Guido Vanden Berghe

Some finite difference methods for computing eigenvalues and eigenvectors of special two-point boundary value problems

Abstract Direct numerical integration methods will be discussed for calculating eigenvalues and eigenvectors of two-point boundary value problems involving the differential equation y″ + (a - p(x))y = 0 with p(x) = p(-x). The derived results are compared with previously derived numerical data and with available exact values.

Symplectic exponentially-fitted four-stage Runge---Kutta methods of the Gauss type

The construction of symmetric and symplectic exponentially-fitted Runge–Kutta methods for the numerical integration of Hamiltonian systems with oscillatory solutions deserves a lot of interest. In previous papers fourth-order and sixth-order symplectic exponentially-fitted integrators of Gauss type, either with fixed or variable nodes, have been derived. In this paper new such integrators of eighth-order are studied and constructed by making use of the six-step procedure of Ixaru and Vanden Berghe (2004). Numerical experiments for some oscillatory problems are presented.

P-stable Obrechkoff methods of arbitrary order for second-order differential equations

Following the ideas of Ananthakrishnaiah we develop a family of P-stable Obrechkoff methods of arbitrary even order. The coefficients of these methods follow from a recursive algorithm. It is also shown that the stability functions of the thus obtained methods can be expressed as Padé approximants of the exponential function with a complex argument. A numerical example is given to illustrate the performance of the methods.Following the ideas of Ananthakrishnaiah we develop a family of P-stable Obrechkoff methods of arbitrary even order. The coefficients of these methods follow from a recursive algorithm. It is also shown that the stability functions of the thus obtained methods can be expressed as Padé approximants of the exponential function with a complex argument. A numerical example is given to illustrate the performance of the methods.

Two-step fourth order methods for linear ODEs of the second order

A family of two step difference schemes of the fourth order has been developed for linear ODEs of the second order. Stability properties for such schemes are discussed and results of numerical tests are given. It is shown how the proposed technique can be extended to non-linear ODEs of second order.

Exponentially-fitted methods: searching the frequency

Exponential fitted algorithms for initial value and boundary value methods and for the calculation of quadrature rules are introduced. Special attention is paid to the form of the leading order term of the error and to the global error term in particular. It is shown in which way a “best value” for the occurring frequency can be selected. Numerical experiments illustrate the proposed strategy.

Properties and eigenvalues of a fifth label generating operator for quadrupole-phonon states

The properties of a previously algebraically derived fifth label generating operator S for quadrupole-phonon states are discussed. This algebraic operator is connected with the operators following from pure group theoretical principles. A method is developed by which it is possible to calculate numerically the eigenvalues of the operator S.

Piecewise constant perturbation methods for the multichannel schrödinger equation

The CPM{P,N} methods form a class of methods specially devised for the propagation of the solution of the one-dimensional Schrodinger equation. Using these CPM{P,N} methods in a shooting procedure, eigenvalues of the boundary value problem are obtained to very high precision. Some recent advances allowed the generalization of the CPM{P,N} methods to systems of coupled Schrodinger equations. Also for these generalised CPM{P,N} methods a shooting procedure can be formulated, solving the multichannel bound state problem.

Linear Multistep Solvers for Ordinary Differential Equations

The solution of the initial value problem for ordinary differential equations is one of the main topics in numerical analysis. The linear multistep methods (algorithms) form a class of methods which benefitted from much attention over the years. The theory of these methods is basically due to Dahlquist, [14], and a series of well-known books which cover both theoretical and practical aspects are available, to mention only the books of Henrici, [22], Lambert, [40], Hairer, Norsett and Wanner, [20], Shampine, [52], Hairer and Wanner, [21], and Butcher, [4].

Construction of EF Formulae for Functions with Oscillatory or Hyperbolic Variation

The considerations of the previous chapter referred to the general features of the exponential fitting technique. When the functions to be approximated are oscillatory or with a variation well described by hyperbolic functions the technique exhibits some helpful features. This chapter aims at presenting these features and at formulating a simple algorithm-like flow chart to be followed in the current practice.

Runge-Kutta Solvers for Ordinary Differential Equations

Since the original papers of Runge [24] and Kutta [17] a great number of papers and books have been devoted to the properties of Runge-Kutta methods. Reviews of this material can be found in [4], [5], [12], [18]. Kutta [17] formulated the general scheme of what is now called a Runge-Kutta method.