共查询到20条相似文献,搜索用时 32 毫秒
1.
M. Vlasák V. Dolejší J. Hájek 《Numerical Methods for Partial Differential Equations》2011,27(6):1456-1482
We deal with the numerical solution of a scalar nonstationary nonlinear convection‐diffusion equation. We employ a combination of the discontinuous Galerkin finite element (DGFE) method for the space as well as time discretization. The linear diffusive and penalty terms are treated implicitly whereas the nonlinear convective term is treated by a special higher order explicit extrapolation from the previous time step, which leads to the necessity to solve only a linear algebraic problem at each time step. We analyse this scheme and derive a priori asymptotic error estimates in the L∞(L2) –norm and the L2(H1) –seminorm with respect to the mesh size h and time step τ. Finally, we present an efficient solution strategy and numerical examples verifying the theoretical results. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1456–1482, 2010 相似文献
2.
Yinnian He 《Numerical Methods for Partial Differential Equations》2009,25(5):1009-1028
We consider a combination of the standard Galerkin method and the subspace decomposition methods for the numerical solution of the two‐dimensional time‐dependent incompressible Navier‐Stokes equations with nonsmooth initial data. Because of the poor smoothness of the solution near t = 0, we use the standard Galerkin method for time interval [0, 1] and the subspace decomposition method time interval [1, ∞). The subspace decomposition method is based on the solution into the sum of a low frequency component integrated using a small time step Δt and a high frequency integrated using a larger time step pΔt with p > 1. From the H1‐stability and L2‐error analysis, we show that the subspace decomposition method can yield a significant gain in computing time. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2009 相似文献
3.
In this paper, we study a numerical scheme to solve coupled Maxwell's equations with a nonlinear conductivity. This model plays an important role in the study of type‐II superconductors. The approximation scheme is based on backward Euler discretization in time and mixed conforming finite elements in space. We will prove convergence of this scheme to the unique weak solution of the problem and develop the corresponding error estimates. As a next step, we study the stability of the scheme in the quasi‐static limit ? → 0 and present the corresponding convergence rate. Finally, we support the theory by several numerical experiments. Copyright © 2012 John Wiley & Sons, Ltd. 相似文献
4.
Jun Zhang Shimin Lin JinRong Wang 《Mathematical Methods in the Applied Sciences》2019,42(5):1596-1608
We design and analyze an efficient numerical approach to solve the coupled Schrödinger equations with space‐fractional derivative. The numerical scheme is based on leap‐frog in time direction and Fourier method in spatial direction. The advantage of the numerical scheme is that only a linear equation needs to be solved for each time step size, and we proved that the energy and mass of space‐fractional coupled Schrödinger equations (SFCSEs) are conserved in the case of full‐discrete scheme. Moreover, we also analyze the error estimate of the numerical scheme, and numerical solutions converge with the order in L2 norm. Numerical examples are illustrated to verify the theoretical results. 相似文献
5.
We study the rate of convergence of some explicit and implicit numerical schemes for the solution of a parabolic stochastic partial differential equation driven by white noise. These include the forward and backward Euler and the Crank–Nicholson schemes. We use the finite element method. We find, as expected, that the rates of convergence are substantially similar to those found for finite difference schemes, at least when the size of the time step k is on the order of the square of the size of the space step h: all the schemes considered converge at a rate on the order of h1/2+k1/4, which is known to be optimal. We also consider cases where k is much greater than h2, and find that only the backward Euler method always attains the optimal rate; other schemes, even though they are stable, can fail to convergence to the true solution if the time step is too long relative to the space step. The Crank–Nicholson scheme behaves particularly badly in this case, even though it is a higher-order method.
Mathematics Subject Classifications (2000) 60H15, 60H35, 65N30, 35R60. 相似文献
6.
Lei Zhao Zhi‐zhong Sun Jian‐ming Liu 《Numerical Methods for Partial Differential Equations》2006,22(3):744-760
In this article, we present a numerical simulation of one‐dimensional problem of quasi‐static contact with an elastic obstacle. A finite difference scheme is derived by the method of reduction of order on uniform meshes. The stability and convergence are proved. The convergence order is of O(τ2 + h2), where τ and h are the time step size and the space step size, respectively. Some numerical examples demonstrate the theoretical results. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005 相似文献
7.
This article studies a numerical solution method for a special class of continuous time linear programming problems denoted
by (SP). We will present an efficient method for finding numerical solutions of (SP). The presented method is a discrete approximation algorithm, however, the main work of computing a numerical solution in
our method is only to solve finite linear programming problems by using recurrence relations. By our constructive manner,
we provide a computational procedure which would yield an error bound introduced by the numerical approximation. We also demonstrate
that the searched approximate solutions weakly converge to an optimal solution. Some numerical examples are given to illustrate
the provided procedure. 相似文献
8.
This paper gives the detailed numerical analysis of mixed finite element method for fractional Navier-Stokes equations.The proposed method is based on the mixed finite element method in space and a finite difference scheme in time.The stability analyses of semi-discretization scheme and fully discrete scheme are discussed in detail.Furthermore,We give the convergence analysis for both semidiscrete and flly discrete schemes and then prove that the numerical solution converges the exact one with order O(h2+k),where h and k:respectively denote the space step size and the time step size.Finally,numerical examples are presented to demonstrate the effectiveness of our numerical methods. 相似文献
9.
In this article, we consider the finite element methods (FEM) for Grwünwald–Letnikov time-fractional diffusion equation, which is obtained from the standard two-dimensional diffusion equation by replacing the first-order time derivative with a fractional derivative (of order α, with 0?h r+1?+?τ2-α), where h, τ and r are the space step size, time step size and polynomial degree, respectively. A numerical example is presented to verify the order of convergence. 相似文献
10.
In this paper, an implicit fractional-step method for numerical solutions of the incompressible Navier–Stokes equations is studied. The time advancement is decomposed into a sequence of two steps, and the first step can be seen as a linear elliptic problem; on the other hand, the second step has the structure of the Stokes problem. The two problems satisfy the full homogeneous Dirichlet boundary conditions on the velocity. At the same time, we introduce a diffusion term −θΔu in all steps of the schemes. It allows to calculate by the large time step and enhance numerical stability by choosing the proper parameter values of θ. The convergence analysis and error estimates for the intermediate velocities, the end-of step velocities and the pressure solution are derived. Finally, numerical experiments show that the feasibility and effectiveness of this method. 相似文献
11.
Luoping Chen Yanping Chen Yunqing Huang 《Numerical Methods for Partial Differential Equations》2019,35(5):1676-1693
In this paper, we will investigate a two grid finite element discretization method for the semi‐linear hyperbolic integro‐differential equations by piecewise continuous finite element method. In order to deal with the semi‐linearity of the model, we use the two grid technique and derive that once the coarse and fine mesh sizes H, h satisfy the relation h = H2 for the two‐step two grid discretization method, the two grid method achieves the same convergence accuracy as the ordinary finite element method. Both theoretical analysis and numerical experiments are given to verify the results. 相似文献
12.
In this work we construct and analyze discrete artificial boundary conditions (ABCs) for different finite difference schemes to solve nonlinear Schrödinger equations. These new discrete boundary conditions are motivated by the continuous ABCs recently obtained by the potential strategy of Szeftel. Since these new nonlinear ABCs are based on the discrete ABCs for the linear problem we first review the well-known results for the linear Schrödinger equation. We present our approach for a couple of finite difference schemes, including the Crank–Nicholson scheme, the Dùran–Sanz-Serna scheme, the DuFort–Frankel method and several split-step (fractional-step) methods such as the Lie splitting, the Strang splitting and the relaxation scheme of Besse. Finally, several numerical tests illustrate the accuracy and stability of our new discrete approach for the considered finite difference schemes. 相似文献
13.
Dissipativity of the linearly implicit Euler scheme for Navier‐Stokes equations with delay 下载免费PDF全文
Wansheng Wang 《Numerical Methods for Partial Differential Equations》2017,33(6):2114-2140
In this article, we study the dissipativity of the linearly implicit Euler scheme for the 2D Navier‐Stokes equations with time delay volume forces (NSD). This scheme can be viewed as an application of the implicit Euler scheme to linearized NSD. Therefore, only a linear system is needed to solve at each time step. The main results we obtain are that this scheme is L2 dissipative for any time step size and H1 dissipative under a time‐step constraint. As a consequence, the existence of a numerical attractor of the discrete dynamical system is established. A by‐product of the dissipativity analysis of the linearly implicit Euler scheme for NSD is that the dissipativity of an implicit‐explicit scheme for the celebrated Navier‐Stokes equations that treats the volume forces term explicitly is obtained.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2114–2140, 2017 相似文献
14.
Summary. A fully discrete modified finite element nonlinear Galerkin method is presented for the two-dimensional equation of Navier-Stokes type. The spatial discretization is based on two finite element spaces XH and Xh defined on a coarse grid with grid size H and a fine grid with grid size h << H, respectively; the time discretization is based on the Euler explicit scheme with respect to the nonlinear term. We analyze the stability and convergence rate of the method. Comparing with the standard finite element Galerkin method and the nonlinear Galerkin method, this method can admit a larger time step under the same convergence rate of same order. Hence this method can save a large amount of computational time. Finally, we provide some numerical tests on this method, the standard finite element Galerkin method, and the nonlinear Galerkin method, which are in a good agreement with the theoretical analysis.Mathematics Subject Classification (2000): 35Q30, 65M60, 65N30, 76D05 相似文献
15.
In this paper, we consider the local discontinuous Galerkin (LDG) finite element method for one-dimensional linear time-fractional Tricomi-type equation (TFTTE), which is obtained from the standard one-dimensional linear Tricomi-type equation by replacing the first-order time derivative with a fractional derivative (of order α, with 1?<?α?≤?2). The proposed LDG is based on LDG finite element method for space and finite difference method for time. We prove that the method is unconditionally stable, and the numerical solution converges to the exact one with order O(h k?+?1?+?τ 2), where h, τ and k are the space step size, time step size, polynomial degree, respectively. The comparison of the LDG results with the exact solutions is made, numerical experiments reveal that the LDG is very effective. 相似文献
16.
Samir Karaa 《Numerical Functional Analysis & Optimization》2013,34(7):750-767
We consider a family of fully discrete finite element schemes for solving a viscous wave equation, where the time integration is based on the Newmark method. A rigorous stability analysis based on the energy method is developed. Optimal error estimates in both time and space are obtained. For sufficiently smooth solutions, it is demonstrated that the maximal error in the L 2-norm over a finite time interval converges optimally as O(h p+1 + Δt s ), where p denotes the polynomial degree, s = 1 or 2, h the mesh size, and Δt the time step. 相似文献
17.
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
18.
We analyze an h-p version Petrov-Galerkin finite element method for linear Volterra integrodifferential equations. We prove optimal a priori error bounds in the L 2- and H 1-norm that are explicit in the time steps, the approximation orders and in the regularity of the exact solution. Numerical experiments confirm the theoretical results. Moreover, we observe that the numerical scheme superconverges at the nodal points of the time partition. 相似文献
19.
Holger Knieling Adrian Lange Gunnar Matthies Ingo Rehberg Reinhard Richter 《PAMM》2007,7(1):4140025-4140026
We investigate the growth of a pattern of liquid crests emerging in a layer of magnetic liquid when subjected to a magnetic field oriented normally to the fluid surface. After a step like increase of the magnetic field, the temporal evolution of the pattern amplitude is measured by means of a Hall-sensor array. The extracted growth rate is compared with predictions from linear stability analysis by taking into account the nonlinear magnetization curve M (H). The remaining discrepancy can be resolved by numerical calculations via the finite element method. By starting with a finite surface perturbation it can reproduce the temporal evolution of the pattern amplitude and the growth rate. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
20.
Shiva Eshaghi Reza Khoshsiar Ghaziani Alireza Ansari 《Mathematical Methods in the Applied Sciences》2019,42(7):2302-2323
In this paper, we studied the stabilization of nonlinear regularized Prabhakar fractional dynamical systems without and with time delay. We establish a Lyapunov stabiliy theorem for these systems and study the asymptotic stability of these systems without design a positive definite function V (without considering the fractional derivative of function V is negative). We design a linear feedback controller to control and stabilize the nonautonomous and autonomous chaotic regularized Prabhakar fractional dynamical systems without and with time delay. By means of the Lyapunov stability, we obtain the control parameters for these type of systems. We further present a numerical method to solve and analyze regularized Prabhakar fractional systems. Furthermore, by employing numerical simulation, we reveal chaotic attractors and asymptotic stability behaviors for four systems to illustrate the presented theorem. 相似文献