Marnix Van Daele

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.

Superconvergent Deferred Correction Methods For First Order Systems of Nonlinear Two-Point Boundary Value Problems

Iterated deferred correction is a widely used approach to the numerical solution of first order systems of nonlinear two-point boundary value problems. Normally the orders of accuracy of the various methods used in a deferred correction scheme differ by 2, and, as a direct result, each time a deferred correction is applied the order of the overall scheme is increased by a maximum of 2. In this paper we consider the construction of mono-implicit Runge--Kutta (MIRK) methods where an increase of four orders of accuracy is obtained for each deferred correction. We develop a very powerful yet rather straightforward theory which allows us to identify the appropriate Runge--Kutta formulae for inclusion in such schemes. In particular, we will focus on the construction of pairs of MIRK formulae of order 4 and 8 which will allow this superconvergence to be realized. We will further show that it is possible to derive formulae of this type for which high order interpolants and accurate error estimates are readily available.

Interpolatory Quadrature Rules for Oscillatory Integrals

In this paper we revisit some quadrature methods for highly oscillatory integrals of the form

Solution of Sturm-Liouville Problems Using Modified Neumann Schemes

\int_{-1}^{1}f(x)e^{\mathrm{i}\omega x}dx,\omega>0

Matslise 2.0: A Matlab Toolbox for Sturm-Liouville Computations

. Exponentially Fitted (EF) rules depend on frequency dependent nodes which start off at the Gauss-Legendre nodes when the frequency is zero and end up at the endpoints of the integral when the frequency tends to infinity. This makes the rules well suited for small as well as for large frequencies. However, the computation of the EF nodes is expensive due to iteration and ill-conditioning. This issue can be resolved by making the connection with Filon-type rules. By introducing some S-shaped functions, we show how Gauss-type rules with frequency dependent nodes can be constructed, which have an optimal asymptotic rate of decay of the error with increasing frequency and which are effective also for small or moderate frequencies. These frequency-dependent nodes can also be included into Filon-Clenshaw-Curtis rules to form a class of methods which is particularly well suited to be implemented in an automatic software package.

Exponentially-fitted methods: searching the frequency

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.

Explicit Runge–Kutta Methods for Stiff Problems with a Gap in Their Eigenvalue Spectrum

The Matslise 2.0 software package is a thorough revision of the successful Matlab package Matslise of 2005. The package can be used to compute the eigenvalues and eigenfunctions of regular and some important classes of singular self-adjoint Sturm-Liouville boundary value problems. The code uses new or improved algorithms, offers some new features, and has an updated graphical user interface.

Functionally Fitted Explicit Two Step Peer Methods

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.

Algorithms and Software for the Sturm‐Liouville Problem

In this paper we consider the numerical solution of stiff problems in which the eigenvalues are separated into two clusters, one containing the “stiff”, or fast, components and one containing the slow components, that is, there is a gap in their eigenvalue spectrum. By using exponential fitting techniques we develop a class of explicit Runge–Kutta methods, that we call stability fitted methods, for which the stability domain has two regions, one close to the origin and the other one fitting the large eigenvalues. We obtain the size of their stability regions as a function of the order and the fitting conditions. We also obtain conditions that the coefficients of these methods must satisfy to have a given stiff order for the Prothero–Robinson test equation. Finally, we construct an embedded pair of stability fitted methods of orders 2 and 1 and show its performance by means of several numerical experiments.