求解对流扩散方程的混合三次B-样条配点法

通过对三次B-样条和三次三角B-样条基函数引入权因子[ω],给出了对流扩散方程的混合三次B-样条配点法。对对流扩散方程空间离散采用混合三次B-样条配点法和时间离散采用向前有限差分,引入参数[θ],建立差分格式。对差分格式的稳定性进行分析,得到稳定性条件。数值实验表明所构造方法的有效性,并且适当调整权因子[ω]和参数[θ]的值,可提高计算的精度。     

Reconstruction of the corrosion boundary for the Laplace equation by using a boundary collocation method

In this paper, we consider the identification of a corrosion boundary for the two-dimensional Laplace equation. A boundary collocation method is proposed for determining the unknown portion of the boundary from the Cauchy data on a part of the boundary. Since the resulting matrix equation is badly ill-conditioned, a regularized solution is obtained by employing the Tikhonov regularization technique, while the regularization parameter is provided by the generalized cross-validation criterion. Numerical examples show that the proposed method is reasonable and feasible.     

An implicit high-order discontinuous Galerkin method for steady and unsteady incompressible flows

This paper presents the latest developments of a discontinuous Galerkin (DG) method for incompressible flows introduced in [Bassi F, Crivellini A, Di Pietro DA, Rebay S. An artificial compressibility flux for the discontinuous Galerkin solution of the incompressible Navier–Stokes equations. J Comput Phys 2006;218(2):794–815] for the steady Navier–Stokes equations and extended in [Bassi F, Crivellini A. A high-order discontinuous Galerkin method for natural convection problems. In: Wesseling P, Oñate E, Periaux J, editors. Electronic proceedings of the ECCOMAS CFD 2006 conference, Egmond aan Zee, The Netherlands, September 5–8; 2006. TU Delft] to the coupled Navier–Stokes and energy equations governing natural convection flows.

The method is fully implicit and applies to the governing equations in primitive variable form. Its distinguishing feature is the formulation of the inviscid interface flux, which is based on the solution of local Riemann problems associated with the artificial compressibility perturbation of the Euler equations. The tight coupling between pressure and velocity so introduced stabilizes the method and allows using equal-order approximation spaces for both pressure and velocity. Since, independently of the amount of artificial compressibility added, the interface flux reduces to the physical one for vanishing interface jumps, the resulting method is strongly consistent.

In this paper, we present a review of the method together with two recently developed issues: (i) the high-order DG discretization of the incompressible Euler equations; (ii) the high-order implicit time integration of unsteady flows. The accuracy and versatility of the method are demonstrated by a suite of computations of steady and unsteady, inviscid and viscous incompressible flows.     

An implicit/explicit spectral method for Burgers' equation

Chebyshev spectral collocation methods for approximating the solution of Burgers' equation are defined and analyzed. Discretization in time by an implicit/explicit single step method is discussed. This method is shown to be stable under a very weak condition on the time step, for the (linear) diffusive part is dealt with implicitly. Besides, fast transform methods can be used to compute the explicit (non linear) convective term. Optimal order error estimates are established in the weighted L²-norm. The research of this author has been partially supported by the U.S. Army through its European Research Office under contract No. DAJA-84-C-0035.     

A numerical method for diffusion-convection equation using high-order difference schemes

In this paper, we propose a simple general form of high-order approximation of O(c²+ch²+h⁴) to solve the two-dimensional parabolic equation αuxx+βuyy=F(x,y,t,u,ux,uy,ut), where α and β are positive constants. We apply the compact form for solving diffusion-convection equation. The results of numerical experiments are presented and compared with analytical solutions to confirm the higher accuracy of the presented scheme.     

Chebyshev collocation method for solving singular integral equation with cosecant kernel

A collocation method based on Chebyshev polynomials is proposed for solving cosecant-type singular integral equations (SIE). For solving SIE, difficulties lie in its singular term. In order to remove singular term, we introduce Gauss–Legendre integration and integral properties of the cosecant kernel. An advantage of this method is to approximate the best uniform approximation by the best square approximation to obtain the unknown coefficients in the method. On the other hand, the convergence is fast and the accuracy is high, which is verified by the final numerical experiments compared with the existing references.     

Multi-symplectic wavelet collocation method for the nonlinear Schrödinger equation and the Camassa–Holm equation

In this paper, we develop a novel multi-symplectic wavelet collocation method for solving multi-symplectic Hamiltonian system with periodic boundary conditions. Based on the autocorrelation function of Daubechies scaling functions, collocation method is conducted for the spatial discretization. The obtained semi-discrete system is proved to have semi-discrete multi-symplectic conservation laws and semi-discrete energy conservation laws. Then, appropriate symplectic scheme is applied for time integration, which leads to full-discrete multi-symplectic conservation laws. Numerical experiments for the nonlinear Schrödinger equation and Camassa–Holm equation show the high accuracy, effectiveness and good conservation properties of the proposed method.     

Spectral collocation method for stochastic Burgers equation driven by additive noise

Almost nothing decisive has been said about collocation methods for solving SPDEs. Among the best of such SPDEs the Burgers equation shows a prototypical model for describing the interaction between the reaction mechanism, convection effect, and diffusion transport. This paper discusses spectral collocation method to reduce stochastic Burgers equation to a system of stochastic ordinary differential equations (SODEs). The resulting SODEs system is then solved by an explicit 3-stage stochastic Runge-Kutta method of strong order one. The convergence rate of Fourier collocation method for Burgers equation is also obtained. Some numerical experiments are included to show the performance of the method.     

Numerical solution of high-order linear Volterra integro-differential equations by using Taylor collocation method

The main purpose of this work is to provide a new direct numerical method for high-order linear Volterra integro-differential equations (VIDEs). An algorithm based on the use of Taylor polynomials is developed for the numerical solution of high-order linear VIDEs. It is shown that this algorithm is convergent. Numerical results are presented and comparisons are made with well-known numerical methods to prove the effectiveness of the presented algorithm.     

Implementation of high-order implicit runge-kutta methods

In this paper, we develop single-Newton iterative schemes for the solution of the stage equations of some implicit Runge-Kutta methods such as the four-stage Gauss and Radau IIA methods and the five-stage Lobatto IIIA formula. We also compare the implementation cost of these schemes with the simplified-Newton iteration and we present some numerical experiments on some well-known stiff test problems to show that the proposed iterations are reliable and efficient.     

A linearized implicit finite-difference method for solving the equal width wave equation

A linearized implicit finite-difference method is presented to find numerical solutions of the equal width wave equation. The method has been used successfully to investigate the motion of a single solitary wave, the development of the interaction of two solitary waves and an undular bore. The obtained results are compared with other numerical results in the literature. A stability analysis of the scheme is also investigated.     

欧拉方程的隐式间断有限元算法研究

针对Euler方程,设计了适合间断Galerkin有限元方法的LU-SGS、GMRES以及修正LU-SGS隐式算法。采用Roe通量以及Van Albada限制器技术实现了经典LU-SGS、GMRES算法,引入高阶项误差补偿,发展了修正LU-SGS算法。以NACA0012、RAE2822翼型为例验证分析了算法的可靠性和高效性。结果表明修正LU-SGS算法存储量较少,程序实现方便,而且计算效率是LU-SGS算法的2.5倍以上,接近于循环GMRES算法。     

Wavelets Galerkin method for solving stochastic heat equation

In this paper, a new computational method based on the second kind Chebyshev wavelets (SKCWs) together with the Galerkin method is proposed for solving a class of stochastic heat equation. For this purpose, a new stochastic operational matrix for the SKCWs is derived. A collocation method based on block pulse functions is employed to derive a general procedure for forming this matrix. The SKCWs and their operational matrices of integration and stochastic Itô-integration are used to transform the under consideration problem into the corresponding linear system of algebraic equations which can be simply solved to achieve the solution of the problem. The proposed method is very convenient for solving such problems, since the initial and boundary conditions are taken into account automatically. Moreover, the efficiency of the proposed method is shown for some concrete examples. The results reveal that the proposed method is very accurate and efficient.     

The implicit equation of a canal surface

A canal surface is an envelope of a one-parameter family of spheres. In this paper we present an efficient algorithm for computing the implicit equation of a canal surface generated by a rational family of spheres. By using Laguerre and Lie geometries, we relate the equation of the canal surface to the equation of a dual variety of a certain curve in 5-dimensional projective space. We define the μ$\mu$-basis for arbitrary dimension and give a simple algorithm for its computation. This is then applied to the dual variety, which allows us to deduce the implicit equations of the dual variety, the canal surface and any offset to the canal surface.     

自适应SW-ADI方法解反应扩散方程的并行实现

根据解反应扩散方程的自适应样条小波-交替方向(SW-ADI)方法,使用MPI、OpenMP两种并行编程模式,对串行程序进行了直接并行化,并在上海大学的高性能计算机自强2000上分别用MPI和OpenMP实现了对方程的求解。对运算结果进行了分析并给出了与串行程序相比较的并行加速比。     

Parallel implementation of the projector augmented plane wave method for charged systems

A parallel implementation of the projector augmented plane wave (PAW) method with the applications to several transition metal complexes is presented. A unique aspect of our PAW code is that it can treat both charged and neutral cluster systems. We discuss how this is achieved via accurate numerical treatment of the Coulomb Green's function with free space boundary conditions. The strategy for parallelizing the PAW code is based on distributing the plane wave basis across processors. This is a versatile approach and is implemented using a parallel three-dimensional Fast Fourier Transformation (FFT). We report parallel performance analysis of our program and of the three-dimensional FFT's and discuss large-scale parallelization issues of the PAW code. Using a series of transition metal monoxides and dioxides, as well as two iron aqueous complexes, it is shown that a free space PAW code can give structural parameters and energies in good accord with Gaussian based methods.