G. Vanden Berghe

Exponentially-fitted explicit Runge–Kutta methods

An exponentially-fitted explicit Runge–Kutta method is constructed, which exactly integrates differential initial-value problems whose solutions are linear combinations of functions of the form exp(ωx) and exp(−ωx) (ω∈R or iR); this method is compared to a previously constructed method of Simos. Numerical experiments show the efficiency of the new method.

Exponentially fitted Runge-Kutta methods

Abstract Exponentially fitted Runge–Kutta methods with s stages are constructed, which exactly integrate differential initial-value problems whose solutions are linear combinations of functions of the form {x j exp (ωx),x j exp (−ωx)} , ( ω∈ R or i R , j=0,1,…,j max ), where 0⩽j max ⩽⌊s/2−1⌋ , the lower bound being related to explicit methods, the upper bound applicable for collocation methods. Explicit methods with s∈{2,3,4} belonging to that class are constructed. For these methods, a study of the local truncation error is made, out of which follows a simple heuristic to estimate the ω-value. Error and step length control is introduced based on Richardson extrapolation ideas. Some numerical experiments show the efficiency of the introduced methods. It is shown that the same techniques can be applied to construct implicit exponentially fitted Runge–Kutta methods.

Frequency evaluation in exponential fitting multistep algorithms for ODEs

We consider the linear multistep algorithms for first order ODEs and examine the problem of how the λ-frequencies should be tuned in order to obtain the maximal benefit from the exponential fitting versions of such algorithms. We find out that the key of the answer consists in analysing the behaviour of the error. On further investigating the simple case of two-step bdf algorithms we produce formulae for the optimal λs and show that, if the optimal λs are used, the order of the method is increased by one unit. The reported numerical illustrations suggest that further investigations along these lines deserve a real attention.

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.

On a new type of mixed interpolation

Abstract We approximate every function f by a function f n ( x ) of the form a cos kx + b sin kx + Σ n −2 i =0 c i x i so that f ( jh ) = f n ( jh ) for the n + 1 equidistant points jh , j = 0,…, n . That interpolation function f n ( x ) is proved to be unique and can be written as the sum of the n th-degree interpolation polynomial based on the same points and two correction terms. The error term is also discussed. The results for this mixed type of interpolation reduce to the known results of the polynomial case as the parameter k is tending to 0. This new interpolation theory will be used in the future for the construction of quadrature rules and multistep methods for ordinary differential equations.

Frequency determination and step-length control for exponentially-fitted Runge---Kutta methods

An exponentially (tted Runge–Kutta (EFRK) (fth-order method with six stages is constructed, which exactly integrates (rst-order di8erential initial-value problems whose solutions are linear combinations of functions of the form {exp(!x); exp(−!x)} ,( ! ∈ R or iR). By combining this EFRK method with an equivalent classical embedded (4,5) Runge–Kutta method, a technique is developed for the estimation of the occurring !-values. Error and step-length control is carried out by using the Richardson extrapolation procedure. Some numerical experiments show the e;ciency of the introduced methods. c � 2001 Elsevier Science B.V. All rights reserved.

Exponentially fitted open Newton–Cotes differential methods as multilayer symplectic integrators

Classical open and closed Newton-Cotes differential methods possessing the characteristics of multilayer symplectic structures have been constructed in the past. In this paper, we study the exponentially fitted open Newton-Cotes differential methods of order two, four, and six. It is shown that these integrators, just as their classical counterparts, preserve the volume in the phase space of a Hamiltonian system. They can be converted into a multilayer symplectic structure so that volume-preserving integrators of a Hamiltonian system are obtained. A numerical example has been carried out to show the effectiveness of the present differential method.Classical open and closed Newton–Cotes differential methods possessing the characteristics of multilayer symplectic structures have been constructed in the past. In this paper, we study the exponentially fitted open Newton–Cotes differential methods of order two, four, and six. It is shown that these integrators, just as their classical counterparts, preserve the volume in the phase space of a Hamiltonian system. They can be converted into a multilayer symplectic structure so that volume-preserving integrators of a Hamiltonian system are obtained. A numerical example has been carried out to show the effectiveness of the present differential method.

Extended) Numerov method for computing eigenvalues of specific Schrodinger equations

Numerovs method and an extended version of it are introduced for computing eigenvalues of Schrodinger equations with potentials V(x) which are even functions with respect to x. Furthermore it is assumed that the wavefunctions tend to zero for x to +or- infinity . The derived results are compared with previously derived numerical data and with available exact values.

SLCPM12 - A program for solving regular Sturm-Liouville problems.

Abstract The code SLCPM12 first converts the original Sturm—Liouville equation into an equation of the Schrodinger form and then it solves the latter by means of a suited highly accurate method. The conversion is done by using Liouvilles transformation and the numerical method for solving the Schrodinger equation is CPM12, 10 developed by Ixaru, De Meyer and Vanden Berghe, CP Methods for the Schrodinger Equation revisited, J. Comput. Appl. Math. (1998). The new code is by far faster and more accurate than other existing codes, e.g. SLEDGE, SLEIGN and SL02F .

A finite difference approach for the calculation of perturbed oscillator energies

A simple numerical method for calculating eigenvalues and corresponding eigenvectors of the Schrodinger equation for a perturbed oscillator is described. The derived results are compared with previously derived numerical data and with available exact values.