M. Van Daele

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.

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.

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.

Exponential fitted Runge--Kutta methods of collocation type: fixed or variable knot points?

Two different classes of exponential fitted Runge-Kutta collocation methods are considered: methods with fixed points and methods with frequency-dependent points. For both cases we have obtained extensions of the classical two-stage Gauss, RadauIIA and LobattoIIIA methods. Numerical examples reveal important differences between both approaches.

Optimal implicit exponentially-fitted Runge-Kutta methods.

Abstract Implicit Runge–Kutta methods for first-order ODEs are considered and the problem of how frequencies should be tuned in order to obtain the maximal benefit from the exponential fitted versions of such algorithms is examined. The key to the answer lies in the analysis of the behaviour of the error. A two-stage implicit Runge–Kutta method is particularly investigated. Formulae for optimal frequencies are produced; in that case the order of the method is increased by one unit. A numerical experiment illustrates the properties of the developed algorithms.

The 600 yr eruptive history of Villarrica Volcano (Chile) revealed by annually laminated lake sediments

Lake sediments contain valuable information about past volcanic and seismic events that have affected the lake catchment, and they provide unique records of the recurrence interval and magnitude of such events. This study uses a multilake and multiproxy analytical approach to obtain reliable and high-resolution records of past natural catastrophes from ~600-yr-old annually laminated (varved) lake sediment sequences extracted from two lakes, Villarrica and Calafquen, in the volcanically and seismically active Chilean Lake District. Using a combination of micro–X-ray fl uorescence (µXRF) scanning, microfacies analysis, grain-size analysis, color analysis, and magnetic-susceptibility measurements, we detect and characterize four different types of event deposits (lacustrine turbidites, tephra-fall layers , runoff cryptotephras, and lahar deposits) and produce a revised eruption record for Villarrica Volcano, which is unprecedented in its continuity and temporal resolution. Glass geochemistry and mineralogy also reveal deposits of eruptions from the more remote Carran–Los Venados volcanic complex, Quetrupillan Volcano, and the Huanquihue Group in the studied lake sediments. Time-series analysis shows 112 eruptions with a volcanic explosivity index (VEI) ≥2 from Villarrica Volcano in the last ~600 yr, of which at least 22 also produced lahars. This signifi cantly expands our knowledge of the eruptive frequency of the volcano in this time window, compared to the previously known eruptive history from historical records. The last VEI ≥2 eruption of Villarrica Volcano occurred in 1991. Based on the last ~500 yr, for which we have a complete record from both lakes, we estimate the probability of the occurrence of future eruptions from Villarrica Volcano and statistically demonstrate that the probability of a 22 yr repose period (anno 2013) without VEI ≥2 eruptions is ≤1.7%. This new perspective on the recurrence interval of eruptions and historical lahar activity will help improve volcanic hazard assessments for this rapidly expanding tourist region, and it highlights how lake records can be used to signifi cantly improve historical eruption records in areas that were previously uninhabited.

A smooth approximation for the solution of a fourth-order boundary value problem based on nonpolynomial splines

We develop a method of second order for the continuous approximation of the solution of a two-point boundary value problem involving a fourth-order differential equation via mixed spline functions consisting of a polynomial part of degree three and a trigonometric part. It will be shown that the angular frequency k of the trigonometric part can be used to raise the order of accuracy of the obtained scheme. Two numerical examples are given to sustain our autonomous parametric spline method and to compare it with a quintic polynomial spline method of second order.

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.