Relja Vulanović

A higher-order scheme for quasilinear boundary value problems with two small parameters

A class of singularly perturbed quasilinear boundary value problems with two small parameters is solved numerically by finite differences on a Shishkin-type mesh. The discretization combines a four-point third-order scheme inside the boundary layers with the standard central scheme outside the layers. This results in an almost third-order accuracy which is uniform with respect to the perturbation parameters. The paper also shows that the Shishkin meshes are more suitable for higher-order schemes than the Bakhvalov meshes, since complicated nonequidistant schemes can be avoided.

A uniform numerical method for quasilinear singular perturbation problems without turning points

A numerical method for singularly perturbed quasilinear boundary value problems without turning points is proposed: the continuous problem is transformed by introducing a special new independent variable and then finite-difference schemes are applied. The first order convergence uniform in the perturbation parameter is proved in the discreteL1-norm. The numerical results show the pointwise convergence, too.ZusammenfassungEin numerisches Verfahren für quasilineare, singulär gestörte Randwertaufgaben ohne Umkehrpunkte wird angegeben: das kontinuierliche Problem wird durch spezielle neue unabhängige Variable transformiert und danach werden Differenzenverfahren verwendet. Die Konvergenz erster Ordnung gleichmäßig bezüglich des Störungsparameters wird in der diskretenL1-Norm bewiesen. Die numerischen Ergebnisse zeigen auch punktweise Konvergenz.

On the extrapolation for a singularly perturbed boundary value problem

We consider the non-linear boundary value problem with small perturbation parameter ε. We find its numerical solution by using Richardson extrapolation on a special non-uniform discretization mesh. Both, high accuracy and uniformity in ε, are obtained. Numerical examples are provided.ZusammenfassungWir betrachten die nichtlineare Randwertaufgabe mit kleinem Störungsparameter ε. Wir finden ihre numerische Lösung unter Anwendung von Richardson-Extrapolation mit einem speziellen nichtäquidistanten Diskretisierungsgitter. Man erhält gleichzeitig große Genauigkeit und gleichmäßige Konvergenz. Numerische Beispiele werden angegeben.

Fourth order algorithms for a semilinear singular perturbation problem

Fourth order finite-difference algorithms for a semilinear singularly perturbed reaction–diffusion problem are discussed and compared both theoretically and numerically. One of them is the method of Sun and Stynes (1995) which uses a piecewise equidistant discretization mesh of Shishkin type. Another one is a simplified version of Vulanovićs method (1993), based on a discretization mesh of Bakhvalov type. It is shown that the Bakhvalov mesh produces much better numerical results.

Numerical solution of quasilinear attractive turning point problems

Abstract A quasilinear singularity perturbed boundary value problem with a turning point of the attractive type is considered. Analytic properties of its solution are established and used for the construction of a numerical method uniform in the perturbation parameter. The paper is an extension of a result obtained previously for the semilinear case.

Higher order schemes and Richardson extrapolation for singular perturbation problems

Semilinear singular perturbation problems are solved numerically by using finite–difference schemes on non-equidistant meshes which are dense in the layers. The fourth order uniform accuracy of the Hermitian approximation is improved by the Richardson extrapolation.

Finite-difference schemes for quasilinear singular perturbation problems

Finite-difference schemes for singularly perturbed quasilinear boundary value problems are considered. The schemes have fitting factors which guarantee their uniform stability with respect to the perturbation parameter. Besides the usual stability in the discrete L1-norm, we consider the maximum norm stability as well. Various schemes are compared numerically.

A modification of the Shishkin discretization mesh for one-dimensional reaction-diffusion problems

In this paper we consider a modification of the Shishkin discretization mesh designed for the numerical solution of one-dimensional singularly perturbed reaction-diffusion problems. The modification consists of a slightly different choice of the transition points between the fine and coarse parts of the mesh. We prove that this change does not affect the order of convergence of the numerical solution obtained by using the central finite-difference scheme. However, due to a better layer-resolving mesh, numerical results show an improvement in the accuracy of the computed solution when compared to the results on the standard Shishkin mesh.

Word Order, Marking, and Parts-of-Speech Systems

Abstract Parts-of-speech systems, as defined by Hengeveld et al. in their 2004 article on parts-of-speech systems and word order, form the framework for this article, in which we investigate how the distribution of languages with respect to certain linguistic features is related to the number of propositional functions and the number of lexeme classes. The linguistic features are the presence or absence of fixed word order and markers, which may disambiguate between different propositional functions. We show that the relation can be modelled by a three-dimensional generalization of the sigmoid, for which we provide theoretical justification.

Model‐based measuring of syntactic change*

Abstract A new method for quantitative representation of syntactic change is introduced. The method is model‐based in the sense that a formal grammatical model is used to describe stages of syntactic change. Then, by measuring some relevant features of the model, syntactic change is represented as a sequence of numerical values. This can be used for establishing the rate of syntactic change and for further investigations and comparisons of various syntactic changes.