共查询到20条相似文献,搜索用时 15 毫秒
1.
Gürhan Gürarslan 《Applied mathematics and computation》2010,216(8):2472-2478
In this work, accurate solutions to linear and nonlinear diffusion equations were introduced. A combination of a sixth-order compact finite difference scheme in space and a low-storage third-order total variation diminishing Runge-Kutta scheme in time have been used for treatment of these equations. The computed results with the use of this technique have been compared with the exact solution to show the accuracy of it. Here, the approximate solution to the diffusion equations has been obtained easily and elegantly with neither transforming nor linearizing the equation. The present method is seen to be a very good alternative method to some existing techniques for realistic problems. 相似文献
2.
We consider streamline diffusion finite element methods applied to a singularly perturbed convection–diffusion two‐point boundary
value problem whose solution has a single boundary layer. To analyse the convergence of these methods, we rewrite them as
finite difference schemes. We first consider arbitrary meshes, then, in analysing the scheme on a Shishkin mesh, we consider
two formulations on the fine part of the mesh: the usual streamline diffusion upwinding and the standard Galerkin method.
The error estimates are given in the discrete L
∞ norm; in particular we give the first analysis that shows precisely how the error depends on the user-chosen parameter τ0 specifying the mesh. When τ0 is too small, the error becomes O(1), but for τ0 above a certain threshold value, the error is small and increases either linearly or quadratically as a function of . Numerical
tests support our theoretical results.
This revised version was published online in August 2006 with corrections to the Cover Date. 相似文献
3.
In this paper, we present fourth-order finite difference method for solving nonlinear one-dimensional Burgers’ equation. This method is unconditionally stable. The convergence analysis of the present method is studied and an upper bound for the error is derived. Numerical comparisons are made with most of the existing numerical methods for solving this equation. 相似文献
4.
An implicit unconditional stable difference scheme is presented for a kind of linear space–time fractional convection–diffusion equation. The equation is obtained from the classical integer order convection–diffusion equations with fractional order derivatives for both space and time. First-order consistency, unconditional stability, and first-order convergence of the method are proven using a novel shifted version of the classical Grünwald finite difference approximation for the fractional derivatives. A numerical example with known exact solution is also presented, and the behavior of the error is examined to verify the order of convergence. 相似文献
5.
Yuezhen Ma 《Applied mathematics and computation》2010,215(9):3408-4070
In this paper, we extend the Sun and Zhang’s [24] work on high order finite difference method, which is based on the Richardson extrapolation technique and an operator interpolation scheme for the one and two dimensional steady convection diffusion equations to the three dimensional case. Firstly, we employ a fourth order compact difference scheme to get the fourth order accurate solution on the fine and the coarse grids. Then, we use the Richardson extrapolation technique by combining the two approximate solutions to get a sixth order accurate solution on coarse grid. Finally, we apply an operator interpolation scheme to achieve the sixth order accurate solution on the fine grid. During this process, we use alternating direction implicit (ADI) method to solve the resulting linear systems. Numerical experiments are conducted to verify the accuracy and effectiveness of the present method. 相似文献
6.
In this paper, a class of finite difference method for solving two-sided space-fractional wave equation is considered. The stability and consistency of the method are discussed by means of Gerschgorin theorem and using the stability matrix analysis. Numerical solutions of some wave fractional partial differential equation models are presented. The results obtained are compared to exact solutions. 相似文献
7.
Xianyi Zeng Md Mahmudul Hasan 《Numerical Methods for Partial Differential Equations》2023,39(1):421-446
In this article we study the stability of explicit finite difference discretization of advection–diffusion equations (ADE) with arbitrary order of accuracy in the context of method of lines. The analysis first focuses on the stability of the system of ordinary differential equations that is obtained by discretizing the ADE in space and then extends to fully discretized methods in combination with explicit Runge–Kutta methods. In particular, we prove that all stable semi-discretization of the ADE leads to a conditionally stable fully discretized method as long as the time-integrator is at least first-order accurate, whereas high-order spatial discretization of the advection equation cannot yield a stable method if the temporal order is too low. In the second half of the article, the analysis and the stability results are extended to a partially dissipative wave system, which serves as a model for common practice in many fluid mechanics applications that incorporate a viscous stress in the momentum equation but no heat dissipation in the energy equation. Finally, the major theoretical predictions are verified by numerical examples. 相似文献
8.
Long Chen Yonggang Wang Jinbiao Wu 《Journal of Computational and Applied Mathematics》2008,220(1-2):712-724
A one-dimensional singularly perturbed problem with a boundary turning point is considered in this paper. Let Vh be the linear finite element space on a suitable grid . A variant of streamline diffusion finite element method is proved to be almost uniform stable in the sense that the numerical approximation uh satisfies u-uh∞C|lnε| infvhVhu-vh∞, where C is independent with the small diffusion coefficient ε and the mesh . Such stability result is applied to layer-adapted grids to obtain almost ε-uniform second order scheme for turning point problems. 相似文献
9.
10.
Jiansong Zhang Danping Yang 《Numerical Methods for Partial Differential Equations》2011,27(4):854-866
Based on the overlapping domain decomposition, an efficient parallel characteristic finite difference scheme is proposed for solving convection‐diffusion equations numerically. We give the optimal convergence order in error estimate analysis, which shows that we just need to iterate once or twice at each time level to reach the optimal convergence order. Numerical experiments also confirm the theoretical analysis. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 854–866, 2011 相似文献
11.
Dong-Yang Shi 《Applied mathematics and computation》2009,213(2):411-418
A low order anisotropic nonconforming rectangular finite element method for the convection-diffusion problem with a modified characteristic finite element scheme is studied in this paper. The O(h2) order error estimate in L2-norm with respect to the space, one order higher than the expanded characteristic-mixed finite element scheme with order O(h), and the same as the conforming case for a modified characteristic finite element scheme under regular meshes, is obtained by use of some distinct properties of the interpolation operator and the mean value technique, instead of the so-called elliptic projection, which is an indispensable tool in the convergence analysis of the previous literature. Lastly, some numerical results of the element are provided to verify our theoretical analysis. 相似文献
12.
In this paper, we construct a kind of novel finite difference (NFD) method for solving singularly perturbed reaction–diffusion problems. Different from directly truncating the high‐order derivative terms of the Taylor's series in the traditional finite difference method, we rearrange the Taylor's expansion in a more elaborate way based on the original equation to develop the NFD scheme for 1D problems. It is proved that this approach not only can highly improve the calculation accuracy but also is uniformly convergent. Then, applying alternating direction implicit technique, the newly deduced schemes are extended to 2D equations, and the uniform error estimation based on Shishkin mesh is derived, too. Finally, numerical experiments are presented to verify the high computational accuracy and theoretical prediction. 相似文献
13.
Guo‐Dong Zhang Yinnian He Yan Zhang 《Numerical Methods for Partial Differential Equations》2014,30(6):1877-1901
In this article, a streamline diffusion finite element method is proposed and analyzed for stationary incompressible magnetohydrodynamics (MHD) equations. This method is stable for any combinations of velocity, pressure, and magnet finite element spaces, without requiring Ladyzenskaja‐Babu?ka‐Brezzi (LBB) condition. The well‐posedness and convergence (at optimal error rate) of this scheme are proved in terms of some conditions. Two numerical experiments are illustrated to validate our theoretical analysis and show the streamline diffusion finite element approach is effective for solving the MHD problems. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1877–1901, 2014 相似文献
14.
Zhitao Li 《Applied mathematics and computation》2009,212(2):318-326
We consider a upwinding mixed element method for a system of first order partial differential equations resulting from the mixed formulation of a general advection diffusion problem. The system can be used to model the transport of a contaminant carried by a flow. We use the lowest order Raviart-Thomas mixed finite element space. We show the first order convergence both for concentration and concentration flux in L2(Ω). 相似文献
15.
YING Long'an 《中国科学A辑(英文版)》2004,47(Z1)
We study a finite difference scheme for a combustion model problem. A projection scheme near the combustion wave, and the standard upwind finite difference scheme away from the combustion wave are applied. Convergence to weak solutions with a combustion wave is proved under the normal Courant-Friedrichs-Lewy condition. Some con- 相似文献
16.
Two-grid methods for characteristic finite volume element solutions are presented for a kind of semilinear convection-dominated diffusion equations. The methods are based on the method of characteristics, two-grid method and the finite volume element method. The nonsymmetric and nonlinear iterations are only executed on the coarse grid (with grid size H). And the fine-grid solution (with grid size h) can be obtained by a single symmetric and linear step. It is proved that the coarse grid can be much coarser than the fine grid. The two-grid methods achieve asymptotically optimal approximation as long as the mesh sizes satisfy H = O(h1/3). 相似文献
17.
由同顺 《高校应用数学学报(A辑)》2016,(4):491-500
使用Arnold等人提出的求解椭圆方程的间断有限元的一般框架及新的处理非线性对流项的方法,得到了非线性对流扩散方程的三层隐-显hp-LDG方法的误差估计.对Burgers方程进行了数值计算,计算结果验证了文中得到的理论结果. 相似文献
18.
19.
M. Mbehou 《Applicable analysis》2013,92(11):2031-2047
This paper is devoted to the study of the finite element method for a class of non-linear nonlocal diffusion problems associated with p-Laplace-type operator. Using the Euler–Galerkin finite element method, the convergence and a priori error estimates for the semi-discrete as well as fully-discrete formulations are established. 相似文献
20.
A. Serghini Mounim 《Numerical Methods for Partial Differential Equations》2012,28(6):1916-1943
A stabilized finite element method (FEM) is presented for solving the convection–diffusion equation. We enrich the linear finite element space with local functions chosen according to the guidelines of the residual‐free bubble (RFB) FEM. In our approach, the bubble part of the solution (the microscales) is approximated via an adequate choice of discontinuous bubbles allowing static condensation. This leads to a streamline‐diffusion FEM with an explicit formula for the stability parameter τK that incorporates the flow direction, has the capability to deal with problems where there is substantial variation of the Péclet number, and gives the same limit as the RFB method. The method produces the same a priori error estimates that are typically obtained with streamline‐upwind Petrov/Galerkin and RFB. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2011 相似文献