共查询到20条相似文献,搜索用时 31 毫秒
1.
In this Letter, we employ finite element method to study a periodic initial value problem for the coupled Schrödinger-KdV equations. For the case of one dimension, this problem is reduced to a system of ordinary differential equations by using a semi-discrete scheme. The conservation properties of this scheme, the existence and uniqueness of the discrete solutions, and error estimates are presented. In numerical experiments, the resulting system of ordinary differential equations are solved by Runge-Kutta method at each time level. The superior accuracy of this scheme is shown by comparing the numerical solutions with the exact solutions. 相似文献
2.
We describe a numerical scheme for computing time-dependent solutions of the incompressible Navier-Stokes equations in the primitive variable formulation. This scheme uses finite elements for the space discretization and operator splitting techniques for the time discretization. The resulting discrete equations are solved using specialized nonlinear optimization algorithms that are computationally efficient and have modest storage requirements. The basic numerical kernel is the preconditioned conjugate gradient method for symmetric, positive-definite, sparse matrix systems, which can be efficiently implemented on the architectures of vector and parallel processing supercomputers. 相似文献
3.
Finite Difference/Element Method for a Two-Dimensional Modified Fractional Diffusion Equation
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We present the finite difference/element method for a
two-dimensional modified fractional diffusion equation. The analysis
is carried out first for the time semi-discrete scheme, and then for
the full discrete scheme. The time discretization is based on the
$L1$-approximation for the fractional derivative terms and the
second-order backward differentiation formula for the classical
first order derivative term. We use finite element method for the
spatial approximation in full discrete scheme. We show that both the
semi-discrete and full discrete schemes are unconditionally stable
and convergent. Moreover, the optimal convergence rate is obtained.
Finally, some numerical examples are tested in the case of one and
two space dimensions and the numerical results confirm our
theoretical analysis. 相似文献
4.
This paper is concerned with a stabilized finite element method
based on two local Gauss integrations for the two-dimensional
non-stationary conduction-convection equations by using the lowest
equal-order pairs of finite elements. This method only offsets the
discrete pressure space by the residual of the simple and symmetry
term at element level in order to circumvent the inf-sup condition.
The stability of the discrete scheme is derived under some
regularity assumptions. Optimal error estimates are obtained by
applying the standard Galerkin techniques. Finally, the numerical
illustrations agree completely with the theoretical expectations. 相似文献
5.
首先,简单介绍了基于粘接元的无重叠区域分裂方法.这种方法利用变分原理,非常适合有限元近似.然后,着重讨论了这种区域分裂方法在求解不可压Navier-Stokes方程中的应用,具体包括等价变分公式的建立、通过算子分裂的时间离散、区域分裂情形下广义Stokes问题的共轭梯度迭代求解方法、空间的有限元离散.最后,以数值实验结果验证了这种区域分裂方法应用于不可压Navier-Stokes方程求解时的可靠性. 相似文献
6.
考察非饱和水流问题的模型方程,利用线性迎风有限体积元方法建立非饱和流动的守恒形式,并获得该方法形式为O(Δt+h)的误差估计,最后给出数值模拟. 相似文献
7.
8.
We consider the application of least-squares finite element models combined with spectral/hp methods for the numerical solution of viscous flow problems. The paper presents the formulation, validation, and application of a spectral/hp algorithm to the numerical solution of the Navier–Stokes equations governing two- and three-dimensional stationary incompressible and low-speed compressible flows. The Navier–Stokes equations are expressed as an equivalent set of first-order equations by introducing vorticity or velocity gradients as additional independent variables and the least-squares method is used to develop the finite element model. High-order element expansions are used to construct the discrete model. The discrete model thus obtained is linearized by Newton’s method, resulting in a linear system of equations with a symmetric positive definite coefficient matrix that is solved in a fully coupled manner by a preconditioned conjugate gradient method. Spectral convergence of the L2 least-squares functional and L2 error norms is verified using smooth solutions to the two-dimensional stationary Poisson and incompressible Navier–Stokes equations. Numerical results for flow over a backward-facing step, steady flow past a circular cylinder, three-dimensional lid-driven cavity flow, and compressible buoyant flow inside a square enclosure are presented to demonstrate the predictive capability and robustness of the proposed formulation. 相似文献
9.
A stable high-order Spectral Difference method for hyperbolic conservation laws on triangular elements 总被引:1,自引:0,他引:1
Numerical schemes using piecewise polynomial approximation are very popular for high order discretization of conservation laws. While the most widely used numerical scheme under this paradigm appears to be the Discontinuous Galerkin method, the Spectral Difference scheme has often been found attractive as well, because of its simplicity of formulation and implementation. However, recently it has been shown that the scheme is not linearly stable on triangles. In this paper we present an alternate formulation of the scheme, featuring a new flux interpolation technique using Raviart–Thomas spaces, which proves stable under a similar linear analysis in which the standard scheme failed. We demonstrate viability of the concept by showing linear stability both in the semi-discrete sense and for time stepping schemes of the SSP Runge–Kutta type. Furthermore, we present convergence studies, as well as case studies in compressible flow simulation using the Euler equations. 相似文献
10.
A numerical study is given on the spectral methods and the high order
WENO finite difference scheme for the solution of linear and nonlinear hyperbolic
partial differential equations with stationary and non-stationary singular sources.
The singular source term is represented by the $δ$-function. For the approximation
of the $δ$-function, the direct projection method is used that was proposed in [6].
The $δ$-function is constructed in a consistent way to the derivative operator. Nonlinear
sine-Gordon equation with a stationary singular source was solved with the
Chebyshev collocation method. The $δ$-function with the spectral method is highly
oscillatory but yields good results with small number of collocation points. The
results are compared with those computed by the second order finite difference
method. In modeling general hyperbolic equations with a non-stationary singular
source, however, the solution of the linear scalar wave equation with the non-stationary
singular source using the direct projection method yields non-physical
oscillations for both the spectral method and the WENO scheme. The numerical
artifacts arising when the non-stationary singular source term is considered on the
discrete grids are explained. 相似文献
11.
This paper proves the optimal estimations of a low-order spatial-temporal fully discrete method for the non-stationary Navier-Stokes Problem. In this paper, the semi-implicit scheme based on Euler method is adopted for time discretization, while the special finite volume scheme is adopted for space discretization. Specifically, the spatial discretization adopts the traditional triangle trial function pair, combined with macro element form to ensure local stability. The theoretical analysis results show that under certain conditions, the full discretization proposed here has the characteristics of local stability, and we can indeed obtain the optimal theoretic and numerical order error estimation of velocity and pressure. This helps to enrich the corresponding theoretical results. 相似文献
12.
13.
非定常Navier-Stokes方程基于完全重叠型区域分解的有限元并行算法 总被引:1,自引:0,他引:1
基于完全重叠型区域分解技巧,提出三种求解非定常Navier-Stokes方程的有限元并行算法.其基本思想是首先对空间施行完全重叠区域分解,然后各个处理器使用向后Euler格式独立并行求解关于时间t的常微分方程;对于非线性的对流项,分别采用半隐格式和全隐格式进行处理.算法中每个处理器所负责的子问题是一个全局问题,它定义在整个求解区域上,但绝大部分自由度来自其所负责的子区域,从而使得算法实现简单,通信需求少.数值算例验证了算法的有效性及其良好的并行性能. 相似文献
14.
15.
流动数值模拟中一种并行自适应有限元算法 总被引:1,自引:0,他引:1
给出了一种流动数值模拟中的基于误差估算的并行网格自适应有限元算法.首先,以初网格上获得的当地事后误差估算值为权,应用递归谱对剖分方法划分初网格,使各子域上总体误差近似相等,以解决负载平衡问题.然后以误差值为判据对各子域内网格进行独立的自适应处理.最后应用基于粘接元的区域分裂法在非匹配的网格上求解N-S方程.区域分裂情形下N-S方程有限元解的误差估算则是广义Stokes问题误差估算方法的推广.为验证方法的可靠性,给出了不可压流经典算例的数值结果. 相似文献
16.
Analysis of an Implicit Fully Discrete Local Discontinuous Galerkin Method for the Time-Fractional Kdv Equation
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Leilei Wei Yinnian He & Xindong Zhang 《advances in applied mathematics and mechanics.》2015,7(4):510-527
In this paper, we consider a fully discrete local discontinuous Galerkin (LDG)
finite element method for a time-fractional Korteweg-de Vries (KdV) equation. The
method is based on a finite difference scheme in time and local discontinuous Galerkin
methods in space. We show that our scheme is unconditionally stable and convergent
through analysis. Numerical examples are shown to illustrate the efficiency and accuracy
of our scheme. 相似文献
17.
We introduce a new high-resolution central scheme for multidimensional Hamilton–Jacobi equations. The scheme retains the simplicity of the non-oscillatory central schemes developed by C.-T. Lin and E. Tadmor (in press, SIAM J. Sci. Comput.), yet it enjoys a smaller amount of numerical viscosity, independent of 1/Δt. By letting Δt↓0 we obtain a new second-order central scheme in the particularly simple semi-discrete form, along the lines of the new semi-discrete central schemes recently introduced by the authors in the context of hyperbolic conservation laws. Fully discrete versions are obtained with appropriate Runge–Kutta solvers. The smaller amount of dissipation enables efficient integration of convection-diffusion equations, where the accumulated error is independent of a small time step dictated by the CFL limitation. The scheme is non-oscillatory thanks to the use of nonlinear limiters. Here we advocate the use of such limiters on second discrete derivatives, which is shown to yield an improved high resolution when compared to the usual limitation of first derivatives. Numerical experiments demonstrate the remarkable resolution obtained by the proposed new central scheme. 相似文献
18.
基于两重网格离散和区域分解技巧,提出三种求解非定常Navier-Stokes方程的有限元并行算法.算法的基本思想是在每一时间迭代步,在粗网格上采用Oseen迭代法求解非线性问题,在细网格上分别并行求解Oseen、Newton、Stokes线性问题以校正粗网格解.对于空间变量采用有限元离散,时间变量采用向后Euler格式离散.数值实验验证了算法的有效性. 相似文献
19.
The Locally Conservative Galerkin (LCG) Method — a Discontinuous Methodology Applied to a Continuous Framework
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Rhodri L. T. Bevan Raoul vanLoon & Perumal Nithiarasu 《advances in applied mathematics and mechanics.》2009,1(3):319-340
This paper presents a comprehensive overview of the element-wise
locally conservative Galerkin (LCG) method. The LCG method was
developed to find a method that had the advantages of the
discontinuous Galerkin methods, without the large computational and
memory requirements. The initial application of the method is
discussed, to the simple scalar transient convection-diffusion
equation, along with its extension to the Navier-Stokes equations
utilising the Characteristic Based Split (CBS) scheme. The
element-by-element solution approach removes the standard finite
element assembly necessity, with an face flux providing continuity
between these elemental subdomains. This face flux provides explicit
local conservation and can be determined via a simple small
post-processing calculation. The LCG method obtains a unique
solution from the elemental contributions through the use of simple
averaging. It is shown within this paper that the LCG method
provides equivalent solutions to the continuous (global) Galerkin
method for both steady state and transient solutions. Several
numerical examples are provided to demonstrate the abilities of the
LCG method. 相似文献