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51.
Maximum norm error bounds of ADI and compact ADI methods for solving parabolic equations 总被引:1,自引:0,他引:1
Alternating direction implicit (ADI) schemes are computationally efficient and widely utilized for numerical approximation of the multidimensional parabolic equations. By using the discrete energy method, it is shown that the ADI solution is unconditionally convergent with the convergence order of two in the maximum norm. Considering an asymptotic expansion of the difference solution, we obtain a fourth‐order, in both time and space, approximation by one Richardson extrapolation. Extension of our technique to the higher‐order compact ADI schemes also yields the maximum norm error estimate of the discrete solution. And by one extrapolation, we obtain a sixth order accurate approximation when the time step is proportional to the squares of the spatial size. An numerical example is presented to support our theoretical results. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010 相似文献
52.
Leijie Qiao Wenlin Qiu Bo Tang 《Numerical Methods for Partial Differential Equations》2023,39(2):1333-1354
In this paper, we investigate the numerical solution of the three-dimensional (3D) nonlinear tempered fractional integrodifferential equation which is subject to the initial and boundary conditions. The backward Euler (BE) method in association with the first-order convolution quadrature rule is employed to discretize this equation for time, and the Galerkin finite element method is applied for space, which is combined with an alternating direction implicit (ADI) algorithm, in order to reduce the computational cost for solving the three-dimensional nonlocal problem. Then a fully discrete BE ADI Galerkin finite element scheme can be obtained by linearizing the non-linear term. Thereafter we prove a positive-type lemma, from which the stability and convergence of the proposed numerical scheme are derived based on the energy method. Numerical experiments are performed to verify the effectiveness of the proposed approach. 相似文献
53.
V. J. Ervin William J. Layton 《Numerical Methods for Partial Differential Equations》1999,15(1):91-110
We propose and analyze a new relaxation scheme for the iterative solution of the linear system arising from the finite difference discretization of convection–diffusion problems. For problems that are convection dominated, the (nondimensionalized) diffusion parameter ϵ is usually several orders of magnitude smaller than computationally feasible mesh widths. Thus, it is of practical importance that approximation methods not degrade for small ϵ. We give a relaxation procedure that is proven to converge uniformly in ϵ to the solution of the linear algebraic system (i.e., “robustly”). The procedure requires, at each step, the solution of one 4 × 4 linear system per mesh cell. Each 4 × 4 system can be independently solved, and the result communicated to the neighboring mesh cells. Thus, on a mesh connected processor array, the communication requirements are four local communications per iteration per mesh cell. An example is given, which illustrates the robustness of the new relaxation scheme. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 91–110, 1999 相似文献
54.
In accordance with the recent experimental research for flow visualization,theunsteady behavior of the starting period is investigated numerically for flow past bluntbodies.Finite difference methods are employed to solve the unsteady two-dimensionalincompressible Navier-Stokes equations.A short discussion is presented of explicit,implicit and ADI methods.Finally,the explicit and ADI schemes are used to study the flowfield in the starting period for flow past mountain-shaped and rectangular bodies. 相似文献
55.
A second-order unconditionally stable ADI scheme has been developed for solving three-dimensional parabolic equations. This scheme reduces three-dimensional problems to a succession of one-dimensional problems. Further, the scheme is suitable for simulating fast transient phenomena. Numerical examples show that the scheme gives an accurate solution for the parabolic equation and converges rapidly to the steady state solution. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14:159–168, 1998 相似文献
56.
Several solution acceleration techniques, used to obtain steady state CFD solutions as quickly as possible, are applied to an implicit, upwind Euler solver to evaluate their effectiveness. The implicit system is solved using either ADI or ILU and the solution acceleration techniques evaluated are quasi-Newton iteration, Jacobian freezing, multigrid and GMRES. ILU is a better preconditioner than ADI because it can use larger time steps. Adding GMRES does not always improve the convergence. However, GMRES preconditioned with ILU and multigrid can take advantage of Jacobian freezing to produce an efficient scheme that is relatively independent of grid size and grid quality. 相似文献
57.
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
58.
An ADI scheme for solving three-dimensional parabolic equations withfirst-order derivatives and variable coefficients has been developed basedon our previous papers and the idea of the modified upwind differencescheme. This ADI scheme is second-order accurate and unconditionallystable. Further, a small parameter can be chosen which makes it suitablefor simulating fast-transient phenomena or for computations on fine spatialmeshes. The method is illustrated with numerical examples. 相似文献
59.
Huan Liu Xiangcheng Zheng Hongfei Fu Hong Wang 《Numerical Methods for Partial Differential Equations》2021,37(1):818-835
In this article, we develop a Crank–Nicolson alternating direction implicit finite volume method for time‐dependent Riesz space‐fractional diffusion equation in two space dimensions. Norm‐based stability and convergence analysis are given to show that the developed method is unconditionally stable and of second‐order accuracy both in space and time. Furthermore, we develop a lossless matrix‐free fast conjugate gradient method for the implementation of the numerical scheme, which only has memory requirement and computational complexity per iteration with N being the total number of spatial unknowns. Several numerical experiments are presented to demonstrate the effectiveness and efficiency of the proposed scheme for large‐scale modeling and simulations. 相似文献
60.