ADI spectral collocation methods for parabolic problems

We discuss the Crank–Nicolson and Laplace modified alternating direction implicit Legendre and Chebyshev spectral collocation methods for a linear, variable coefficient, parabolic initial-boundary value problem on a rectangular domain with the solution subject to non-zero Dirichlet boundary conditions. The discretization of the problems by the above methods yields matrices which possess banded structures. This along with the use of fast Fourier transforms makes the cost of one step of each of the Chebyshev spectral collocation methods proportional, except for a logarithmic term, to the number of the unknowns. We present the convergence analysis for the Legendre spectral collocation methods in the special case of the heat equation. Using numerical tests, we demonstrate the second order accuracy in time of the Chebyshev spectral collocation methods for general linear variable coefficient parabolic problems.     

Chebyshev collocation method for solving the radiative transfer equation

A Chebyshev collocation method is presented for solving the spherical harmonics approximation to the equation of radiative transfer in a plane-parallel, homogeneous medium. As a result of test computations performed for Rayleigh and Henyey-Greenstein phase functions, it was found that the proposed method can be used to solve transfer problems stably and accurately, even for an optically thick case, and that the accuracy of computations is greatly improved by increasing the order of Chebyshev polynomials in expansion.     

A multi-domain Chebyshev collocation method for predicting ultrasonic field parameters in complex material geometries

The use of ultrasound to measure elastic field parameters as well as to detect cracks in solid materials has received much attention, and new important applications have been developed recently, e.g., the use of laser generated ultrasound in non-destructive evaluation (NDE). To model such applications requires a realistic calculation of field parameters in complex geometries with discontinuous, layered materials. In this paper we present an approach for solving the elastic wave equation in complex geometries with discontinuous layered materials. The approach is based on a pseudospectral elastodynamic formulation, giving a direct solution of the time-domain elastodynamic equations. A typical calculation is performed by decomposing the global computational domain into a number of subdomains. Every subdomain is then mapped on a unit square using transfinite blending functions and spatial derivatives are calculated efficiently by a Chebyshev collocation scheme. This enables that the elastodynamic equations can be solved within spectral accuracy, and furthermore, complex interfaces can be approximated smoothly, hence avoiding staircasing. A global solution is constructed from the local solutions by means of characteristic variables. Finally, the global solution is advanced in time using a fourth order Runge-Kutta scheme. Examples of field prediction in discontinuous solids with complex geometries are given and related to ultrasonic NDE.     

Time-domain simulation of acoustic wave propagation and interaction with flexible structures using Chebyshev collocation method

A time-domain Chebyshev collocation (ChC) method is used to simulate acoustic wave propagation and its interaction with flexible structures in ducts. The numerical formulation is described using a two-dimensional duct noise control system, which consists of an expansion chamber and a tensioned membrane covering the side-branch cavity. Full coupling between the acoustic wave and the structural vibration of the tensioned membrane is considered in the modelling. A systematic method of solution is developed for the discretized differential equations over multiple physical domains. The time-domain ChC model is tested against analytical solutions under two conditions: one with an initial state of wave motion; the other with a time-dependent acoustic source. Comparisons with the finite-difference time-domain (FDTD) method are also made. Results show that the time-domain ChC method is highly accurate and computationally efficient for the time-dependent solution of duct acoustic problems. For illustrative purposes, the time-domain ChC method is applied to investigate the acoustic performance of three typical duct noise control devices: the expansion chamber, the quarter wavelength resonator and the drum silencer. The time-dependent simulation of the sound-structure interaction in the drum silencer reveals the delicate role of the membrane mass and tension in its sound reflection capability.     

Efficient multi-dimensional solution of PDEs using Chebyshev spectral methods

A robust methodology is presented for efficiently solving partial differential equations using Chebyshev spectral techniques. It is well known that differential equations in one dimension can be solved efficiently with Chebyshev discretizations, O(N) operations for N unknowns, however this efficiency is lost in higher dimensions due to the coupling between modes. This paper presents the “quasi-inverse“ technique (QIT), which combines optimizations of one-dimensional spectral differentiation matrices with Kronecker matrix products to build efficient multi-dimensional operators. This strategy results in O(N^2D?1) operations for N^D unknowns, independent of the form of the differential operators. QIT is compared to the matrix diagonalization technique (MDT) of Haidvogel and Zang [D.B. Haidvogel, T. Zang, The accurate solution of Poisson’s equation by expansion in Chebyshev polynomials, J. Comput. Phys. 30 (1979) 167–180] and Shen [J. Shen, Efficient spectral-Galerkin method. II. Direct solvers of second- and fourth-order equations using Chebyshev polynomials, SIAM J. Sci. Comp. 16 (1) (1995) 74–87]. While the cost for MDT and QIT are the same in two dimensions, there are significant differences. MDT utilizes an eigenvalue/eigenvector decomposition and can only be used for relatively simple differential equations. QIT is based upon intrinsic properties of the Chebyshev polynomials and is adaptable to linear PDEs with constant coefficients in simple domains. We present results for a standard suite of test problems, and discuss of the adaptability of QIT to more complicated problems.     

Compact optimal quadratic spline collocation methods for the Helmholtz equation

Quadratic spline collocation methods are formulated for the numerical solution of the Helmholtz equation in the unit square subject to non-homogeneous Dirichlet, Neumann and mixed boundary conditions, and also periodic boundary conditions. The methods are constructed so that they are: (a) of optimal accuracy, and (b) compact; that is, the collocation equations can be solved using a matrix decomposition algorithm involving only tridiagonal linear systems. Using fast Fourier transforms, the computational cost of such an algorithm is O(N² log N) on an N × N uniform partition of the unit square. The results of numerical experiments demonstrate the optimal global accuracy of the methods as well as superconvergence phenomena. In particular, it is shown that the methods are fourth-order accurate at the nodes of the partition.     

Schur-decomposition for 3D matrix equations and its application in solving radiative discrete ordinates equations discretized by Chebyshev collocation spectral method

The Schur-decomposition for three-dimensional matrix equations is developed and used to directly solve the radiative discrete ordinates equations which are discretized by Chebyshev collocation spectral method. Three methods, say, the spectral methods based on 2D and 3D matrix equation solvers individually, and the standard discrete ordinates method, are presented. The numerical results show the good accuracy of spectral method based on direct solvers. The CPU time cost comparisons against the resolutions between these three methods are made using MATLAB and FORTRAN 95 computer languages separately. The results show that the CPU time cost of Chebyshev collocation spectral method with 3D Schur-decomposition solver is the least, and almost only one thirtieth to one fiftieth CPU time is needed when using the spectral method with 3D Schur-decomposition solver compared with the standard discrete ordinates method.     

The numerical solution of problems in calculus of variation using Chebyshev finite difference method

The Chebyshev finite difference method is used for finding the solution of the ordinary differential equations which arise from problems of calculus of variations. Our approach consists of reducing the problem to a set of algebraic equations. This method can be regarded as a non-uniform finite difference scheme. Some numerical results are also given to demonstrate the validity and applicability of the presented technique. The method is easy to implement and yields very accurate results.     

Variational methods in radiative transfer problems

An efficient variational-iterative method is applied to the problem of diffuse reflection by a plane-parallel inhomogeneous atmosphere with isotropic scattering. The emergent intensity I(τ = 0; μ, μ₀) with μ = μ₀ corresponds to the maximum of an associated functional. It is, however, shown that I(τ = 0; μ, μ₀) computed by the variational method alone has relatively large errors when μ ≠ μ₀. Such deficiencies are removed by a combined variational-iterative method. The interdependence of the iterative and variational methods is also investigated. They are shown to play a complementary role to each other. The proper choice of trial functions is emphasized in light of computational efficiency and flexibility. Two distinct classes of trial functions: the polynomials, and the step functions are investigated as possible choices of trial functions. The latter choice is shown to be far more efficient in computation. Numerical results for both approximate emergent intensities and source functions are presented and found to be in good agreement with the exact solutions. Simple analytic two-step function approximations of the source function and intensities are also presented for the case of a two-layer inhomogeneous model.     

Spectral collocation methods using sine functions for a rotating Bose–Einstein condensation in optical lattices

We study spectral-Galerkin methods (SGM) and spectral collocation methods (SCM) for parameter-dependent problems, where the Fourier sine functions are used as the basis functions. When the SGM and the SCM are incorporated in the context of a Taylor predictor–inexact Newton corrector continuation algorithm for tracing solution curves of the Gross–Pitaevskii equation (GPE), they can efficiently provide accurate numerical solutions for the GPE. We show how the inexact Newton method outperforms the classical Newton method in the continuation algorithm. In our numerical experiments, the centered difference method (CDM), the SGM and SCM are exploited to compute energy levels and wave functions of a rotating Bose–Einstein condensation (BEC) and a rotating BEC in optical lattices in 2D. Sample numerical results are reported.     

A block Chebyshev–Davidson method with inner–outer restart for large eigenvalue problems

We propose a block Davidson-type subspace iteration using Chebyshev polynomial filters for large symmetric/hermitian eigenvalue problem. The method consists of three essential components. The first is an adaptive procedure for constructing efficient block Chebyshev polynomial filters; the second is an inner–outer restart technique inside a Chebyshev–Davidson iteration that reduces the computational costs related to using a large dimension subspace; and the third is a progressive filtering technique, which can fully employ a large number of good initial vectors if they are available, without using a large block size. Numerical experiments on several Hamiltonian matrices from density functional theory calculations show the efficiency and robustness of the proposed method.     

A scalable framework for the solution of stochastic inverse problems using a sparse grid collocation approach

Experimental evidence suggests that the dynamics of many physical phenomena are significantly affected by the underlying uncertainties associated with variations in properties and fluctuations in operating conditions. Recent developments in stochastic analysis have opened the possibility of realistic modeling of such systems in the presence of multiple sources of uncertainties. These advances raise the possibility of solving the corresponding stochastic inverse problem: the problem of designing/estimating the evolution of a system in the presence of multiple sources of uncertainty given limited information.A scalable, parallel methodology for stochastic inverse/design problems is developed in this article. The representation of the underlying uncertainties and the resultant stochastic dependant variables is performed using a sparse grid collocation methodology. A novel stochastic sensitivity method is introduced based on multiple solutions to deterministic sensitivity problems. The stochastic inverse/design problem is transformed to a deterministic optimization problem in a larger-dimensional space that is subsequently solved using deterministic optimization algorithms. The design framework relies entirely on deterministic direct and sensitivity analysis of the continuum systems, thereby significantly enhancing the range of applicability of the framework for the design in the presence of uncertainty of many other systems usually analyzed with legacy codes. Various illustrative examples with multiple sources of uncertainty including inverse heat conduction problems in random heterogeneous media are provided to showcase the developed framework.     

Four-point transformation methods in approximation and the smoothing problems

The main goal of the adaptive local strategy consists in reducing the complexity of computational problems. We propose a new approach to curve approximation and smoothing based on 4-point transformations or Discrete Projective Transform (DPT). In the framework of DPT, the variable point is related to three data points (accompanying points). The variable y-ordinate is expressed via the convolution of accompanying y-ordinates and weight functions that are defined as cross-ratio functions of four x-coordinates. DPT has some attractive properties (natural norming, scale invariance, threefold symmetry, and “4-point” orthogonality), which are useful in designing new algorithms. Diverse methods and algorithms based on DPT have been developed. The text was submitted by the author in English.     

静态电磁场边值问题计算方法

介绍常用的电磁场分析方法——解析法、有限差分法、有限元法.以静态电磁场边值问题为例,分别用自己编写的MATLAB有限差分程序及大型有限元分析软件ANSYS求出了数值解,并与解析法得到的精确解进行比较,得出了用数值法求解电磁场问题基本满足工程需要的结论.采用数值计算时,对如何减小计算机内存,取消冗余数据,也做了进一步的讨论.     

Solving optimization problems with mean field methods

A brief review is given for the use of feed-back artificial neural networks (ANN) to obtain good approximate solutions to combinatorial optimization problems. The key element is the mean field approximation (MFT), which differs from conventional methods and “feels” its ways towards good solutions rather than fully or partly exploring different possible solutions. The methodology, which is illustrated for the graphs bisection and knapsack problems, is easily generalized to Potts systems. The latter is related to the deformable templates method, which is illustrated with the track finding problem.

The mean field approximation is based on a variational principle, which also turns out to be very profitable when computing correlations in polymers.     

Fourier-modal methods applied to waveguide computational problems

Rigorous coupled-wave analysis (also called the Fourier-modal method) is an efficient tool for the numerical analysis of grating diffraction problems. We show that, with only a few modifications, this method can be used efficiently for the numerical analysis of aperiodic diffraction problems, including photonic crystal waveguides, Bragg mirrors, and grating couplers. We thus extend the domain of applications of grating theories.     

Semi-spectral Chebyshev method in quantum mechanics

Traditionally, finite differences and finite element methods have been by many regarded as the basic tools for obtaining numerical solutions in a variety of quantum mechanical problems emerging in atomic, nuclear and particle physics, astrophysics, quantum chemistry, etc. In recent years, however, an alternative technique based on the semi-spectral methods has focused considerable attention. The purpose of this work is first to provide the necessary tools and subsequently examine the efficiency of this method in quantum mechanical applications. Restricting our interest to time-independent two-body problems, we obtained the continuous and discrete spectrum solutions of the underlying Schrödinger or Lippmann-Schwinger equations in both, the coordinate and momentum space. In all of the numerically studied examples we had no difficulty in achieving the machine accuracy and the semi-spectral method showed exponential convergence combined with excellent numerical stability.     

On solving continuum-mechanics problems by fast multipole methods

It is shown analytically and numerically that the kernel-independent fast multipole method provides the accuracy and computational complexity of the analytical method if circular (spherical in 3D) equivalent surfaces are used.