近似,在空间方向上用四阶精度的紧致差分离散二阶空间导数得到全离散紧致差分格式.基于与卷积求积相对应的实二次型的非负性,利用能量方法研究了差分格式的稳定性和收敛性,理论结果表明紧致差分格式的收敛阶为O(k~(a+1)+h_1~4+h_2~4),其中k为时间步长,h_1和h_2分别是空间x和y方向的步长.最后,数值算例验证了理论分析的正确性. 相似文献
6.
7.
8.
提出了数值求解一维非定常对流扩散反应方程的一种高精度紧致隐式差分格式,其截断误差为O(τ~4+τ~2h~2+h~4),即格式整体具有四阶精度.差分方程在每一时间层上只用到了三个网格节点,所形成的代数方程组为三对角型,可采用追赶法进行求解,最后通过数值算例验证了格式的精确性和可靠性. 相似文献
9.
该文的主要目的是研究Extended Fisher-Kolmogorov(EFK)方程的一类低阶非协调元混合有限元方法.首先引入一个中间变量v=-△u将原方程分裂为两个二阶方程,建立了一个非协调混合元逼近格式,并通过构造一个李雅普诺夫泛函证明了半离散格式逼近解的一个先验估计并证明了解的存在唯一性.在半离散格式下,利用这个先验估计和单元的性质,证明了原始变量u和中间变量v的H~1-模意义下的最优误差估计.进一步地,借助高精度技巧得到了O(h~2)阶的超逼近性质.其次,建立了一个新的线性化的向后Euler全离散格式,通过对相容误差和非线性项采用新的分裂技术,导出了u和v的H~1-模意义下具有O(h+τ)和O(h~2+τ)的最优误差估计和超逼近结果.这里,h,τ分别表示空间剖分参数和时间步长.最后,给出了一个数值算例,计算结果验证了理论分析的正确性,该文的分析为利用非协调混合有限元研究其它四阶初边值问题提供了一个可借鉴的途径. 相似文献
10.
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. 相似文献
11.
Guang‐Hua Gao Zhi‐Zhong Sun 《Numerical Methods for Partial Differential Equations》2013,29(5):1459-1486
This is the further work on compact finite difference schemes for heat equation with Neumann boundary conditions subsequent to the paper, [Sun, Numer Methods Partial Differential Equations (NMPDE) 25 (2009), 1320–1341]. A different compact difference scheme for the one‐dimensional linear heat equation is developed. Truncation errors of the proposed scheme are O(τ2 + h4) for interior mesh point approximation and O(τ2 + h3) for the boundary condition approximation with the uniform partition. The new obtained scheme is similar to the one given by Liao et al. (NMPDE 22 (2006), 600–616), while the major difference lies in no extension of source terms to outside the computational domain any longer. Compared with ones obtained by Zhao et al. (NMPDE 23 (2007), 949–959) and Dai (NMPDE 27 (2011), 436–446), numerical solutions at all mesh points including two boundary points are computed in our new scheme. The significant advantage of this work is to provide a rigorous analysis of convergence order for the obtained compact difference scheme using discrete energy method. The global accuracy is O(τ2 + h4) in discrete maximum norm, although the spatial approximation order at the Neumann boundary is one lower than that for interior mesh points. The analytical techniques are important and can be successfully used to solve the open problem presented by Sun (NMPDE 25 (2009), 1320–1341), where analyzed theoretical convergence order of the scheme by Liao et al. (NMPDE 22 (2006), 600–616) is only O(τ2 + h3.5) while the numerical accuracy is O(τ2 + h4), and convergence order of theoretical analysis for the scheme by Zhao et al. (NMPDE 23 (2007), 949–959) is O(τ2 + h2.5), while the actual numerical accuracy is O(τ2 + h3). Following the procedure used for the new obtained difference scheme in this work, convergence orders of these two schemes can be proved rigorously to be O(τ2 + h4) and O(τ2 + h3), respectively. Meanwhile, extension to the case involving the nonlinear reaction term is also discussed, and the global convergence order O(τ2 + h4) is proved. A compact ADI difference scheme for solving two‐dimensional case is derived. Finally, several examples are given to demonstrate the numerical accuracy of new obtained compact difference schemes. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013 相似文献
12.
Dongdong He 《Numerical Algorithms》2016,72(4):1103-1117
The telegraph equation is one of the important models in many physics and engineering. In this work, we discuss the high-order compact finite difference method for solving the two-dimensional second-order linear hyperbolic equation. By using a combined compact finite difference method for the spatial discretization, a high-order alternating direction implicit method (ADI) is proposed. The method is O(τ2 + h6) accurate, where τ, h are the temporal step size and spatial size, respectively. Von Neumann linear stability analysis shows that the method is unconditionally stable. Finally, numerical examples are used to illustrate the high accuracy of the new difference scheme. 相似文献
13.
A Fast Conservative Scheme for the Space Fractional Nonlinear Schrödinger Equation with Wave Operator 下载免费PDF全文
Mustafa Almushaira & Fei Liu 《数学研究》2021,54(4):407-426
A new efficient compact difference scheme is proposed for solving a space fractional nonlinear Schrödinger equation with wave operator. The scheme is proved to conserve the total mass and total energy in a discrete sense. Using the energy method, the proposed scheme is proved to be unconditionally stable and its convergence order is shown to be of $ \mathcal{O}( h^6 + \tau^2) $ in the discrete $ L_2 $ norm with mesh size $ h $ and the time step $ \tau $. Moreover, a fast difference solver is developed to speed up the numerical computation of the scheme. Numerical experiments are given to support the theoretical analysis and to verify the efficiency, accuracy, and discrete conservation laws. 相似文献
14.
Yiping Fu 《计算数学(英文版)》2008,26(1):98-111
In this paper, two fourth-order accurate compact difference schemes are presented for solving the Helmholtz equation in two space dimensions when the corresponding wave numbers are large. The main idea is to derive and to study a fourth-order accurate compact difference scheme whose leading truncation term, namely, the O(h^4) term, is independent of the wave number and the solution of the Helmholtz equation. The convergence property of the compact schemes are analyzed and the implementation of solving the resulting linear algebraic system based on a FFT approach is considered. Numerical results are presented, which support our theoretical predictions. 相似文献
15.
16.
For solving complex flow field with multi-scale structure higher order accurate schemes are preferred. Among high order schemes
the compact schemes have higher resolving efficiency. When the compact and upwind compact schemes are used to solve aerodynamic
problems there are numerical oscillations near the shocks. The reason of oscillation production is because of non-uniform
group velocity of wave packets in numerical solutions. For improvement of resolution of the shock a parameter function is
introduced in compact scheme to control the group velocity. The newly developed method is simple. It has higher accuracy and
less stencil of grid points 相似文献
17.
曾文平 《高等学校计算数学学报》2004,26(4):378-384
In this paper, we present two classes of symplectic schemes with high order accuracy for solving four-order rod vibration equation utt uxxxx=0 via the third type generating function method. First, the equation of four order rod vibration is written into the canonical Hamilton system; second, overcoming successfully the essential difficult on the calculus of high order variations derivative, we get the semi-discretization with arbitrary order of accuracy in time direction for the PDEs by the third type generating function method. Furthermore the discretization of the related modified equation of original equation is obtained. Finally, arbitrary order accuracy symplectic schemes are obtained. Numerical results are also presented to show the effectiveness of the scheme, high order accuracy and properties of excellent long-time numerical behavior. 相似文献
18.
In this paper, a compact finite difference method is proposed for the solution of time fractional advection-dispersion equation which appears extensively in fluid dynamics. In this approach the time fractional derivative of mentioned equation is approximated by a scheme of order O(τ 2???α ), 0?<?α?<?1, and spatial derivatives are replaced with a fourth order compact finite difference scheme. We will prove the unconditional stability and solvability of proposed scheme. Also we show that the method is convergence with convergence order O(τ 2???α ?+?h 4). Numerical examples confirm the theoretical results and high accuracy of proposed scheme. 相似文献