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非定常对流扩散问题的非协调局部投影有限元方法 总被引:1,自引:1,他引:0
本文将近年来基于协调有限元逼近提出的涡旋粘性法推广应用到非协调有限元逼近,对非定常的对流占优扩散问题,空间采用非协调Crouzeix-Raviart元逼近,时间用Crank-Nicolson差分离散格式,提出了Crank-Nicolson差分-局部投影法稳定化有限元格式,我们对稳定性和误差估计给出了详细的分析,得出了最... 相似文献
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对非定常Oseen方程最优控制问题分析了一种新型L~2投影稳定化方法.空间采用工程上好用的多项式有限元P_l/P_l(l≥1)逼近,时间采用中心差分离散.该稳定化方法对速度和压力分别采用全局或局部L~2投影,不仅绕开了inf-sup条件对等阶元的束缚,而且克服了雷诺数较大,对流占优造成的解的震荡.该方法特点是,所有计算只需要在同一套网格上执行,不需要嵌套的网格或将速度和压力的梯度投影到粗网格上进行计算.给出了详细的误差分析,误差结果与雷诺数一致,且数值解的L~2误差与雷诺数无关. 相似文献
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《数学的实践与认识》2013,(16)
讨论了对流占优问题稳定化的扩展混合元数值模拟.把稳定化的思想与扩展混合元方法相结合,既可以高精度逼近未知函数,未知函数的梯度及伴随向量函数,又能保证格式的稳定性.理论分析表明,方法是有效的,具有最优L2逼近精度. 相似文献
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将Matthies,Skrzypacz和Tubiska的思想从线性的Oseen方程拓展到了非线性的Navier-Stokes方程,针对不可压缩的定常Navier-Stokes方程,提出了一种局部投影稳定化有限元方法.该方法既克服了对流占优,又绕开了inf-sup条件的限制.给出的局部投影空间既可以定义在两种不同网格上,又可以定义在相同网格上.与其他两级方法相比,定义在同一网格空间上的局部投影稳定化格式更紧凑.在同一网格上,除了给出需要bubble函数来增强的逼近空间外,还特别考虑了两种不需要用bubble函数来增强的新的空间.基于一种特殊的插值技巧,给出了稳定性分析和误差估计.最后,还列举了两个数值算例,进一步验证了理论结果的正确性. 相似文献
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本文介绍稳态对流扩散问题的稳定化有限元方法.该方法的主要难点在于,当对流占优时可能出现边界层,导致传统有限元方法在边界层内失去稳定性,从而产生剧烈振荡.在拟均匀网格下,稳定化有限元方法可分为两类:迎风型方法和指数拟合方法.前者利用对流速度的信息在变分形式中增加稳定化项,而后者利用边界层解的特征将指数函数引入到格式设计中.这两类方法对于设计电磁场等新型对流扩散问题的数值方法起到重要指导作用. 相似文献
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本文将局部投影稳定化(LPS)方法和连续时空有限元方法相结合研究对流扩散反应方程,给出稳定化连续时空有限元离散格式.与传统的时空有限元研究思路不同,时间方向利用Lagrange插值多项式,解耦时间和空间变量,降低时空有限元解的维数,具有减少计算量和简化理论分析的优点.通过引入Legendre多项式给出了有限元解的稳定性分析,进一步引进Lobatto多项式证明了有限元解的全局L∞(L2)和局部L2(Jn;LPS)范数误差估计.最后给出数值算例验证理论分析的正确性,以及稳定化格式的可行性和有效性. 相似文献
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一类非线性对流占优抛物型方程组的特征交替方向有限元方法 总被引:1,自引:0,他引:1
对一类非线性对流占优抛物型方程组建立时间离散的Patch逼近特征交替方向有限元格式 ,并给出了最优阶的L2 和H1模误差估计 . 相似文献
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Haisen Zhang 《Numerical Functional Analysis & Optimization》2013,34(6):752-776
In this article, a stochastic theta method for a reflected stochastic differential equation is proposed. When the parameter θ = 0, this method coincides with the projection Euler scheme; while when the parameter θ = 1, it is called an implicit projection Euler scheme which is first proposed in this article. Under some conditions, the strong convergence and the A-stability of this numerical scheme are proved. 相似文献
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Qianqian Ding Xiaoming He Xiaonian Long Shipeng Mao 《Numerical Methods for Partial Differential Equations》2023,39(2):1449-1477
In this paper, we develop and analyze a finite element projection method for magnetohydrodynamics equations in Lipschitz domain. A fully discrete scheme based on Euler semi-implicit method is proposed, in which continuous elements are used to approximate the Navier–Stokes equations and H ( curl ) conforming Nédélec edge elements are used to approximate the magnetic equation. One key point of the projection method is to be compatible with two different spaces for calculating velocity, which leads one to obtain the pressure by solving a Poisson equation. The results show that the proposed projection scheme meets a discrete energy stability. In addition, with the help of a proper regularity hypothesis for the exact solution, this paper provides a rigorous optimal error analysis of velocity, pressure and magnetic induction. Finally, several numerical examples are performed to demonstrate both accuracy and efficiency of our proposed scheme. 相似文献
13.
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
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Long’an Ying 《中国科学A辑(英文版)》2004,47(1):121-135
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 conditions on the ignition temperature are given to guarantee the solution containing a strong detonation wave or a weak detonation wave. Convergence to strong detonation wave solutions for the random projection method is also proved. 相似文献
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In this paper a Godunov-type projection method for computing approximate solutions of the zero Froude number (incompressible)
shallow water equations is presented. It is second-order accurate and locally conserves height (mass) and momentum. To enforce
the underlying divergence constraint on the velocity field, the predicted numerical fluxes, computed with a standard second
order method for hyperbolic conservation laws and applied to an auxiliary system, are corrected in two steps. First, a MAC-type
projection adjusts the advective velocity divergence. In a second projection step, additional momentum flux corrections are
computed to obtain new time level cell-centered velocities, which satisfy another discrete version of the divergence constraint.
The scheme features an exact and stable second projection. It is obtained by a Petrov–Galerkin finite element ansatz with
piecewise bilinear trial functions for the unknown height and piecewise constant test functions. The key innovation compared
to existing finite volume projection methods is a correction of the in-cell slopes of the momentum by the second projection.
The stability of the projection is proved using a generalized theory for mixed finite elements. In order to do so, the validity
of three different inf-sup conditions has to be shown. The results of preliminary numerical test cases demonstrate the method’s
applicability. On fixed grids the accuracy is improved by a factor four compared to a previous version of the scheme. 相似文献
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Based on a modified line search scheme, this paper presents a new derivative-free projection method for solving nonlinear monotone equations with convex constraints, which can be regarded as an extension of the scaled conjugate gradient method and the projection method. Under appropriate conditions, the global convergence and linear convergence rate of the proposed method is proven. Preliminary numerical results are also reported to show that this method is promising. 相似文献
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非定常不可压Navier-Stokes方程的高效和稳健的差分格式Ⅱ 总被引:3,自引:3,他引:0
A second order accurate implicit finite difference scheme CNMT2 is proposed in this paper for the unsteady incompressible Navier-Stokes equations. It is proved that the scheme is unconditionally nonlinearly stable on smoothly nonuniform halfstaggered meshes; this stability also holds for this scheme with the pressure correction projection method. However, it is found that the pressure correction projection method may lead to deviation problems in practical simulation of high Re flow;the reason and the cure is given in this paper in terms of differential-algebraic equations. 相似文献
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我们提出和分析了一种求解Stokes方程的数值方法.新方法基于空间上的Legendre谱离散,时间上则采用投影/方向分裂格式.更确切地说,时间离散的出发点是旋度形式的压力校正投影法,在此基础上进一步应用方向分裂法,把速度和压力方程分裂为一系列一维的椭圆型子问题.然后生成的这些一维子问题用Legendre谱方法进行空间离散.另外,我们证明了全离散格式的稳定性.一些数值实验验证了收敛性和方法的有效性. 相似文献
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The incremental projection scheme and its enhanced version, the rotational projection scheme are powerful and commonly used approaches producing efficient numerical algorithms for solving the Navier–Stokes equations. However, the much improved rotational projection scheme cannot be used on models with non-homogeneous viscosity, imposing the use of the less accurate incremental projection. This paper presents a projection method for the Navier–Stokes equations for fluids having variable viscosity, giving a consistent pressure and increased accuracy in pressure when compared to the incremental projection. The accuracy of the method will be illustrated using a manufactured solution. 相似文献
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H. Kojima 《BIT Numerical Mathematics》2016,56(4):1317-1337
Numerical integration of ordinary differential equations with some invariants is considered. For such a purpose, certain projection methods have proved its high accuracy and efficiency. Unfortunately, however, sometimes they can exhibit instability. In this paper, a new, highly efficient projection method is proposed based on explicit Runge–Kutta methods. The key there is to employ the idea of the perturbed collocation method, which gives a unified way to incorporate scheme parameters for projection. Numerical experiments confirm the stability of the proposed method. 相似文献