Anisotropic adaptive resolution of boundary layers for heat conduction problems

We deal with the numerical solution of heat conduction problems featuring steep gradients. In order to solve the associated partial differential equation a finite volume technique is used and unstructured grids are employed. A discrete maximum principle for triangulations of a Delaunay type is developed. To capture thin boundary layers incorporating steep gradients an anisotropic mesh adaptation technique is implemented. Computational tests are performed for an academic problem where the exact solution is known as well as for a real world problem of a computer simulation of the thermoregulation of premature infants.

Modellierung und numerische Simulation der Thermoregulation von Früh- und Neugeborenen

Bei frühgeborenen Säuglingen spielt die Thermoregulation zur Aufrechterhaltung einer überlebenswichtigen Körpertemperatur durch Wärmeproduktion, -abgabe bzw. -aufnahme eine entscheidende Rolle. Der Einsatz moderner Inkubatoren soll die körpereigenen Thermoregulatoren unterstützen, und es ist im Hinblick auf verschiedene medizinische Fragestellungen wünschenswert, diesen Prozess modellieren zu können. Wir stellen ein einfaches Modell auf der Basis von partiellen Differentialgleichungen vor und beschreiben detailliert die numerische Simulation mit Hilfe einer Finite-Volumen-Methode. Dazu wird ein zweidimensionales Modell eines Frühgeborenen trianguliert und das Modell diskretisiert. Zahlreiche numerische Resultate zeigen die Qualität unseres Modells.

Stability of preconditioned finite volume schemes at low Mach numbers

In [4], Guillard and Viozat propose a finite volume method for the simulation of inviscid steady as well as unsteady flows at low Mach numbers, based on a preconditioning technique. The scheme satisfies the results of a single scale asymptotic analysis in a discrete sense and comprises the advantage that this can be derived by a slight modification of the dissipation term within the numerical flux function. Unfortunately, it can be observed by numerical experiments that the preconditioned approach combined with an explicit time integration scheme turns out to be unstable if the time step Dt does not satisfy the requirement to be O(M2) as the Mach number M tends to zero, whereas the corresponding standard method remains stable up to Dt=O(M), M to 0, which results from the well-known CFL-condition. We present a comprehensive mathematical substantiation of this numerical phenomenon by means of a von Neumann stability analysis, which reveals that in contrast to the standard approach, the dissipation matrix of the preconditioned numerical flux function possesses an eigenvalue growing like M-2 as M tends to zero, thus causing the diminishment of the stability region of the explicit scheme. Thereby, we present statements for both the standard preconditioner used by Guillard and Viozat [4] and the more general one due to Turkel [21]. The theoretical results are after wards confirmed by numerical experiments.

Preconditioner updates applied to CFD model problems

In the present paper we concentrate on solving sequences of nonsymmetric linear systems with block structure arising from compressible flow problems. We attempt to improve the solution process by sharing part of the computational effort throughout the sequence. This is achieved by application of a cheap updating technique for preconditioners which we adapted in order to be used for our applications. Tested on three benchmark compressible flow problems, the strategy speeds up the entire computation with an acceleration being particularly pronounced in phases of instationary behavior.

Elementaria mathematicae

Diese Lecture Note beinhaltet drei Beiträge zur lehramts-orientierten elementaren Mathematik. Ein 1. Beitrag liefert Module der elementaren Mengenlehre und Ordnungstheorie, Algebra und Informatik, Geometrie und Stochastik, ein 2.Beitrag liefert Zitate großer Meister und großer Geister und ein 3.Beitrag ist den Elementen der Analysis gewidmet.