Catálogo de publicaciones - libros

The standard models for groups of interacting and moving individuals (from cell biology to vertebrate population dynamics) are reaction-diffusion models. They base on Brownian motion, which is characterized by one single parameter (diffusion coefficient). In particular for moving bacteria and (slime mold) amoebae, detailed information on individual movement behavior is available (speed, run times, turn angle distributions). If such information is entered into models for populations, then reaction-transport equations or hyperbolic equations (telegraph equations, damped wave equations) result.

The goal of this review is to present some basic applications of transport equations and hyperbolic systems and to illustrate the connections between transport equations, hyperbolic models, and reaction-diffusion equations. Applied to chemosensitive movement (chemotaxis) functional estimates for the nonlinearities in the classical chemotaxis model (Patlak-Keller-Segel) can be derived, based on the individual behavior of cells and attractants.

A detailed review is given on two methods of reduction for transport equations. First the construction of parabolic limits (diffusion limits) for linear and non-linear transport equations and then a moment closure method based on energy minimization principles. We illustrate the moment closure method on the lowest non-trivial case (two-moment closure), which leads to Cattaneo systems.

Moreover we study coupled dynamical systems and models with quiescent states. These occur naturally if it is assumed that different processes, like movement and reproduction, do not occur simultaneously. We report on travelling front problems, stability, epidemic modeling, and transport equations with resting phases.

We discuss several results on the existence of continuous travelling wave solutions in systems of conservation laws with nonlinear source terms.

In the first part we show how waves with oscillatory tails can emerge from the combination of a strictly hyperbolic system of conservation laws and a source term possessing a stable line of equilibria. Two-dimensional manifolds of equilibria can lead to Takens-Bogdanov bifurcations without parameters. In this case there exist several families of small heteroclinic waves connecting different parts of the equilibrium manifold.

The second part is concerned with large heteroclinic waves for which the wave speed is characteristic at some point of the profile. This situation has been observed numerically for shock profiles in extended thermodynamics. We discuss the desingularization of the resulting quasilinear implicit differential-algebraic equations and possible bifurcations. The results are illustrated using the -system with source and the 14-moment system of extended thermodynamics.

Our viewpoint is from dynamical systems and bifurcation theory. Local normal forms at singularities are used and the dynamics is described with the help of blowup transformations and invariant manifolds.

We provide a full description of the Jacobian to the discontinuous Galerkin discretization of the compressible Euler equations, one of the key ingredients of the adaptive discontinuous Galerkin methods recently developed in [7, 8]. We demonstrate the use of this Jacobian within an implicit solver for the approximation of the (primal) stationary flow problems as well as in the adjoint (dual) problems that occur in the context of error estimation and adaptive mesh refinement. In particular, we show that the (stationary) compressible Euler equations can efficiently be solved by the Newton method. Full quadratic Newton convergence is achieved on higher order elements as well as on locally refined meshes.

Substellar atmospheres are observed to be irregularly variable for which the formation of dust clouds is the most promising candidate explanation. The atmospheric gas is convectively unstable and, last but not least, colliding convective cells are seen as cause for a turbulent fluid field. Since dust formation depends on the local properties of the fluid, turbulence influences the dust formation process and may even allow the dust formation in an initially dust-hostile gas.

A regime-wise investigation of dust forming substellar atmospheric situations reveals that the largest scales are determined by the interplay between gravitational settling and convective replenishment which results in a dust-stratified atmosphere. The regime of small scales is determined by the interaction of turbulent fluctuations. Resulting lane-like and curled dust distributions combine to larger and larger structures. We compile necessary criteria for a subgrid model in the frame of large scale simulations as result of our study on small scale turbulence in dust forming gases.

In this article, two meshfree methods for the numerical solution of conservation laws are considered. The Finite Volume Particle Method (FVPM) generalizes the Finite Volume approach and the Finite Pointset Method (FPM) is a Finite Difference scheme which can work on unstructured and moving point clouds. Details of the derivation and numerical examples are presented for the case of incompressible, viscous, two-phase flow. In the case of FVPM, our main focus lies on the derivation of stability estimates.

Type Ia supernovae, i.e. stellar explosions which do not have hydrogen in their spectra, but intermediate-mass elements such as silicon, calcium, cobalt, and iron, have recently received considerable attention because it appears that they can be used as ”standard candles” to measure cosmic distances out to billions of light years away from us. Observations of type Ia supernovae seem to indicate that we are living in a universe that started to accelerate its expansion when it was about half its present age. These conclusions rest primarily on phenomenological models which, however, lack proper theoretical understanding, mainly because the explosion process, initiated by thermonuclear fusion of carbon and oxygen into heavier elements, is difficult to simulate even on supercomputers.

Here, we investigate a new way of modeling turbulent thermonuclear deflagration fronts in white dwarfs undergoing a type Ia supernova explosion. Our approach is based on a level set method which treats the front as a mathematical discontinuity and allows for full coupling between the front geometry and the flow field. New results of the method applied to the problem of type Ia supernovae are obtained. It is shown that in 2-D with high spatial resolution and a physically motivated subgrid scale model for the nuclear flames numerically “converged” results can be obtained, but for most initial conditions the stars do not explode. In contrast, simulations in 3-D do give the desired explosions and many of their properties, such as the explosion energies, lightcurves and nucleosynthesis products, are in very good agreement with observed type Ia supernovae.

The charge conservation laws in general are not strictly obeyed in computational electromagnetics and Magnetohydrodynamics (), due to the presence of various types of numerical errors. In this paper, a field theoretical method for the treatment of the often violated charge conservation laws in computational electrodynamics and has been investigated, which reduces to the well-known hyperbolic Generalized Lagrange Multplier () scheme under particular constraints. The central idea of our divergence correction scheme is the implementation of the physically consistent counter terms to Maxwell and equations, for the restoration of the charge conservation laws. The underlying idea has been verified by numerical experiments for Maxwell-Vlasov and shallow water systems.

A numerical technique for the simulation of accelerating turbulent premixed flames in large scale geometries is presented. It is based on a hybrid capturing/tracking method. It resembles a tracking scheme in that the front geometry is explicitly computed and propagated using a level set method. The basic flow properties are provided by solving the reactive Euler equations. The flame-flow-coupling is achieved by an in-cell-reconstruction technique, i.e., in cells cut by the flame the discontinuous solution is reconstructed from given cell-averages by applying Rankine-Hugoniot type jump conditions. Then the reconstructed states and again the front geometry are used to define accurate effective numerical fluxes across grid cell interfaces intersected by the front during the time step considered. Hence the scheme also resembles a capturing scheme in that only cell averages of conserved quantities are updated. To enable the modelling of inherently unsteady effects, like quenching, reignition, etc., during flame acceleration, the new key idea is to provide a local, quasi-onedimensional flame structure model and to extend the Rankine-Hugoniot conditions so as to allow for inherently unsteady flame structure evolution. A source term appearing in the modified jump conditions is computed by evaluating a suitable functional on the basis of a onedimensional flame structure module, that is attached in normal direction to the flame front. This module additionally yields quantities like the net mass burning rate, necessary for the propagation of the level set, and the specific heat release important for the energy release due to the consumption of fuel. Generally the local flame structure calculation takes into account internal (small scale) physical effects which are not active in the (large scale) outer flow but essential for the front motion and its feedback on the surrounding fluid. If a suitable set of different (turbulent) combustion models to compute the flame structure is provided, the new numerical technique allows us to consistently represent laminar deflagrations, fast turbulent deflagrations as well as detonation waves. Supplemented with suitable criteria that capture the essence of a Deflagration-to-Detonation-Transition (DDT), the complete evolution of such an event can be implemented in principle.

In this article, we use the recently developed framework of state and flux decompositions to point out some interesting connections and differences between several Riemann-solver free numerical approaches for systems of hyperbolic conservation laws. We include a numerical comparison of Fey's Method of Transport with a second order version of the HLL scheme and prove an interesting property of the former scheme for linear waves contained in the equations of ideal gas dynamics.

Relaxation dynamics, scaling limits, and relaxation schemes are three main topics on hyperbolic relaxation problems that, remarkably, can be well understood with one model equation. The criterion that leads to desired results for the three problems is the so called “sub-characteristic condition”. The criterion of this nature is also pivotal in the study of general hyperbolic relaxation problems.

In this article we review the recent research development in hyperbolic relaxation problems. The emphasis is on contributions associated with our own project within ANumE priority research program. We will first review some basic properties and notions for hyperbolic relaxation problems, and then focus our investigation on three main topics associated with the underlying relaxation model: relaxation dynamics, scaling limits as well as convergence theory of relaxation schemes.