Evaluation of Chebyshev pseudospectral methods for third order differential equations

When the standard Chebyshev collocation method is used to solve a third order differential equation with one Neumann boundary condition and two Dirichlet boundary conditions, the resulting differentiation matrix has spurious positive eigenvalues and extreme eigenvalue already reaching O(N ⁵ for N = 64. Stable time-steps are therefore very small in this case. A matrix operator with better stability properties is obtained by using the modified Chebyshev collocation method, introduced by Kosloff and Tal Ezer [3]. By a correct choice of mapping and implementation of the Neumann boundary condition, the matrix operator has extreme eigenvalue less than O(N ⁴. The pseudospectral and modified pseudospectral methods are implemented for the solution of one-dimensional third-order partial differential equations and the accuracy of the solutions compared with those by finite difference techniques. The comparison verifies the stability analysis and the modified method allows larger time-steps. Moreover, to obtain the accuracy of the pseudospectral method the finite difference methods are substantially more expensive. Also, for the small N tested, N ⩽ 16, the modified pseudospectral method cannot compete with the standard approach. This revised version was published online in August 2006 with corrections to the Cover Date.     

Finite‐difference approximation for the u(k)‐derivative with O(hM−k+1) accuracy: An analytical expression

An approximation of function u(x) as a Taylor series expansion about a point x₀ at M points x_i, ～ i = 1,2,…,M is used where x_i are arbitrary‐spaced. This approximation is a linear system for the derivatives u^(k) with an arbitrary accuracy. An analytical expression for the inverse matrix A ^?1 where A = [A_ik] = (x_i ? x₀)^k is found. A finite‐difference approximation of derivatives u^(k) of a given function u(x) at point x₀ is derived in terms of the values u(x_i). © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

The homogeneous projective transformation of general quadratic matrix equations

^** Email: vassilios.tsachouridis{at}ieee.org^*** Email: basil.kouvaritakis{at}eng.ox.ac.uk Algebraic quadratic equations are a special case of a singlegeneralized algebraic quadratic matrix equation (GQME). Hence,the importance of that equation in science and engineering isevident. This paper focus on the study of solutions of thatGQME and a unified framework for the characterization and identificationof solutions at infinity and of finite solutions of generalquadratic algebraic matrix equations is presented. The analysisis based on the concept of homogeneous projective transformationfor general polynomial systems (Morgan, 1986). In addition,a numerical error analysis for the computed solutions is providedfor the assessment of numerical accuracy, stability and conditioningof the computed solutions. The proposed framework is independentof any numerical method and therefore it can be used along withvarious possible numerical methods for the GQME solution, especiallymatrix flow-based algorithms (Chu, 1994) (e.g. continuation/homotopy,Morgan, 1989).     

具有周期振荡系数二阶椭圆型方程的双尺度高精度算法

本文考虑了一类具有周期振荡系数二阶椭圆型方程边值问题,提出了基于双尺度渐近展开式的高精度有限元算法,并给出严格的证明。     

抛物型方程的一种高精度区域分解有限差分算法

1引言 近年来,区域分解算法以可以将大型问题分解为一系列小型问题以减少计算规模及算法可高度并行实现等特点受到了人们的广泛关注.前人也做了很多很好的工作:参考文献[1]中C.N.Dawson等人提出了显一隐格式的区域分解算法,在时间层不分层的内边界点采用大步长向前-中心差分显格式及在内点采用古典隐格式,取得的精度为O(△t+h2+H3).参考文献[2]中给出了[1]中区域分解算法对于内边界点为等距分布的多子区域时的新的误差估计,使含H3误差项的系数比[1]中缩小了一倍.还将采用大步长日的saul'yev的非对称差分格式应用于内边界点,并给出了两个子区域和多个子区域情形下差分解的先验误差估计.     

High-Order Leap-Frog Based Discontinuous Galerkin Method for the Time-Domain Maxwell Equations on Non-Conforming Simplicial Meshes

A high-order leap-flog based non-dissipative discontinuous Galerkin timedomain method for solving Maxwell's equations is introduced and analyzed. The proposed method combines a centered approximation for the evaluation of fluxes at the interface between neighboring elements, with a Nth-order leap-frog time scheme. Moreover, the interpolation degree is defined at the element level and the mesh is refined locally in a non-conforming way resulting in arbitrary level hanging nodes. The method is proved to be stable under some CFL-like condition on the time step. The convergence of the semi-discrete approximation to Maxwell's equations is established rigorously and bounds on the global divergence error are provided. Numerical experiments with highorder elements show the potential of the method.     

一种新的组合模型在信用评分中的应用

针对信用评分的特点,首先对传统的Bagging算法进行改进,然后基于不同的基分类器构建多个改进的Bagging组合模型,最后根据一种全新的方式将其组合,得到双层组合模型.通过实证分析得出以下结论:一、与原Bagging算法相比,改进算法能有效降低第二类错误率;二、与单层组合模型相比,多层组合模型能进一步降低第二类错误率和损失率.     

Higher-order time integration with a local Lax–Wendroff procedure for a central scheme on an overlapping grid

We have implemented a high-order Lax–Wendroff type time integration for a central scheme on an overlapping grid for conservation law problems. Using a local iterative approach presented by Dumbser et al. (JCP, 2008) [12], we extend a local high-order spatial reconstruction on each cell to a local higher-order space–time polynomial on the cell. We rewrite the central scheme in a fully discrete form to avoid volume integration in the space–time domain. The fluxes at cell interfaces are calculated directly via integrating a higher-order space–time reconstruction of the flux. We compare this approach with the corresponding multi-stage Runge–Kutta time integration (RK). Numerical results show that the new time integration is more cost-effective.     

A highly accurate numerical solution of a biharmonic equation

The coefficients for a nine–point high–order accuracy discretization scheme for a biharmonic equation ∇ ⁴u = f(x, y) (∇² is the two–dimensional Laplacian operator) are derived. The biharmonic problem is defined on a rectangular domain with two types of boundary conditions: (1) u and ∂²u/∂n² or (2) u and ∂u/part;n (where ∂/part;n is the normal to the boundary derivative) are specified at the boundary. For both considered cases, the truncation error for the suggested scheme is of the sixth-order O(h⁶) on a square mesh (h_x = h_y = h) and of the fourth-order O(h⁴_xh²_xh²_y h⁴_y) on an unequally spaced mesh. The biharmonic equation describes the deflection of loaded plates. The advantage of the suggested scheme is demonstrated for solving problems of the deflection of rectangular plates for cases of different boundary conditions: (1) a simply supported plate and (2) a plate with built-in edges. In order to demonstrate the high–order accuracy of the method, the numerical results are compared with exact solutions. © John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 375–391, 1997