Exponentially-fitted methods and their stability functions

Is it possible to determine the stability function of an exponentially-fitted Runge-Kutta method, without actually constructing the method itself? This question was answered in a recent paper and examples were given for one-stage methods. In this paper we summarize the results and we focus on two-stage methods

Multiparameter exponentially-fitted methods applied to second-order boundary value problems

Second-order boundary value problems are solved by means of a new type of exponentially-fitted methods that are modifications of the Numerov method. These methods depend upon a set of parameters which can be tuned to solve the problem at hand more accurately. Their values can be fixed over the entire integration interval, but they can also be determined locally from the local truncation error. A numerical example is given to illustrate the ideas

On the Leading Error Term of Exponentially Fitted Numerov Methods

Second-order boundary value problems are solved with exponentially-fitted Numerov methods. In order to attribute a value to the free parameter in such a method, we look at the leading term of the local truncation error. By solving the problem in two phases, a value for this parameter can be found such that the tuned method behaves like a sixth order method. Furthermore, guidelines to choose between multi le possible values for this parameter are given

Three-stage two-parameter symplectic, symmetric exponentially-fitted Runge-Kutta methods of Gauss type

We construct an exponentially-fitted variant of the well-known three stage Runge-Kutta method of Gauss-type. The new method is symmetric and symplectic by construction and it contains two parameters, which can be tuned to the problem at hand. Some numerical experiments are given

Application of exponential fitting techniques to numerical methods for solving differential equations

Ever since the work of Isaac Newton and Gottfried Leibniz in the late 17th century, differential equations (DEs) have been an important concept in many branches of science. Differential equations arise spontaneously in i.a. physics, engineering, chemistry, biology, economics and a lot of fields in between. From the motion of a pendulum, studied by high-school students, to the wave functions of a quantum system, studied by brave scientists: differential equations are common and unavoidable. It is therefore no surprise that a large number of mathematicians have studied, and still study these equations. The better the techniques for solving DEs, the faster the fields where they appear, can advance. Sadly, however, mathematicians have yet to find a technique (or a combination of techniques) that can solve all DEs analytically. Luckily, in the meanwhile, for a lot of applications, approximate solutions are also sufficient. The numerical methods studied in this work compute such approximations. Instead of providing the hypothetical scientist with an explicit, continuous recipe for the solution to their problem, these methods give them an approximation of the solution at a number of discrete points. Numerical methods of this type have been the topic of research since the days of Leonhard Euler, and still are. Nowadays, however, the computations are performed by digital processors, which are well-suited for these methods, even though many of the ideas predate the modern digital computer by almost a few centuries. The ever increasing power of even the smallest processor allows us to devise newer and more elaborate methods. In this work, we will look at a few well-known numerical methods for the solution of differential equations. These methods are combined with a technique called exponential fitting, which produces exponentially fitted methods: classical methods with modified coefficients. The original idea behind this technique is to improve the performance on problems with oscillatory solutions

Andermaal Romeins en vroegmiddeleeuws langs de Zandstraat. te Sint-Andries/Brugge (prov. West-Vlaanderen)

De jongste jaren wordt langs de Zandstraat in de Brugse deelgemeente Sint- Andries op systematische wijze preventief archeologisch onderzoek uitgevoerd op een aantal binnengronden bedreigd door woonverkavelingen en andere bouwprojecten. Na de grootschalige opgravingen achter de Refuge, verplaatste het onderzoek zich in 1997 naar de geplande verkaveling Molendorp, zowat 1.500 m meer westwaarts, tussen de Oudstrijderslaan en de Korte Molenstraat. Ook voerde de Brugse Stedelijke Archeologische Dienst onderzoek uit aan de andere zijde van de Korte Molenstraat. Bij deze opgravingen kwamen, naast verscheidene geïsoleerde structuren uit de metaaltijden en de resten van een kleine Romeinse begraafplaats, overwegend vroegmiddeleeuwse nederzettingssporen aan het licht. Deze vormen niet enkel een belangrijke aanvulling voor het onderzoek verricht in de vlakbij gelegen verkaveling Molendorp maar werpen ook nieuw licht op de resultaten verkregen bij de opgraving van 1996 in de aangrenzende Jabbeekse deelgemeente Varsenare of deze uit het Oudenburgse

Clinical and radiological characteristics of 82 solitary benign peripheral nerve tumours

Benign peripheral nerve tumours are rare lesions. The surgical treatment and clinical outcomes depend on the resectability. The aim of this retrospective study was to identify clinical or radiological features that may predict the surgical technique that should be used to improve clinical outcome. Eighty-two patients were diagnosed with solitary benign peripheral nerve tumours. Fifty-five tumours were surgically resectable, and 27 were nonresectable. Pre-operative magnetic resonance imaging and ultrasound were used, which were predictive of the neural origin of the tumours in 87% (39/45) of cases imaged. In 78% (50/64) of cases imaged, an origin from the nerve sheath (peripheral nerve sheath tumour), or from non-neural elements was possible. However, no imaging or clinical criteria were identified that could determine tumour resectability preoperatively. The diagnosis of solitary peripheral nerve tumour still relies on the macroscopic appearance and definitive histology after epineurotomy