Application of Galerkin Weighted Residual Method to 2nd, 3rd and 4th order Sturm-Liouville Problems

The aim of this paper is to compute the eigenvalues for a class of linear Sturm-Liouville problems (SLE) with Dirichlet and mixed boundary conditions applying Galerkin Weighted Residual methods. We use Legendre polynomials over [0,1] as trial functions to approximate the solutions of second, third and fourth order SLE problems. We derive rigorous matrix formulations and special attention is given about how the polynomials satisfy the corresponding homogeneous form of Dirichlet boundary conditions of Sturm-Liouville problems. The obtained approximate eigenvalues are compared with the previous computational studies by various methods available in literature. Keywords: Sturm-Liouville problems, eigenvalue, Legendre polynomials, Galerkin method.

Numerical Solution of Linear Ordinary Differential Equations and Differential-Algebraic Equations by Spectral Methods

This thesis involves the implementation of spectral methods, for numerical solution of linear Ordinary Differential Equations (ODEs) and linear Differential-Algebraic Equations (DAEs). First we consider ODEs with some ordinary problems, and then, focus on those problems in which the solution function or some coefficient functions have singularities. Then, by expressing weak and strong aspects of spectral methods to solve these kinds of problems, a modified pseudo-spectral method which is more efficient than other spectral methods is suggested and tested on some examples. We extend the pseudo-spectral method to solve a system of linear ODEs and linear DAEs and compare this method with other methods such as Backward Difference Formulae (BDF), and implicit Runge-Kutta (RK) methods using some numerical examples. Furthermore, by using appropriate choice of Gauss-Chebyshev-Radau points, we will show that this method can be used to solve a linear DAE whenever some of coefficient functions have singularities by providing some examples. We also used some problems that have already been considered by some authors by finite difference methods, and compare their results with ours. Finally, we present a short survey of properties and numerical methods for solving DAE problems and then we extend the pseudo-spectral method to solve DAE problems with variable coefficient functions. Our numerical experience shows that spectral and pseudo-spectral methods and their modified versions are very promising for linear ODE and linear DAE problems with solution or coefficient functions having singularities. In section 3.2, a modified method for solving an ODE is introduced which is new work. Furthermore, an extension of this method for solving a DAE or system of ODEs which has been explained in section 4.6 of chapter four is also a new idea and has not been done by anyone previously. In all chapters, wherever we talk about ODE or DAE we mean linear

Spectral methods for limited area models

1984 Fall.Includes bibliographical references (pages143-150).This study investigates the usefulness of Chebyshev spectral methods in limited area atmospheric modeling. Basic concepts of spectral methods and properties of Chebyshev polynomials are reviewed. Chebyshev spectral methods are illustrated by applying them to the linear advection equation in one dimension. Numerical results demonstrate the high accuracy obtained compared to finite difference methods. The nonlinear shallow water equations on a bounded domain in two dimensions are then considered as a more realistic prototype model. Characteristic boundary conditions based on Reimann invariants are developed, and contrasted with wall conditions and boundary conditions based on the assumption of balanced flow. Chebyshev tau and collocation methods are developed for this model. Results from one-dimensional tests show the superiority of the characteristic conditions in most situations. Results from two-dimensional tests are also presented. Comparison of the tau and collocation methods shows that each has its own advantages and both are practical. Time differencing schemes for Chebyshev spectral methods are studied. The stability condition obtained with explicit time differencing, often thought to be "severe", is shown to be less severe than the corresponding condition for finite difference methods. Numerical results and asymptotic estimates show that time steps may in fact be limited by accuracy rather than stability, in which case simple explicit time differencing is practical and efficient. Two modified explicit schemes are reviewed, and implicit time differencing is also discussed. A Chebyshev spectral method is also used to solve the vertical structure problem associated with vertical normal mode transforms in a hydrostatic atmosphere. Numerical results demonstrate the accuracy of the method, and illustrate the aliasing which can occur unless the vertical levels at which data is supplied are carefully chosen. Vertical transforms of observed forcings of tropical wind and mass fields are presented. The results of this study indicate that Chebyshev spectral methods are a practical alternative to finite difference methods for limited area modeling, especially when high accuracy is desired. Spectral methods require less storage than finite difference methods, are more efficient when high enough accuracy is desired, and are at least as easy to program.Supported by the National Science Foundation - ATM-8207563.Supported by the Office of Naval Research - N00014-84-C-0591.Acknowledgment to the National Center for Atmospheric Research, sponsored by the National Science Foundation, for computer time

Spectral Methods for Numerical Relativity

Version published online by Living Reviews in Relativity.International audienceEquations arising in General Relativity are usually too complicated to be solved analytically and one has to rely on numerical methods to solve sets of coupled partial differential equations. Among the possible choices, this paper focuses on a class called spectral methods where, typically, the various functions are expanded onto sets of orthogonal polynomials or functions. A theoretical introduction on spectral expansion is first given and a particular emphasis is put on the fast convergence of the spectral approximation. We present then different approaches to solve partial differential equations, first limiting ourselves to the one-dimensional case, with one or several domains. Generalization to more dimensions is then discussed. In particular, the case of time evolutions is carefully studied and the stability of such evolutions investigated. One then turns to results obtained by various groups in the field of General Relativity by means of spectral methods. First, works which do not involve explicit time-evolutions are discussed, going from rapidly rotating strange stars to the computation of binary black holes initial data. Finally, the evolutions of various systems of astrophysical interest are presented, from supernovae core collapse to binary black hole mergers

Rational spectral methods for PDEs involving fractional Laplacian in unbounded domains

Many PDEs involving fractional Laplacian are naturally set in unbounded domains with underlying solutions decay very slowly, subject to certain power laws. Their numerical solutions are under-explored. This paper aims at developing accurate spectral methods using rational basis (or modified mapped Gegenbauer functions) for such models in unbounded domains. The main building block of the spectral algorithms is the explicit representations for the Fourier transform and fractional Laplacian of the rational basis, derived from some useful integral identites related to modified Bessel functions. With these at our disposal, we can construct rational spectral-Galerkin and direct collocation schemes by pre-computing the associated fractional differentiation matrices. We obtain optimal error estimates of rational spectral approximation in the fractional Sobolev spaces, and analyze the optimal convergence of the proposed Galerkin scheme. We also provide ample numerical results to show that the rational method outperforms the Hermite function approach