Veerle Ledoux

MATSLISE: A MATLAB package for the numerical solution of Sturm-Liouville and Schrödinger equations

MATSLISE is a graphical MATLAB software package for the interactive numerical study of regular Sturm-Liouville problems, one-dimensional Schrödinger equations, and radial Schrödinger equations with a distorted Coulomb potential. It allows the fast and accurate computation of the eigenvalues and the visualization of the corresponding eigenfunctions. This is realized by making use of the power of high-order piecewise constant perturbation methods, a technique described by Ixaru. For a well-outlined class of problems, the implemented algorithms are more efficient than the well-established SL-solvers SL02f, SLEDGE, SLEIGN, and SLEIGN2, which are included by Pryce in the SLDRIVER code that has been built on top of SLTSTPAK.

CP methods of higher order for Sturm–Liouville and Schrödinger equations

Abstract The algorithm upon which the code SLCPM12, described in Computer Physics Communications 118 (1999) 259–277, is based, is extended to higher order. The implementation of the original algorithm, which was of order { 12 , 10 } (meaning order 12 at low energies and order 10 at high energies), was more efficient than the well-established codes SL02F, SLEDGE and SLEIGN. In the new algorithm the orders { 14 , 12 } , { 16 , 14 } and { 18 , 16 } are introduced. Besides regular Sturm–Liouville and one-dimensional Schrodinger problems also radial Schrodinger equations are considered with potentials of the form V ( r ) = S ( r ) / r + R ( r ) , where S ( r ) and R ( r ) are well behaved functions which tend to some (not necessarily equal) constants when r → 0 and r → ∞ . Numerical illustrations are given showing the accuracy, the robustness and the CPU-time gain of the proposed algorithms.

Solution of the Schrödinger equation by a high order perturbation method based on a linear reference potential

The paper is devoted to the enhancement of the accuracy of the line-based perturbation method via the introduction of the perturbation corrections. We effectively construct the first and the second order corrections. We also perform the error analysis to predict that the introduction of successive corrections substantially enhances the order of the method from four, for the zeroth order version, to six and ten when the first and the second-order corrections are included. In order to remove the effect of the accuracy loss due to near-cancellation of like-terms when evaluating the perturbation corrections we construct alternative asymptotic formulae using a Maple code. We also propose a procedure for choosing the step size in terms of the preset accuracy and give a number of numerical illustrations.

Grassmannian spectral shooting

We present a new numerical method for computing the pure-point spectrum associated with the linear stability of coherent structures. In the context of the Evans function shooting and matching approach, all the relevant information is carried by the flow projected onto the underlying Grassmann manifold. We show how to numerically construct this projected flow in a stable and robust manner. In particular, the method avoids representation singularities by, in practice, choosing the best coordinate patch representation for the flow as it evolves. The method is analytic in the spectral parameter and of complexity bounded by the order of the spectral problem cubed. For large systems it represents a competitive method to those recently developed that are based on continuous orthogonalization. We demonstrate this by comparing the two methods in three applications: Boussinesq solitary waves, autocatalytic travelling waves and the Ekman boundary layer.

Efficient computation of high index Sturm–Liouville eigenvalues for problems in physics

Abstract 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 behavior of the solutions corresponding to high eigenvalues, which forces a naive integrator to take increasingly smaller steps. We will discuss the most used approaches to the numerical solution of the Sturm–Liouville problem: finite differences and variational methods, both leading to a matrix eigenvalue problem; shooting methods using an initial-value solver; and coefficient approximation methods. Special attention will be paid to techniques that yield uniform approximation over the whole eigenvalue spectrum and that allow large steps even for high eigenvalues.

Computing stability of multidimensional traveling waves

We present a numerical method for computing the pure-point spectrum associated with the linear stability of multidimensional traveling fronts to parabolic nonlinear systems. Our method is based on the Evans function shooting approach. Transverse to the direction of propagation we project the spectral equations onto a finite Fourier basis. This generates a large, linear, one-dimensional system of equations for the longitudinal Fourier coefficients. We construct the stable and unstable solution subspaces associated with the longitudinal far-field zero boundary conditions, retaining only the information required for matching, by integrating the Riccati equations associated with the underlying Grassmannian manifolds. The Evans function is then the matching condition measuring the linear dependence of the stable and unstable subspaces and thus determines eigenvalues. As a model application, we study the stability of two-dimensional wrinkled front solutions to a cubic autocatalysis model system. We compare our ...

Solution of the schrodinger equation over an infinite integration interval by perturbation methods, revisited

We consider the solution of the one-dimensional Schrodinger problem over an infinite integration interval. The infinite problem is regularized by truncating the integration interval and imposing the appropriate boundary conditions at the truncation points. The Schrodinger problem is then solved on the truncated integration interval using one of the piecewise perturbation methods developed for the regular Schrodinger problem.We select the truncation points using a procedure based on the WKB approximation. However for problems which behave as a Coulomb problem both around the origin and in the asymptotic range, a more accurate treatment of the numerical boundaries is possible. Taking into account the asymptotic form of the Coulomb equation, adapted boundary conditions can be constructed and as a consequence smaller truncation points can be chosen. To deal with the singularity of the Coulomb-like problem around the origin, a special perturbative algorithm is applied in a small interval around the origin.

Finite difference approach for the two-dimensional Schrodinger equation with application to scission-neutron emission

We present a grid-based procedure to solve the eigenvalue problem for the two-dimensional Schrodinger equation in cylindrical coordinates. The Hamiltonian is discretized by using adapted finite difference approximations of the derivatives and this leads to an algebraic eigenvalue problem with a large (sparse) matrix, which is solved by the method of Arnoldi. By this procedure the single particle eigenstates of nuclear systems with arbitrary deformations can be obtained. As an application we have considered the emission of scission neutrons from fissioning nuclei.

A numerical procedure to solve the multichannel Schrodinger eigenvalue problem

Abstract We discuss the numerical solution of eigenvalue problems for systems of (regular) coupled Schrodinger equations. Using a high order CPM (abbreviation for piecewise Constant (reference potential) Perturbation Method) in a shooting procedure, eigenvalues can be computed accurately. A generalization of the Prufer method for scalar Sturm–Liouville problems makes the whole procedure more robust and allows us to specify the required eigenvalue by its index.

CPM{P, N} methods extended for the solution of coupled channel Schrödinger equations

Abstract The successful CPM { P , N } methods for the one-dimensional time-independent Schrodinger problem are generalized to the coupled channel case. The derivation of the formulae is discussed and a Maple program code is included which allows to determine the analytic expressions of the perturbation corrections needed to construct methods of different orders. Some numerical illustrations are given showing the accuracy, the robustness and the CPU-time gain of the proposed algorithms over the sixth order LILIX method which was presented in [L.Gr. Ixaru, LILIX—A package for the solution of the coupled channel Schrodinger equation, Comput. Phys. Comm. 147 (2002) 834–852].