Legendre spectral method for solving integral and integro-differential equations

A new spectral approximation of an integral based on Legendre approximation at the zeros of the first term of the residual is presented. The method is used to solve integral and integro-differential equations. The method generates approximations to the lower order derivatives of the function through successive integrations of the Legendre polynomial approximation to the highest order derivative. Numerical results are included to confirm the efficiency and accuracy of the method.     

Error analysis of the Chebyshev collocation method for linear second-order partial differential equations

The purpose of this study is to apply the Chebyshev collocation method to linear second-order partial differential equations (PDEs) under the most general conditions. The method is given with a priori error estimate which is obtained by polynomial interpolation. The residual correction procedure is modified to the problem so that the absolute error may be estimated. Finally, the effectiveness of the method is illustrated in several numerical experiments such as Laplace and Poisson equations. Numerical results are overlapped with the theoretical results.     

A multi-domain Legendre spectral collocation method for nonlinear neutral equations with piecewise continuous argument

In this paper, for the neutral equations with piecewise continuous argument, we construct a spectral collocation method by combining the shifted Legendre–Gauss–Radau interpolation and a multi-domain division. Based on the non-classical Lipschitz condition, the convergence results of the method are derived. The results show that the method can arrive at high accuracy under the suitable conditions. Several numerical examples further illustrate the obtained theoretical results and the computational effectiveness of the method.     

Discrete Legendre spectral Galerkin method for Urysohn integral equations

In this paper, we consider the discrete Legendre spectral Galerkin method to approximate the solution of Urysohn integral equation with smooth kernel. The convergence of the approximate and iterated approximate solutions to the actual solution is discussed and the rates of convergence are obtained. In particular we have shown that, when the quadrature rule is of certain degree of precision, the superconvergence rates for the iterated Legendre spectral Galerkin method are maintained in the discrete case.     

An efficient method based on the second kind Chebyshev wavelets for solving variable-order fractional convection diffusion equations

In this paper, a class of variable-order fractional convection diffusion equations have been solved with assistance of the second kind Chebyshev wavelets operational matrix. The operational matrix of variable-order fractional derivative is derived for the second kind Chebyshev wavelets. By implementing the second kind Chebyshev wavelets functions and also the associated operational matrix, the considered equations will be reduced to the corresponding Sylvester equation, which can be solved by some appropriate iterative solvers. Also, the convergence analysis of the proposed numerical method to the exact solutions and error estimation are given. A variety of numerical examples are considered to show the efficiency and accuracy of the presented technique.     

A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order

We are concerned with linear and nonlinear multi-term fractional differential equations (FDEs). The shifted Chebyshev operational matrix (COM) of fractional derivatives is derived and used together with spectral methods for solving FDEs. Our approach was based on the shifted Chebyshev tau and collocation methods. The proposed algorithms are applied to solve two types of FDEs, linear and nonlinear, subject to initial or boundary conditions, and the exact solutions are obtained for some tested problems. Numerical results with comparisons are given to confirm the reliability of the proposed method for some FDEs.     

An efficient Chebyshev-Tau spectral method for Ginzburg-Landau-Schrödinger equations

We propose an efficient time-splitting Chebyshev-Tau spectral method for the Ginzburg-Landau-Schrödinger equation with zero/nonzero far-field boundary conditions. The key technique that we apply is splitting the Ginzburg-Landau-Schrödinger equation in time into two parts, a nonlinear equation and a linear equation. The nonlinear equation is solved exactly; while the linear equation in one dimension is solved with Chebyshev-Tau spectral discretization in space and Crank-Nicolson method in time. The associated discretized system can be solved very efficiently since they can be decoupled into two systems, one for the odd coefficients, the other for the even coefficients. The associated matrices have a quasi-tridiagonal structure which allows a direction solution to be obtained. The computation cost of the method in one dimension is O(Nlog(N)) compared with that of the non-optimized one, which is O(N²). By applying the alternating direction implicit (ADI) technique, we extend this efficient method to solve the Ginzburg-Landau-Schrödinger equation both in two dimensions and in three dimensions, respectively. Numerical accuracy tests of the method in one dimension, two dimensions and three dimensions are presented. Application of the method to study the semi-classical limits of Ginzburg-Landau-Schrödinger equation in one dimension and the two-dimensional quantized vortex dynamics in the Ginzburg-Landau-Schrödinger equation are also presented.     

Accuracy improvement of a multistep splitting method for nonlinear viscous wave equations

This paper focuses on a multistep splitting method for a class of nonlinear viscous equations in two spaces, which uses second-order backward differentiation formula (BDF2) combined with approximation factorization for time integration, and second-order centred difference approximation to second derivatives for spatial discretization. By the discrete energy method, it is shown that this splitting method can attain second-order accuracy in both time and space with respect to the discrete L²- and H¹-norms. Moreover, for improving computational efficiency, we introduce a Richardson extrapolation method and obtain extrapolation solution of order four in both time and space. Numerical experiments illustrate the accuracy and performance of our algorithms.     

差分进化与有理谱方法求解奇异摄动问题

讨论一种数值求解奇异摄动问题的高精度有理谱配点法。用sinh变换的有理谱配点法使Chebyshev节点在边界层处加密,只需较少的节点即可达到较高的精度。为了获得sinh变换中边界层的宽度,设计了一个以误差最小为目标函数的无约束的非线性优化问题,并给出了求解该优化问题的差分进化算法。数值实验表明,与其他的智能算法和传统的优化算法相比,差分进化算法在sinh变换中的参数优化方面具有明显的优势。     

Chebyshev series approximation for solving differential–algebraic equations (DAEs)

The numerical solution of differential–algebraic equations (DAEs) using the Chebyshev series approximation is considered in this article. Two different problems are solved using the Chebyshev series approximation and the solutions are compared with the exact solutions. First, we calculate the power series of a given equation system and then transform it into Chebyshev series form, which gives an arbitrary order for solving the DAE numerically.     

Chebyshev pseudospectral method for wave equation with absorbing boundary conditions that does not use a first order hyperbolic system

The analysis and solution of wave equations with absorbing boundary conditions by using a related first order hyperbolic system has become increasingly popular in recent years. At variance with several methods which rely on this transformation, we propose an alternative method in which such hyperbolic system is not used. The method consists of approximation of spatial derivatives by the Chebyshev pseudospectral collocation method coupled with integration in time by the Runge-Kutta method. Stability limits on the timestep for arbitrary speed are calculated and verified numerically. Furthermore, theoretical properties of two methods by Jackiewicz and Renaut are derived, including, in particular, a result that corrects some conclusions of these authors. Numerical results that verify the theory and illustrate the effectiveness of the proposed approach are reported.     

Lanczos–Chebyshev pseudospectral methods for wave-propagation problems

The pseudospectral approach is a well-established method for studies of the wave propagation in various settings. In this paper, we report that the implementation of the pseudospectral approach can be simplified if power-series expansions are used. There is also an added advantage of an improved computational efficiency. We demonstrate how this approach can be implemented for two-dimensional (2D) models that may include material inhomogeneities. Physically relevant examples, taken from optics, are presented to show that, using collocations at Chebyshev points, the power-series approximation may give very accurate 2D soliton solutions of the nonlinear Schrödinger (NLS) equation. To find highly accurate numerical periodic solutions in models including periodic modulations of material parameters, a real-time evolution method (RTEM) is used. A variant of RTEM is applied to a system involving the copropagation of two pulses with different carrier frequencies, that cannot be easily solved by other existing methods.     

Semiconductor nanodevice simulation by multidomain spectral method with Chebyshev, prolate spheroidal and Laguerre basis functions

A new approach based on spectral method with efficient basis functions is proposed in the paper to simulate semiconductor nanodevice by solving the Schrödinger equation. The computational domain is partitioned at heterojunctions into a number of subdomains. The envelope functions in subdomains are expanded by various efficient basis functions and then patched by the BenDaniel-Duke boundary conditions to preserve exponential order of accuracy. Importantly, the consideration to choose the basis functions depends on the oscillatory characteristics of envelope functions. Three kinds of basis functions including prolate spheroidal wave functions, Chebyshev polynomials, and Laguerre-Gaussian functions are used according to the mathematical features in this work. In addition, the determinations of optimum values of scaling factor in Laguerre-Gaussian functions and bandwidth parameter in prolate spheroidal wave functions are also discussed in detail. Several quantum well examples are simulated to validate the effectiveness of the present scheme. The relative errors of energy levels achieve the order of 10⁻¹² requiring merely a few grid points.     

Incomplete Jacobian Newton method for nonlinear equations

This paper presents a new modified Newton method for nonlinear equations. This method uses a part of elements of the Jacobian matrix to obtain the next iteration point and is refereed to as the incomplete Jacobian Newton (IJN) method. The IJN method may be fit for solving large scale nonlinear equations with dense Jacobian. The conditions of linear, superlinear and quadratic convergence of the IJN method are given and the local convergence results are analyzed and proved. Some special IJN algorithms are designed and numerical experiments are given. The results show that the IJN method is promising.     

求解任意波数的三维Helmholtz方程

提出通过Adomian分解法求解任意波数的三维Helmholtz方程。通过Adomian分解法可以把三维Helmholtz微分方程转换成递归代数公式,并进一步把其边界条件转换成适用符号计算的简单代数公式。利用边界条件可以很容易得到方程的解析解表达式。Adomian分解法的主要特点在于计算简单快速,并且不需要进行线性化或离散化。最后通过数值计算以验证Adomian分解法求解任意波数下三维Helmholtz方程的有效性。数值计算结果表明：Adomian分解法的计算结果非常接近精确解,并且该方法在大波数情况下还具有良好的收敛性。     

切比契夫谱元素局部混合基函数构造

针对切比契夫谱方法,该文首次构造了两类局部混合基函数,据此发展了一种新的谱元素方法:在元素端点采用局部拉格朗日插值基,元素内部采用经调整后的切比契夫多项式。这里的两类混合基函数在计算精度上可与传统的拉格朗日基相媲美,而且元素矩阵具有稀疏特征和数据重用性。该文给出的局部混合基函数对传统的谱元素方法进行了扩充。     

Convergence of the Kleiser Schumann method for the Navier-Stokes equations

Calcolo - We analyze the Kleiser-Schumann metho for the numerical approximation of Navier-Stokes' equations with two directions of periodicity. In these directions a pseudo-spectral Galerkin...     

An incomplete factorization iterative technique for elliptic equations

This paper describes efficient iterative techniques for solving the large sparse symmetric linear systems that arise from application of finite difference approximations to self-adjoint elliptic equations. We use an incomplete factorization technique with the method of D'Yakonov type, generalized conjugate gradient and Chebyshev semi-iterative methods. We compare these methods with numerical examples. Bounds for the 4-norm of the error vector of the Chebyshev semi-iterative method in terms of the spectral radius of the iteration matrix are derived.