黏弹性方程的Wilson元收敛性分析

对一类黏弹性方程利用Wilson元提出新的半离散和全离散逼近格式.基于单元的性质,通过定义新的双线性型,在不需要外推和插值后处理技术的前提下,分别得到了比传统的H~1-范数更大的模意义下相应的O(h~2)阶和O(h~2+τ~2)阶的误差分析结果,正好比通常的关于Wilson元的误差估计高出一阶.这里,h,τ表示空间剖分参数和时间步长.     

抛物方程的基于BB型对偶剖分的有限体积元法

本文讨论了抛物方程的基于三角形剖分和BB型对偶剖分的有限体积元法,给出了半离散及全离散有限体积元格式的最佳阶L2和H1误差估计.     

滩海地区运移聚集的精细数值模拟和分析

对滩海地区三层油资源运移聚集进行高精度精细平行数值模拟,提出数学模型和精细平行算子分裂隐式迭代格式,设计了并行计算程序,提出了并行计算的信息传递和交替方向网格剖分方法. 并对不同的CPU组合进行并行计算和分析,对滩海地区数值模拟结果和实际情况吻合.对模型问题进行数值分析,得到最佳阶误差估计,成功地解决了这一困难问题．     

Hamilton系统下基于相位误差的精细辛算法

Hamilton系统是一类重要的动力系统，辛算法（如生成函数法、SRK法、SPRK法、多步法等）是针对Hamilton系统所设计的具有保持相空间辛结构不变或保Hamilton函数不变的算法.但是，时域上，同阶的辛算法与Runge-Kutta法具有相同的数值精度，即辛算法在计算过程中也存在相位误差，导致时域上解的数值精度不高.经过长时间计算后，计算结果在时域上也会变得“面目全非”.为了提高辛算法在时域上解的精度，将精细算法引入到辛差分格式中，提出了基于相位误差的精细辛算法（HPD-symplectic method），这种算法满足辛格式的要求，因此在离散过程中具有保Hamilton系统辛结构的优良特性.同时，由于精细化时间步长，极大地减小了辛算法的相位误差，大幅度提高了时域上解的数值精度，几乎可以达到计算机的精度，误差为O（10-13）.对于高低混频系统和刚性系统，常规的辛算法很难在较大的步长下同时实现对高低频精确仿真，精细辛算法通过精细计算时间步长，在大步长情况下，没有额外增加计算量，实现了高低混频的精确仿真.数值结果验证了此方法的有效性和可靠性.     

可混溶驱动问题的超收敛性

本文讨论多孔介质中两相可混溶渗流驱动问题的有限元方法，采用一致网格剖分、指标为k的Raviart－Thomas空间对压力作混合有限元逼近，用正则剖分、逼近阶为l的标准有限元方法处理浓度方程，通过核函数对有限元解作卷积进行局部平均确定非线性项的系数，得到了浓度误差H1范数的超收敛估计，经高阶插值，得到了整体高精度的逼近．     

一维非线性抛物问题两层网格有限体积元逼近

该文主要研究一维非线性抛物问题两层网格有限体积元逼近.对一维非线性抛物问题有限体积元解的存在性进行了讨论,给出了最优阶L~2-模和H~1-模误差估计结果,并研究了其两层网格算法.证明了当粗细网格步长满足h=O(H~2)时两层网格算法具有最优阶H~1-模误差估计.数值算例验证了理论结果.     

四阶强阻尼波动方程的混合控制体积法

本文利用混合控制体积方法在三角网格剖分下求解四阶强阻尼波动方程.通过使用最低阶Raviart-Thomas混合有限元空间和引入迁移算子把解函数空间映射成试探函数空间,构造了半离散和全离散的混合控制体积格式,得到了最优阶误差估计.     

各向异性网格下的双三次Hermite元的超逼近分析

研究了四阶重调和问题在各向异性网格下的双三次Hermite元的有限元方法.通过引入新的思路与技巧,得到了与传统的正则剖分下完全相同的收敛性和超逼近结果.并且给出了相应的数值算例,验证了理论分析的正确性.其结果说明传统有限元分析中的剖分正则性条件不是必要的,从而对进一步设计四阶问题的自适应算法和后验估计具有参考价值.     

三维热传导型半导体问题的差分方法和分析

对2类边值问题提出特征差分格式,利用粗细网块组合、乘积型叁二次插值,处理边界的时间变步长方法、变分形式、先验估计理论和技巧,得到了最佳阶I²误差估计.     

关于一个二元B样条基

连接矩形网剖分中每一矩形的两条对角线得到一个三角剖分,将它记为△_mn。当k≥3时,△_mn上不存在k—1阶光滑度的分片k次非平凡局部支集二元样条函数,所以本文给出了均匀剖分下的具有最小对称支集的二元二次一阶光滑度的B样条基。此外,作为一元样条的Marsden恒等式的推广,我们还得到了二元样条的相应形式以及其它一些恒等式。利用这些恒等式,我们在整个剖分△_mn的二次C¹样条函数空间上建立逼近误差估计以及相应的渐近公式。     

VON NEUMANN STABILITY ANALYSIS OF SYMPLECTIC INTEGRATORS APPLIED TO HAMILTONIAN PDEs

Symplectic integration of separable Hamiltonian ordinary and partial differential equations is discussed. A von Neumann analysis is performed to achieve general linear stability criteria for symplectic methods applied to a restricted class of Hamiltonian PDEs. In this treatment, the symplectic step is performed prior to the spatial step, as opposed to the standard approach of spatially discretising the PDE to form a system of Hamiltonian ODEs to which a symplectic integrator can be applied. In this way stability criteria are achieved by considering the spectra of linearised Hamiltonian PDEs rather than spatial step size.     

Implicit Solution of Hyperbolic Equations with Space-Time Adaptivity

Adaptivity in space and time is introduced to control the error in the numerical solution of hyperbolic partial differential equations. The equations are discretised by a finite volume method in space and an implicit linear multistep method in time. The computational grid is refined in blocks. At the boundaries of the blocks, there may be jumps in the step size. Special treatment is needed there to ensure second order accuracy and stability. The local truncation error of the discretisation is estimated and is controlled by changing the step size and the time step. The global error is obtained by integration of the error equations. In the implicit scheme, the system of linear equations at each time step is solved iteratively by the GMRES method. Numerical examples executed on a parallel computer illustrate the method.     

Geometric integration of non-autonomous linear Hamiltonian problems

Symplectic integration of autonomous Hamiltonian systems is a well-known field of study in geometric numerical integration, but for non-autonomous systems the situation is less clear, since symplectic structure requires an even number of dimensions. We show that one possible extension of symplectic methods in the autonomous setting to the non-autonomous setting is obtained by using canonical transformations. Many existing methods fit into this framework. We also perform experiments which indicate that for exponential integrators, the canonical and symmetric properties are important for good long time behaviour. In particular, the theoretical and numerical results support the well documented fact from the literature that exponential integrators for non-autonomous linear problems have superior accuracy compared to general ODE schemes.     

On finite element methods for convection dominated phenomena

The aim of this work is to give theoretical justification of several types of finite element approximations to the initial-boundary value problems of first order linear hyperbolic equations. Our approximate scheme is obtained by the piecewise linear continuous finite element method for space variable, x, and the Euler type step by step integration method for time variable, t. An artificial viscosity technique, up-stream type methods are considered within the frame work of L²-theory. The convergence and the error estimate of the approximate solutions to the true one are discussed.     

Richardson-extrapolated sequential splitting and its application

During numerical time integration, the accuracy of the numerical solution obtained with a given step size often proves unsatisfactory. In this case one usually reduces the step size and repeats the computation, while the results obtained for the coarser grid are not used. However, we can also combine the two solutions and obtain a better result. This idea is based on the Richardson extrapolation, a general technique for increasing the order of an approximation method. This technique also allows us to estimate the absolute error of the underlying method. In this paper we apply Richardson extrapolation to the sequential splitting, and investigate the performance of the resulting scheme on several test examples.     

New open modified Newton Cotes type formulae as multilayer symplectic integrators

New modified open Newton Cotes integrators are introduced in this paper. For the new proposed integrators the connection between these new algorithms, differential methods and symplectic integrators is studied. Much research has been done on one step symplectic integrators and several of them have obtained based on symplectic geometry. However, the research on multistep symplectic integrators is very poor. Zhu et al. [1] studied the well known open Newton Cotes differential methods and they presented them as multilayer symplectic integrators. Chiou and Wu [2] studied the development of multistep symplectic integrators based on the open Newton Cotes integration methods. In this paper we introduce a new open modified numerical method of Newton Cotes type and we present it as symplectic multilayer structure. The new obtained symplectic schemes are applied for the solution of Hamilton’s equations of motion which are linear in position and momentum. An important remark is that the Hamiltonian energy of the system remains almost constant as integration proceeds. We have applied also efficiently the new proposed method to a nonlinear orbital problem and an almost periodic orbital problem.     

Performance and numerical behavior of the second‐order scheme of precise time‐step integration for transient dynamic analysis

Spurious high‐frequency responses resulting from spatial discretization in time‐step algorithms for structural dynamic analysis have long been an issue of concern in the framework of traditional finite difference methods. Such algorithms should be not only numerically dissipative in a controllable manner, but also unconditionally stable so that the time‐step size can be governed solely by the accuracy requirement. In this article, the issue is considered in the framework of the second‐order scheme of the precise integration method (PIM). Taking the Newmark‐β method as a reference, the performance and numerical behavior of the second‐order PIM for elasto‐dynamic impact‐response problems are studied in detail. In this analysis, the differential quadrature method is used for spatial discretization. The effects of spatial discretization, numerical damping, and time step on solution accuracy are explored by analyzing longitudinal vibrations of a shock‐excited rod with rectangular, half‐triangular, and Heaviside step impact. Both the analysis and numerical tests show that under the framework of the PIM, the spatial discretization used here can provide a reasonable number of model types for any given error tolerance. In the analysis of dynamic response, an appropriate spatial discretization scheme for a given structure is usually required in order to obtain an accurate and meaningful numerical solution, especially for describing the fine details of traction responses with sharp changes. Under the framework of the PIM, the numerical damping that is often required in traditional integration schemes is found to be unnecessary, and there is no restriction on the size of time steps, because the PIM can usually produce results with machine‐like precision and is an unconditionally stable explicit method. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

A unified discontinuous Galerkin framework for time integration

We introduce a new discontinuous Galerkin approach for time integration. On the basis of the method of weighted residual, numerical quadratures are employed in the finite element time discretization to account for general nonlinear ordinary differential equations. Many different conditions, including explicit, implicit, and symplectic conditions, are enforced for the test functions in the variational analysis to obtain desirable features of the resulting time‐stepping scheme. The proposed discontinuous Galerkin approach provides a unified framework to derive various time‐stepping schemes, such as low‐order one‐step methods, Runge–Kutta methods, and multistep methods. On the basis of the proposed framework, several explicit Runge–Kutta methods of different orders are constructed. The derivation of symplectic Runge–Kutta methods has also been realized. The proposed framework allows the optimization of new schemes in terms of several characteristics, such as accuracy, sparseness, and stability. The accuracy optimization is performed on the basis of an analytical form of the error estimation function for a linear test initial value problem. Schemes with higher formal order of accuracy are found to provide more accurate solutions. We have also explored the optimization potential of sparseness, which is related to the general compressive sensing in signal/imaging processing. Two critical dimensions of the stability region, that is, maximal intervals along the imaginary and negative real axes, are employed as the criteria for stability optimization. This gives the largest Courant–Friedrichs–Lewy time steps in solving hyperbolic and parabolic partial differential equations, respectively. Numerical experiments are conducted to validate the optimized time‐stepping schemes. Copyright © 2013 John Wiley & Sons, Ltd.     

Symplectic integration of Hamiltonian wave equations

Summary The numerical integration of a wide class of Hamiltonian partial differential equations by standard symplectic schemes is discussed, with a consistent, Hamiltonian approach. We discretize the Hamiltonian and the Poisson structure separately, then form the the resulting ODE's. The stability, accuracy, and dispersion of different explicit splitting methods are analyzed, and we give the circumstances under which the best results can be obtained; in particular, when the Hamiltonian can be split into linear and nonlinear terms. Many different treatments and examples are compared.