315 research outputs found
Using a (Higher-Order) Magnus Method to Solve the Sturm-Liouville Problem
The main purpose of this paper is to describe techniques for the numerical solution of a Sturm-Liouville equation (in its Schrodinger form) by employing a Magnus expansion. With a suitable method to approximate the highly oscillatory integrals which appear in the Magnus series, high order schemes can be constructed. A method of order ten is presented. Even when the solution is highly-oscillatory, the scheme can accurately integrate the problem using stepsizes typically much larger than the solution "wavelength". This makes the method well suited to be applied in a shooting process to locate the eigenvalues of a boundary value problem
Efficient computation of high index Sturm-Liouville eigenvalues for problems in physics
Finding the eigenvalues of a Sturm-Liouville problem can be a computationally
challenging task, especially when a large set of eigenvalues is computed, or
just when particularly large eigenvalues are sought. This is a consequence of
the highly oscillatory behaviour of the solutions corresponding to high
eigenvalues, which forces a naive integrator to take increasingly smaller
steps. We will discuss some techniques that yield uniform approximation over
the whole eigenvalue spectrum and can take large steps even for high
eigenvalues. In particular, we will focus on methods based on coefficient
approximation which replace the coefficient functions of the Sturm-Liouville
problem by simpler approximations and then solve the approximating problem. The
use of (modified) Magnus or Neumann integrators allows to extend the
coefficient approximation idea to higher order methods
On CP, LP and other piecewise perturbation methods for the numerical solution of the Schrödinger equation
The piecewise perturbation methods (PPM) have proven to be very efficient for the numerical solution of the linear time-independent Schrödinger equation. The underlying idea is to replace the potential function piecewisely by simpler approximations and then to solve the approximating problem. The accuracy is improved by adding some perturbation corrections. Two types of approximating potentials were considered in the literature, that is piecewise constant and piecewise linear functions, giving rise to the so-called CP methods (CPM) and LP methods (LPM). Piecewise polynomials of higher degree have not been used since the approximating problem is not easy to integrate analytically. As suggested by Ixaru (Comput Phys Commun 177:897–907, 2007), this problem can be circumvented using another perturbative approach to construct an expression for the solution of the approximating problem. In this paper, we show that there is, however, no need to consider PPM based on higher-order polynomials, since these methods are equivalent to the CPM. Also, LPM is equivalent to CPM, although it was sometimes suggested in the literature that an LP method is more suited for problems with strongly varying potentials. We advocate that CP schemes can (and should) be used in all cases, since it forms the most straightforward way of devising PPM and there is no advantage in considering other piecewise polynomial perturbation methods
Evaluating the Evans function: Order reduction in numerical methods
We consider the numerical evaluation of the Evans function, a Wronskian-like
determinant that arises in the study of the stability of travelling waves.
Constructing the Evans function involves matching the solutions of a linear
ordinary differential equation depending on the spectral parameter. The problem
becomes stiff as the spectral parameter grows. Consequently, the
Gauss--Legendre method has previously been used for such problems; however more
recently, methods based on the Magnus expansion have been proposed. Here we
extensively examine the stiff regime for a general scalar Schr\"odinger
operator. We show that although the fourth-order Magnus method suffers from
order reduction, a fortunate cancellation when computing the Evans matching
function means that fourth-order convergence in the end result is preserved.
The Gauss--Legendre method does not suffer from order reduction, but it does
not experience the cancellation either, and thus it has the same order of
convergence in the end result. Finally we discuss the relative merits of both
methods as spectral tools.Comment: 21 pages, 3 figures; removed superfluous material (+/- 1 page), added
paragraph to conclusion and two reference
Solution of Sturm-Liouville problems using modified Neumann schemes
The main purpose of this paper is to describe the extension of the successful modified integral series methods for Schrodinger problems to more general Sturm-Liouville eigenvalue problems. We present a robust and reliable modified Neumann method which can handle a wide variety of problems. This modified Neumann method is closely related to the second-order Pruess method but provides for higher-order approximations. We show that the method can be successfully implemented in a competitive automatic general-purpose software package
The Magnus expansion and some of its applications
Approximate resolution of linear systems of differential equations with
varying coefficients is a recurrent problem shared by a number of scientific
and engineering areas, ranging from Quantum Mechanics to Control Theory. When
formulated in operator or matrix form, the Magnus expansion furnishes an
elegant setting to built up approximate exponential representations of the
solution of the system. It provides a power series expansion for the
corresponding exponent and is sometimes referred to as Time-Dependent
Exponential Perturbation Theory. Every Magnus approximant corresponds in
Perturbation Theory to a partial re-summation of infinite terms with the
important additional property of preserving at any order certain symmetries of
the exact solution. The goal of this review is threefold. First, to collect a
number of developments scattered through half a century of scientific
literature on Magnus expansion. They concern the methods for the generation of
terms in the expansion, estimates of the radius of convergence of the series,
generalizations and related non-perturbative expansions. Second, to provide a
bridge with its implementation as generator of especial purpose numerical
integration methods, a field of intense activity during the last decade. Third,
to illustrate with examples the kind of results one can expect from Magnus
expansion in comparison with those from both perturbative schemes and standard
numerical integrators. We buttress this issue with a revision of the wide range
of physical applications found by Magnus expansion in the literature.Comment: Report on the Magnus expansion for differential equations and its
applications to several physical problem
Raising and lowering operators, factorization and differential/difference operators of hypergeometric type
Starting from Rodrigues formula we present a general construction of raising
and lowering operators for orthogonal polynomials of continuous and discrete
variable on uniform lattice. In order to have these operators mutually adjoint
we introduce orthonormal functions with respect to the scalar product of unit
weight. Using the Infeld-Hull factorization method, we generate from the
raising and lowering operators the second order self-adjoint
differential/difference operator of hypergeometric type.Comment: LaTeX, 24 pages, iopart style (late submission
Spectral Properties and Linear Stability of Self-Similar Wave Maps
We study co--rotational wave maps from --Minkowski space to the
three--sphere . It is known that there exists a countable family
of self--similar solutions. We investigate their stability under linear
perturbations by operator theoretic methods. To this end we study the spectra
of the perturbation operators, prove well--posedness of the corresponding
linear Cauchy problem and deduce a growth estimate for solutions. Finally, we
study perturbations of the limiting solution which is obtained from by
letting .Comment: Some extensions added to match the published versio
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