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1.
反应扩散方程的紧交替方向差分格式 总被引:9,自引:0,他引:9
本文研究二维常系数反应扩散方程的紧交替方向隐式差分格式.首先综合应用降阶法和降维法导出了紧差分格式,并给出了差分格式截断误差的表达式.其次引进过渡层变量,给出了紧交替方向隐式差分格式算法.接着用能量分析方法给出了紧交替方向隐式差分格式的解在离散H^1范数下的先验估计式,证明了差分格式的可解性、稳定性和收敛性,在离散H^1范数下收敛阶为O(r^2 H^4).然后将Rechardson外推法应用于紧交替方向隐式差分格式,外推一次得到具有O(r^4 H^6)阶精度的近似解.最后给出了数值例子,数值结果和理论结果是吻合的. 相似文献
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构造了浅水方程组的二阶精度的TVD格式。格式由简单的TVD Runge-Kutta型时间离散和有坡度限制的空间对称离散格式组成。数值耗散项用局部棱柱化河道流的特征变量构造。格式的主要优点是能够计算天然河道中浅水方程组的弱解并且构造简单。格式能够求出天然河道或非平底部渠道中的精确静水解。给出了渠道溃坝问题数值解与解析解的比较,验证格式精度高。实际天然河道型梯级水库溃坝的数值实验表明格式稳定,适应性强。 相似文献
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首先给出二维非饱和土壤水流问题基于Crank-Nicolson(CN)方法的具有时间二阶精度的半离散化格式,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有限元格式,并给出误差估计,最后用数值例子说明全离散化CN有限元格式的优越性.这种方法可以绕开关于空间变量的半离散化格式的讨论,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. 相似文献
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《数学的实践与认识》2015,(9)
首先给出二维土壤溶质输运问题时间二阶精度的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有限元格式,并给出CN有限元解的误差分析,最后用数值例子验证全离散化CN有限元格式的优越性.这种方法提高了时间离散的精度,并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.而且方法绕开对空间变量半离散化有限元格式的讨论,使得理论研究更简便. 相似文献
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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. 相似文献
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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. 相似文献
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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. 相似文献
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Our objective in this article is to present some numerical schemes for the approximation of the 2‐D Navier–Stokes equations with periodic boundary conditions, and to study the stability and convergence of the schemes. Spatial discretization can be performed by either the spectral Galerkin method or the optimum spectral non‐linear Galerkin method; time discretization is done by the Euler scheme and a two‐step scheme. Our results show that under the same convergence rate the optimum spectral non‐linear Galerkin method is superior to the usual Galerkin methods. Finally, numerical example is provided and supports our results. Copyright © 2001 John Wiley & Sons, Ltd. 相似文献
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A higher order uniformly convergent method for singularly perturbed parabolic turning point problems
Swati Yadav Pratima Rai Kapil K. Sharma 《Numerical Methods for Partial Differential Equations》2020,36(2):342-368
In this article, we study numerical approximation for a class of singularly perturbed parabolic (SPP) convection-diffusion turning point problems. The considered SPP problem exhibits a parabolic boundary layer in the neighborhood of one of the sides of the domain. Some a priori bounds are given on the exact solution and its derivatives, which are necessary for the error analysis. A numerical scheme comprising of implicit finite difference method for time discretization on a uniform mesh and a hybrid scheme for spatial discretization on a generalized Shishkin mesh is proposed. Then Richardson extrapolation method is applied to increase the order of convergence in time direction. The resulting scheme has second-order convergence up to a logarithmic factor in space and second-order convergence in time. Numerical experiments are conducted to demonstrate the theoretical results and the comparative study is done with the existing schemes in literature to show better accuracy of the proposed schemes. 相似文献
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1引言设ΩСR2为一个凸的有界开集,边界为ЭΩ;T为一个正常数.我们考虑如下基于Maxwell模型的二维粘弹性固体介质波传导问题。 相似文献
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lIEYINNIAN LIKAITAI 《高校应用数学学报(英文版)》1996,11(2):137-152
Abstract. Ogr object in this artlcle is to describe tbe Galerkln scheme and nonlin-eax Galerkin scheme for the approximation of nonlinear evolution equations, and tostudy the stability of these schemes. Spatial discretizatlon can be pedormed by eitherGalerkln spectral method or nonlinear Galerldn spectral method; time discretizatlort isdone hy Euler sin.heine wklch is explicit or implicit in the nonlinear terms. According tothe stability analysis of the above schemes, the stability of nonllneex Galerkln methodis better than that of Galexkln method. 相似文献
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S. S. Ravindran 《Numerical Functional Analysis & Optimization》2013,34(1):48-79
Extrapolated two-step backward difference (BDF2) in time and finite element in space discretization for the unsteady penetrative convection model is analyzed. Penetrative convection model employs a nonlinear equation of state making the problem more nonlinear. Optimal order error estimates are derived for the semi-discrete finite element spatial discretization. Two time discretization schemes based on linear extrapolation are proposed and analyzed, namely a coupled and a decoupled scheme. In particular, we show that although both schemes are unconditionally nonlinearly stable, the decoupled scheme converges unconditionally whereas coupled scheme requires that the time step be sufficiently small for convergence. These time discretization schemes can be implemented efficiently in practice, saving computational memory. Numerical computations and numerical convergence checks are presented to demonstrate the efficiency and the accuracy of the schemes. 相似文献
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《Journal of Computational and Applied Mathematics》2006,194(1):141-155
In this study, an implicit semi-discrete higher order compact (HOC) scheme, with an averaged time discretization, has been presented for the numerical solution of unsteady two-dimensional (2D) Schrödinger equation. The scheme is second order accurate in time and fourth order accurate in space. The results of numerical experiments are presented, and are compared with analytical solutions and well established numerical results of some other finite difference schemes. In all cases, the present scheme produces highly accurate results with much better computational efficiency. 相似文献
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The discretization of transient magneto-dynamic field problems with geometric discretization schemes such as the Finite Integration Technique or the Finite-Element Method based on Whitney form functions results in nonlinear differential-algebraic systems of equations of index 1. Their time integration with embedded s-stage singly diagonal implicit Runge–Kutta methods requires the solution of s nonlinear systems within one time step. Accelerated solution of these schemes is achieved with techniques following so-called 3R-strategies (“reuse, recycle, reduce”). This involves e.g. the solution of the linear(-ized) equations in each time step where the solution process of the iterative preconditioned conjugate gradient method reuses and recycles spectral information of linear systems from previous stages. Additionally, a combination of an error controlled spatial adaptivity and an error controlled implicit Runge–Kutta scheme is used to reduce the number of unknowns for the algebraic problems effectively and to avoid unnecessary fine grid resolutions both in space and time. First numerical results for 2D nonlinear magneto-dynamic problems validate the presented approach and its implementation. The space discretization in the numerical examples is done by Lagrangian nodal finite elements but the presented algorithms also work in combination with other discretization schemes for the Maxwell equations such as the Whitney vector finite elements. 相似文献
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Faruk Civan 《Numerical Methods for Partial Differential Equations》2009,25(2):347-379
Methodology for development of compact numerical schemes by the practical finite‐analytic method (PFAM) is presented for spatial and/or temporal solution of differential equations. The advantage and accuracy of this approach over the conventional numerical methods are demonstrated. In contrast to the tedious discretization schemes resulting from the original finite‐analytic solution methods, such as based on the separation of variables and Laplace transformation, the practical finite‐analytical method is proven to yield simple and convenient discretization schemes. This is accomplished by a special universal determinant construction procedure using the general multi‐variate power series solutions obtained directly from differential equations. This method allows for direct incorporation of the boundary conditions into the numerical discretization scheme in a consistent manner without requiring the use of artificial fixing methods and fictitious points, and yields effective numerical schemes which are operationally similar to the finite‐difference schemes. Consequently, the methods developed for numerical solution of the algebraic equations resulting from the finite‐difference schemes can be readily facilitated. Several applications are presented demonstrating the effect of the computational molecule, grid spacing, and boundary condition treatment on the numerical accuracy. The quality of the numerical solutions generated by the PFAM is shown to approach to the exact analytical solution at optimum grid spacing. It is concluded that the PFAM offers great potential for development of robust numerical schemes. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009 相似文献
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Recently, two Local Linearization (LL) schemes for the numerical integration of random differential equation have been proposed,
which differ with respect to the algorithm that is used for the numerical implementation of the Local Linear discretization.
However, in contrast with the Local Linear discretization, the order of convergence of the LL schemes have not been studied
so far. In this paper, a general theorem about this matter is presented and, on that base, additional results are derived
for each particular scheme.
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