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Finite element modelling of some incompressible fluid flow problems

Burda, Pavel, Novotný, Jaroslav, Šístek, Jakub, Damašek, Alexandr (2008)

Programs and Algorithms of Numerical Mathematics

We deal with modelling of flows in channels or tubes with abrupt changes of the diameter. The goal of this work is to construct the FEM solution in the vicinity of these corners as precise as desired. We present two ways. The first approach makes use of a posteriori error estimates and the adaptive strategy. The second approach is based on the asymptotic behaviour of the exact solution in the vicinity of the corner and on the a priori error estimate of the FEM solution. Then we obtain the solution...

Geodesic metrics for RBF approximation of some physical quantities measured on sphere

Karel Segeth (2024)

Applications of Mathematics

The radial basis function (RBF) approximation is a rapidly developing field of mathematics. In the paper, we are concerned with the measurement of scalar physical quantities at nodes on sphere in the 3D Euclidean space and the spherical RBF interpolation of the data acquired. We employ a multiquadric as the radial basis function and the corresponding trend is a polynomial of degree 2 considered in Cartesian coordinates. Attention is paid to geodesic metrics that define the distance of two points...

Highly anisotropic nonlinear temperature balance equation and its numerical solution using asymptotic-preserving schemes of second order in time

Alexei Lozinski, Jacek Narski, Claudia Negulescu (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

This paper deals with the numerical study of a nonlinear, strongly anisotropic heat equation. The use of standard schemes in this situation leads to poor results, due to the high anisotropy. An Asymptotic-Preserving method is introduced in this paper, which is second-order accurate in both, temporal and spacial variables. The discretization in time is done using an L-stable Runge−Kutta scheme. The convergence of the method is shown to be independent of the anisotropy parameter , and this for fixed...

Iterative Einschliessungen von Lösungen nichtlinearer Differentialgleichungen durch Newton-ähnliche Iterationsverfahren

Rudolf L. Voller (1986)

Aplikace matematiky

In der vorliegenden Arbeit untersuchen wir monoton einschliessende Newton-ähnliche Iterationsverfahren zur näherungsweisen Lösung verschiedener Klassen vonnichtlinearen Differentialgleichungen. Die behandelten Methoden sind auch für nichtkonvexe Nichtlinearitäten anwendbar. Ferner konstruieren wir einschliessende Startnäherungen für diese Verfahren, so dass wir die Existenz der Lösungen der gegebenen Differentialgleichungen sichern können. Die Konvergenz der Verfahren wird auch für den Fall bewiesen,...

Linear stability of Euler equations in cylindrical domain

Čermák, Libor (2008)

Programs and Algorithms of Numerical Mathematics

The linear stability problem of inviscid incompressible steady flow between two concentric cylinders is investigated. Linearizing the transient behavior around a steady state solution leads to an eigenvalue problem for linearized Euler equations. The discrete eigenvalue problem is obtained by the spectral element method. The algorithm is implemented in MATLAB. The developed program serves as a simple tool for numerical experimenting. It enables to state rough dependency of the stability on various...

Low-rank tensor representation of Slater-type and Hydrogen-like orbitals

Martin Mrovec (2017)

Applications of Mathematics

The paper focuses on a low-rank tensor structured representation of Slater-type and Hydrogen-like orbital basis functions that can be used in electronic structure calculations. Standard packages use the Gaussian-type basis functions which allow us to analytically evaluate the necessary integrals. Slater-type and Hydrogen-like orbital functions are physically more appropriate, but they are not analytically integrable. A numerical integration is too expensive when using the standard discretization...

Mathematical modeling of antigenicity for HIV dynamics

François Dubois, Hervé V.J. Le Meur, Claude Reiss (2010)

MathematicS In Action

This contribution is devoted to a new model of HIV multiplication motivated by the patent of one of the authors. We take into account the antigenic diversity through what we define “antigenicity”, whether of the virus or of the adapted lymphocytes. We model the interaction of the immune system and the viral strains by two processes. On the one hand, the presence of a given viral quasi-species generates antigenically adapted lymphocytes. On the other hand, the lymphocytes kill only viruses for which...

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