求解奇异摄动问题的一个耦合差分格式

本文考察奇异摄动问题(1.1).在一特殊的非均匀网格上,将不稳定、二阶精度的中心差格式和稳定、一阶精度的Abrahamsson-Keller-Kreiss箱子格式相耦合,得到了一个二阶一致收敛的差分格式.最后给出了数值结果.     

一类满足熵增条件的流体力学方程守恒型格式

Lax,Wandtoff曾经证明:对于与守恒律方程组相容的守恒型差分格式,如果其差分解几乎处处有界收敛,那么极限函数是原方程组的一个弱解,并且提出了二阶精度的L-W格式.但是,一些数值计算表明,用二阶守恒型格式(如L-W格式及Mac Corma-ck格式),可能得到非物理解的计算结果.通常称满足熵条件的弱解为物理解.对     

抛物型方程基于POD方法的时间二阶中心差的时间二阶精度简化有限元格式

本文将特征正交分解(Proper Orthogonal Decomposition,简记为POD)方法应用于抛物型方程通常时间二阶中心差的时间二阶精度有限元格式(简称为通常格式),简化其为一个自由度极少但具有时间二阶精度的有限元格式,并给出简化的时间二阶中心差的时间二阶精度有限元格式(简称为简化格式)解的误差分析.数值...     

二维分数阶发展型方程的正式的二阶BDF交替方向隐式紧致差分格式

该文将研究二维分数阶发展型方程的正式的二阶向后微分公式(BDF)的交替方向隐式(ADI)紧致差分格式.在时间方向上用二阶向后微分公式离散一阶时间导数,积分项用二阶卷积求积公式近似,在空间方向上用四阶精度的紧致差分离散二阶空间导数得到全离散紧致差分格式.基于与卷积求积相对应的实二次型的非负性,利用能量方法研究了差分格式的稳定性和收敛性,理论结果表明紧致差分格式的收敛阶为O(k~(a+1)+h_1~4+h_2~4),其中k为时间步长,h_1和h_2分别是空间x和y方向的步长.最后,数值算例验证了理论分析的正确性.     

时间空间均为二阶的新型NND差分格式

张涵信的研究表明,为了避免激波前后差分解的波动,在差分格式的改型方程中三阶导数的系数在激波上游必须是正的,而在激波下游则必须是负的.据此提出了一种新型的无波动、无自由参数耗散性的差分格式,它对时间和空间都是二阶的.证明了此格式是TVD的,而且是推广的二阶Годунов格式.在处理有激波的流场时,此格式是Lax-Wendroff格式的改进和推广.给出了若干算例,计算结果表明,此格式不仅无波动,而且具有形式紧凑、应用方便、分辨率高、稳定性准则中的Courant数较大的优点.     

粘性流动有限差分计算的新策略 *

对粘性流动计算 ,提出有限离散单元流动的流体分析 (理论 )和耦合离散流体理论(CDFT)的差分格式 .利用CDFT差分格式计算Burgers方程和计算激波边界层干扰流动的数值实验表明 :对计算精度和计算效率的提高 ,CDFT格式比提高常用差分格式 (即离散流体力学方程得到的格式 )精度和改进常用格式形式等更有效 ,且运算量小 .     

非饱和土壤水流方程的CN广义差分法

首先给出二维非饱和土壤水流方程时间二阶精度的Crank-Nicolson(CN)时间半离散化格式,然后直接从CN时间半离散化格式出发,建立具有时间二阶精度的全离散化CN广义差分格式,并给出误差分析,最后用数值例子验证全离散化CN广义差分格式的优越性.这种方法能提高时间离散的精度,极大地减少时间方向的迭代步,从而减少实际计算中截断误差的积累,提高计算精度和计算效率.而且该方法可以绕开对空间变量的半离散化广义差分格式的讨论,使得理论研究更简便.     

广义sine-Gordon方程高精度隐式紧致差分方法

本文研究一类二维非线性的广义sine-Gordon(简称SG)方程的有限差分格式.首先构造三层时间的紧致交替方向隐式差分格式,并用能量分析法证明格式具有二阶时间精度和四阶空间精度.然后应用改进的Richardson外推算法将时间精度提高到四阶.最后,数值算例证实改进后的算法在空间和时间上均达到四阶精度.     

求解二阶波动方程的三次样条差分方法

有限差分法在求解二阶波动方程初边值问题过程中通常受到精度和稳定性的限制.本文对二阶波动方程的时间、空间项分别采用三次样条公式进行离散,推导出精度分别为O(τ2+h2),0(τ2+h4),O(τ4+h2)和O(τ4+h4)的四种三层隐式差分格式,以及与之相匹配的第一个时间步的同阶离散格式,并采用Fourier方法分析了格...     

抛物型方程基于POD方法的时间二阶精度CN有限元降维格式

将特征正交分解(proper orthogonal decomposition, 简记为POD) 方法应用于抛物型方程通常的时间二阶精度Crank-Nicolson (简记为CN) 有限元格式, 简化其为一个自由度极少的时间二阶精度CN 有限元降维格式, 并给出简化的时间二阶精度CN 有限元解的误差分析. 数值例子表明在简化的时间二阶精度CN 有限元解和通常的时间二阶精度CN 有限元解之间的误差足够小的情况下, 简化的时间二阶精度CN 有限元格式能大大地节省自由度, 而且时间步长可以比时间一阶精度的格式取大10 倍, 以至能更快计算到所要时刻数值解, 减少计算机计算过程的截断误差, 提高计算速度和计算精度,从而验证降维时间二阶精度CN 有限元格式用于解类似于抛物型方程的时间依赖方程是很有效的.     

A high‐order finite difference discretization strategy based on extrapolation for convection diffusion equations

We propose a new high‐order finite difference discretization strategy, which is based on the Richardson extrapolation technique and an operator interpolation scheme, to solve convection diffusion equations. For a particular implementation, we solve a fine grid equation and a coarse grid equation by using a fourth‐order compact difference scheme. Then we combine the two approximate solutions and use the Richardson extrapolation to compute a sixth‐order accuracy coarse grid solution. A sixth‐order accuracy fine grid solution is obtained by interpolating the sixth‐order coarse grid solution using an operator interpolation scheme. Numerical results are presented to demonstrate the accuracy and efficacy of the proposed finite difference discretization strategy, compared to the sixth‐order combined compact difference (CCD) scheme, and the standard fourth‐order compact difference (FOC) scheme. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 18–32, 2004.     

An efficient TVD‐WENO method for conservation laws

In this article, we present a high‐resolution hybrid scheme for solving hyperbolic conservation laws in one and two dimensions. In this scheme, we use a cheap fourth order total variation diminishing (TVD) scheme for smooth region and expensive seventh order weighted nonoscillatory (WENO) scheme near discontinuities. To distinguish between the smooth parts and discontinuities, we use an efficient adaptive multiresolution technique. For time integration, we use the third order TVD Runge‐Kutta scheme. The accuracy of the resulting hybrid high order scheme is comparable with these of WENO, but with significant decrease of the CPU cost. Numerical demonstrates that the proposed scheme is comparable to the high order WENO scheme and superior to the fourth order TVD scheme. Our scheme has the added advantage of simplicity and computational efficiency. Numerical tests are presented which show the robustness and effectiveness of the proposed scheme.© 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Integrated fast and high‐accuracy computation of convection diffusion equations using multiscale multigrid method

We present an explicit sixth‐order compact finite difference scheme for fast high‐accuracy numerical solutions of the two‐dimensional convection diffusion equation with variable coefficients. The sixth‐order scheme is based on the well‐known fourth‐order compact (FOC) scheme, the Richardson extrapolation technique, and an operator interpolation scheme. For a particular implementation, we use multiscale multigrid method to compute the fourth‐order solutions on both the coarse grid and the fine grid. Then, an operator interpolation scheme combined with the Richardson extrapolation technique is used to compute a sixth‐order accurate fine grid solution. We compare the computed accuracy and the implementation cost of the new scheme with the standard nine‐point FOC scheme and Sun–Zhang's sixth‐order method. Two convection diffusion problems are solved numerically to validate our proposed sixth‐order scheme. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Commutativity of association schemes of prime square order having non-trivial thin closed subsets

Through a study of the structure of the modular adjacency algebra over a field of positive characteristic p for a scheme of prime order p and utilizing the fact that every scheme of prime order is commutative, we show that every association scheme of prime square order having a non-trivial thin closed subset is commutative. The second author was supported by Korea Research Foundation Grant (KRF-2006-003-00008).     

High order upwind scheme for modelling turbulent shallow water flow in hydraulic structures

An unstructured finite volume model for quasi-2D free surface flow with wet-dry fronts and turbulence modelling is presented. The convective flux is discretised with either a an hybrid second-order/first-order scheme, or a fully second order scheme, both of them upwind Godunov's schemes based on Roe's average. The hybrid scheme uses a second order discretisation for the two unit discharge components, whilst keeping a first order discretisation for the water depth [2]. In such a way the numerical diffusion is much reduced, without a significant reduction on the numerical stability of the scheme, obtaining in such a way accurate and stable results. It is important to keep the numerical diffusion to a minimum level without loss of numerical stability, specially when modelling turbulent flows, because the numerical diffusion may interfere with the real turbulent diffusion. In order to avoid spurious oscillations of the free surface when the bathymetry is irregular, an upwind discretisation of the bed slope source term [4] with second order corrections is used [2]. In this way a fully second order scheme which gives an exact balance between convective flux and bed slope in the hydrostatic case is obtained. The k – ε equations are solved with either an hybrid or a second order scheme. In all the numerical simulations the importance of using a second order upwind spatial discretisation has been checked [1]. A first order scheme may give rather good predictions for the water depth, but it introduces too much numerical diffusion and therefore, it excessively smooths the velocity profiles. This is specially important when comparing different turbulence models, since the numerical diffusion introduced by a first order upwind scheme may be of the same order of magnitude as the turbulent diffusion. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical experiments on a hybrid WENO5 filter for shock‐capturing

In the present paper, a hybrid filter is introduced for high accurate numerical simulation of shock‐containing flows. The fourth‐order compact finite difference scheme is used for the spatial discretization and the third‐order Runge–Kutta scheme is used for the time integration. After each time‐step, the hybrid filter is applied on the results. The filter is composed of a linear sixth‐order filter and the dissipative part of a fifth‐order weighted essentially nonoscillatory scheme (WENO5). The classic WENO5 scheme and the WENO5 scheme with adaptive order (WENO5‐AO) are used to form the hybrid filter. Using a shock‐detecting sensor, the hybrid filter reduces to the linear sixth‐order filter in smooth regions for damping high frequency waves and reduces to the WENO5 filter at shocks in order to eliminate unwanted oscillations produced by the nondissipative spatial discretization method. The filter performance and accuracy of the results are examined through several test cases including the advection, Euler and Navier–Stokes equations. The results are compared with that of a hybrid second‐order filter and also that of the WENO5 and WENO5‐AO schemes.     

Weak order stochastic Runge–Kutta methods for commutative stochastic differential equations

A new explicit stochastic Runge–Kutta scheme of weak order 2 is proposed under a commutativity condition, which is derivative-free and which attains order 4 for ordinary differential equations. The weak order conditions are derived by utilizing multi-colored rooted tree analysis and a solution is found in a transparent way. The scheme is compared with other derivative-free and weak second order schemes in numerical experiments.     

An oscillation-free high order TVD/CBC-based upwind scheme for convection discretization

An oscillation-free high order scheme is presented for convection discretization by using the normalized-variable formulation in the finite volume framework. It adopts the cubic upwind interpolation scheme as the basic scheme so as to obtain high order accuracy in smooth solution domain. In order to avoid unphysical oscillations of numerical solutions, the present scheme is designed on the TVD (total variational diminishing) constraint and CBC (convection boundedness criterion) condition. Numerical results of several linear and nonlinear convection equations with smooth or discontinuous initial distributions demonstrate the present scheme possesses second-order accuracy, good robustness and high resolution.     

A scheme of adaptive control

A new scheme of adaptive control is proposed. This scheme does not require a priori knowledge of the structure of the plant to be controlled. The principal part of the scheme is a procedure which decides the order of the model of the plant. A criterion for the order determination is developed. Using this criterion, we can decide whether to keep the current controller or to adopt a new controller based on the information gathered during the operation of the system. The effectiveness of the scheme is illustrated by a numerical example. The Institute of Statistical Mathematics