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We propose and study a strongly coupled PDE-ODE system with tissue-dependent degenerate diffusion and haptotaxis that can serve as a model prototype for cancer cell invasion through the
extracellular matrix. We prove the global existence of weak solutions and illustrate the model behaviour by numerical simulations for a two-dimensional setting.

We propose and study a strongly coupled PDE-ODE-ODE system modeling cancer cell invasion through a tissue network
under the go-or-grow hypothesis asserting that cancer cells can either move or proliferate. Hence our setting features
two interacting cell populations with their mutual transitions and involves tissue-dependent degenerate diffusion and
haptotaxis for the moving subpopulation. The proliferating cells and the tissue evolution are characterized by way of ODEs
for the respective densities. We prove the global existence of weak solutions and illustrate the model behaviour by
numerical simulations in a two-dimensional setting.

These lecture notes give a completely self-contained introduction to the control theory of linear time-invariant systems. No prior knowledge is requried apart from linear algebra and some basic familiarity with ordinary differential equations. Thus, the course is suited for students of mathematics in their second or third year, and for theoretically inclined engineering students. Because of its appealing simplicity and elegance, the behavioral approch has been adopted to a large extend. A short list of recommended text books on the subject has been added, as a suggestion for further reading.

Die Theorie der mehrdimensionalen Systeme ist ein relativ junges Forschungsgebiet innerhalb der Systemtheorie, erste Arbeiten stammen aus den 70er Jahren. Hauptmotiv für das Studium multidimensionaler Systeme war die Notwendigkeit einer Erweiterung der Theorie der digitalen Filter, die in der klassischen, eindimensionalen Signalverarbeitung (zeitabhängige Signale) Anwendung finden, auf den Bereich der Bildverarbeitung, also auf zweidimensionale Signale.; Die Vorlesung beschäftigt sich daher in ihrem ersten Teil mit skalaren zweidimensionalen Systemen und beschränkt sich im wesentlichen auf den linearen Fall. Untersucht werden zweidimensionale Filter, ihre wichtigsten Eigenschaften, Kausalität und Stabilität, sowie ihre Zustandsraum- realisierungen, etwa die Modelle von Roesser und Fornasini-Marchesini. Parallelen und Unterschiede zur eindimensionalen Systemtheorie werden betont.; Im zweiten Teil der Vorlesung werden allgemeine höherdimensionale und multivariable Systeme behandelt. Für diese Systeme erweist sich der von Jan Willems begründete Zugang zur Systemtheorie, der sogenannte behavioral approach, als zweckmäßig. Grundlegende Ideen dieses Ansatzes sowie eine der wichtigsten Methoden zum Rechnen mit Polynomen in mehreren Variablen, die Theorie der Gröbnerbasen, werden vorgestellt.

Sudakov's typical marginals, random linear functionals and a conditional central limit theorem
(1997)

V.N. Sudakov [Sud78] proved that the one-dimensional marginals of a highdimensional second order measure are close to each other in most directions. Extending this and a related result in the context of projection pursuit of P. Diaconis and D. Freedman [Dia84], we give for a probability measure P and a random (a.s.) linear functional F on a Hilbert space simple sufficient conditions under which most of the one-dimensional images of P under F are close to their canonical mixture which turns out to be almost a mixed normal distribution. Using the concept of approximate conditioning we deduce a conditional central limit theorem (theorem 3) for random averages of triangular arrays of random variables which satisfy only fairly weak asymptotic orthogonality conditions.

Bekanntlich gibt es keinen befriedigenden unendlich dimensionalen Ersatz für das Lebesgue-Mass. Andererseits lassen sich viele Techniken klassischer Analysis auch auf unendlich dimensionale Situationen übertragen. Eine Möglichkeit hierzu gibt die Theorie differenzierbarer Masse. Man definiert Richtungsableitungen für Masse ähnlich wie für Funktionen. Eines der zentralen Beispiele ist das Wiener-Mass. Stochastische Integration bezüglich der Brownschen Bewegung, insbesondere das Skorokhod-Integral ergeben sich in natürlicher Weise durch diesen Ansatz und auch die Grundideen des MalliavinKalküls lassen sich in diesem Rahmen einfach erläutern. Die Vorträge geben die meisten Beweise.

Starting from the uniqueness question for mixtures of distributions this review centers around the question under which formally weaker assumptions one can prove the existence of SPLIFs, in other words perfect statistics and tests. We mention a couple of positive and negative results which complement the basic contribution of David Blackwell in 1980. Typically the answers depend on the choice of the set theoretic axioms and on the particular concepts of measurability.

We study a model for learning periodic signals in recurrent neural networks proposed by Doya and Yoshizawa [7] that can be considered as a model for temporal pattern memory in animal motoric systems. A network receives an external oscillatory input and adjusts its weights so that this signal can be reproduced approximately as the network output after some time. We use tools from adaptive control theory to derive an algorithm for weight matrices with special structure. If the input is generated by a network of the same structure the algorithm converges globally and does not exhibit the deficiencies of the back-propagation based approach of Doya and Yoshizawa under a persistency of excitation condition. This simple algorithm can also be used for open loop identification under quite restructive assumptions. The persistency of excitation condition cannot be proven even for the matrices with special structure but for a 3d system. For higher dimensional systems we give connections to the theory of linear time-varying systems where this condition is generically true (under assumption which are also needed in the time-invariant case). However, we cannot show that the linearized system related to the nonlinear neural network fulfills these generic assumptions.

The edge enhancement property of a nonlinear diffusion equation with a suitable expression for the diffusivity is an important feature for image processing. We present an algorithm to solve this equation in a wavelet basis and discuss its one dimensional version in some detail. Sample calculations demonstrate principle effects and treat in particular the case of highly noise perturbed signals. The results are discussed with respect to performance, efficiency, choice of parameters and are illustrated by a large number of figures. Finally, a comparison with a Fourier method and a finite volume method is performed.

In spite of its lack of theoretical justification, nonlinear diffusion filtering has become a powerful image enhancement tool in the recent years. The goal of the present paper is to provide a mathematical foundation for nonlinear diffusion filtering as a scale-space transformation which is flexible enough to simplify images without loosing the capability of enhancing edges. By stuying the Lyapunow functional, it is shown that nonlinear diffusion reduces Lp norms and central moments and increases the entropy of images. The proposed anisotropic class utilizes a diffusion tensor which may be adapted to the image structure. It permits existence, uniqueness and regularity results, the solution depends continuously on the initial image, and it fulfills an extremum principle. All considerations include linear and certain nonlinear isotropic models and apply to m-dimensional vector-valued images. The results are juxtaposed to linear and morphological scale-spaces.

A way to derive consistently kinetic models for vehicular traffic from microscopic follow the leader models is presented. The obtained class of kinetic equations is investigated. Explicit examples for kinetic models are developed with a particular emphasis on obtaining models, that give realistic results. For space homogeneous traffic flow situations numerical examples are given including stationary distributions and fundamental diagrams.

In this paper we present a method for nonlinear frequency response analysis of mechanical vibrations of 3-dimensional solid structures.
For computing nonlinear frequency response to periodic excitations, we employ the well-established harmonic balance method.
A fundamental aspect for allowing a large-scale application of the method is model order reduction of the discretized equation of motion. Therefore we propose the utilization of a modal projection method enhanced with modal derivatives, providing second-order information.
For an efficient spatial discretization of continuum mechanics nonlinear partial differential equations, including large deformations and hyperelastic material laws, we use the isogeometric finite element method, which has already been shown to possess advantages over classical finite element discretizations in terms of higher accuracy of numerical approximations in the fields of linear vibration and static large deformation analysis.
With several computational examples, we demonstrate the applicability and accuracy of the modal derivative reduction method for nonlinear static computations and vibration analysis.
Thus, the presented method opens a promising perspective on application of nonlinear frequency analysis to large-scale industrial problems.

In this paper we analyze the vibrations of nonlinear structures by means of the novel approach of isogeometric finite elements. The fundamental idea of isogeometric finite elements is to apply the same functions, namely B-Splines and NURBS (Non-Uniform Rational B-Splines), for describing the geometry and for representing the numerical solution. In case of linear vibrational analysis, this approach has already been shown to possess substantial advantages over classical finite elements, and we extend it here to a nonlinear framework based on the harmonic balance principle.
As application, the straight nonlinear Euler-Bernoulli beam is used, and overall, it is demonstrated that isogeometric finite elements with B-Splines in combination with the harmonic balance method are a powerful means for the analysis of nonlinear structural vibrations. In particular, the smoother k-method provides higher accuracy than the p-method for isogeometric nonlinear vibration analysis.

The efficient numerical treatment of the Boltzmann equation is a very important task in many fields of application. Most of the practically relevant numerical schemes are based on the simulation of large particle systems that approximate the evolution of the distribution function described by the Boltzmann equation. In particular, stochastic particle systems play an important role in the construction of various numerical algorithms.

If \(A\) generates a bounded cosine function on a Banach space \(X\) then the negative square root \(B\) of \(A\) generates a holomorphic semigroup, and this semigroup is the conjugate potential transform of the cosine function. This connection is studied in detail, and it is used for a characterization of cosine function generators in terms of growth conditions on the semigroup generated by \(B\). This characterization relies on new results on the inversion of the vector-valued conjugate potential transform.

\(C^0\)-scalar-type spectrality criterions for operators \(A\), whose resolvent set contains the negative reals, are provided. The criterions are given in terms of growth conditions on the resolvent of \(A\) and the semi-group generated by \(A\).These criterions characterize scalar-type operators on the Banach space \(X\), if and only if \(X\) has no subspace isomorphic to the space of complex null-sequences.

In the Banach space co there exists a continuous function of bounded semivariation which does not correspond to a countably additive vector measure. This result is in contrast to the scalar case, and it has consequences for the characterization of scalar-type operators. Besides this negative result we introduce the notion of functions of unconditionally bounded variation which are exactly the generators of countably additive vector measures.

The thermal equilibrium state of a bipolar, isothermal quantum fluid confined to a bounded domain \(\Omega\subset I\!\!R^d,d=1,2\) or \( d=3\) is the minimizer of the total energy \({\mathcal E}_{\epsilon\lambda}\); \({\mathcal E}_{\epsilon\lambda}\) involves the squares of the scaled Planck's constant \(\epsilon\) and the scaled minimal Debye length \(\lambda\). In applications one frequently has \(\lambda^2\ll 1\). In these cases the zero-space-charge approximation is rigorously justified. As \(\lambda \to 0 \), the particle densities converge to the minimizer of a limiting quantum zero-space-charge functional exactly in those cases where the doping profile satisfies some compatibility conditions. Under natural additional assumptions on the internal energies one gets an differential-algebraic system for the limiting \((\lambda=0)\) particle densities, namely the quantum zero-space-charge model. The analysis of the subsequent limit \(\epsilon \to 0\) exhibits the importance of quantum gaps. The semiclassical zero-space-charge model is, for small \(\epsilon\), a reasonable approximation of the quantum model if and only if the quantum gap vanishes. The simultaneous limit \(\epsilon =\lambda \to 0\) is analyzed.

Caloric Restriction (CR) is the only intervention proven to retard aging and extend maximum lifespan in mammalians. A possible mechanism for the beneficial effects of CR is that the mild metabolic stress associated with CR induces cells to express stress proteins that increase their resistance to disease processes. In this article we therefore model the retardation of aging by dietary restriction within a mathematical framework. The resulting model comprises food intake, stress proteins, body growth and survival. We successfully applied our model to growth and survival data of mice exposed to different food levels.

Universal Shortest Paths
(2010)

We introduce the universal shortest path problem (Univ-SPP) which generalizes both - classical and new - shortest path problems. Starting with the definition of the even more general universal combinatorial optimization problem (Univ-COP), we show that a variety of objective functions for general combinatorial problems can be modeled if all feasible solutions have the same cardinality. Since this assumption is, in general, not satisfied when considering shortest paths, we give two alternative definitions for Univ-SPP, one based on a sequence of cardinality contrained subproblems, the other using an auxiliary construction to establish uniform length for all paths between source and sink. Both alternatives are shown to be (strongly) NP-hard and they can be formulated as quadratic integer or mixed integer linear programs. On graphs with specific assumptions on edge costs and path lengths, the second version of Univ-SPP can be solved as classical sum shortest path problem.

It is well known that the greedy algorithm solves matroid base problems for all linear cost functions and is, in fact, correct if and only if the underlying combinatorial structure of the problem is a matroid. Moreover, the algorithm can be applied to problems with sum, bottleneck, algebraic sum or \(k\)-sum objective functions.

The shortest path problem in which the \((s,t)\)-paths \(P\) of a given digraph \(G =(V,E)\) are compared with respect to the sum of their edge costs is one of the best known problems in combinatorial optimization. The paper is concerned with a number of variations of this problem having different objective functions like bottleneck, balanced, minimum deviation, algebraic sum, \(k\)-sum and \(k\)-max objectives, \((k_1, k_2)-max, (k_1, k_2)\)-balanced and several types of trimmed-mean objectives. We give a survey on existing algorithms and propose a general model for those problems not yet treated in literature. The latter is based on the solution of resource constrained shortest path problems with equality constraints which can be solved in pseudo-polynomial time if the given graph is acyclic and the number of resources is fixed. In our setting, however, these problems can be solved in strongly polynomial time. Combining this with known results on \(k\)-sum and \(k\)-max optimization for general combinatorial problems, we obtain strongly polynomial algorithms for a variety of path problems on acyclic and general digraphs.

Laser-induced thermotherapy (LITT) is an established minimally invasive percutaneous technique of tumor ablation. Nevertheless, there is a need to predict the effect of laser applications and optimizing irradiation planning in LITT. Optical attributes (absorption, scattering) change due to thermal denaturation. The work presents the possibility to identify these temperature dependent parameters from given temperature measurements via an optimal control problem. The solvability of the optimal control problem is analyzed and results of successful implementations are shown.

In this paper we study a particular class of \(n\)-node recurrent neural networks (RNNs).In the \(3\)-node case we use monotone dynamical systems theory to show,for a well-defined set of parameters, that,generically, every orbit of the RNN is asymptotic to a periodic orbit.Then, within the usual 'learning' context of NeuralNetworks, we investigate whether RNNs of this class can adapt their internal parameters soas to 'learn' and then replicate autonomously certain external periodic signals.Our learning algorithm is similar to identification algorithms in adaptivecontrol theory. The main feature of the adaptation algorithm is that global exponential convergenceof parameters is guaranteed. We also obtain partial convergence results in the \(n\)-node case.

Convex Operators in Vector Optimization: Directional Derivatives and the Cone of Decrease Directions
(1999)

The paper is devoted to the investigation of directional derivatives and the cone of decrease directions for convex operators on Banach spaces. We prove a condition for the existence of directional derivatives which does not assume regularity of the ordering cone K. This result is then used to prove that for continuous convex operators the cone of decrease directions can be represented in terms of the directional derivatices . Decrease directions are those for which the directional derivative lies in the negative interior of the ordering cone K. Finally, we show that the continuity of the convex operator can be replaced by its K-boundedness.

Abstract: Evacuation problems can be modeled as flow problems in dynamic networks. A dynamic network is defined by a directed graph G = (N,A) with sources, sinks and non-negative integral travel times and capacities for every arc (i,j) e A. The earliest arrival flow problem is to send a maximum amount of dynamic flow reaching the sink not only for the given time horizon T, but also for any time T' < T . This problem mimics the evacuation problem of public buildings where occupancies may not known. For the buildings where the number of occupancies is known and concentrated only in one source, the quickest flow model is used to find the minimum egress time. We propose in this paper a solution procedure for evacuation problems with a single source of the building where the occupancy number is either known or unknown. The possibility that the flow capacity may change due to the increasing of smoke density or fire obstructions can be mirrored in our model. The solution procedure looks iteratively for the shortest conditional augmenting path (SCAP) from source to sink and compute the time intervals in which flow reaches the sink via this path.

In this paper we present a domain decomposition approach for the coupling of Boltzmann and Euler equations. Particle methods are used for both equations. This leads to a simple implementation of the coupling procedure and to natural interface conditions between the two domains. Adaptive time and space discretizations and a direct coupling procedure leads to considerable gains in CPU time compared to a solution of the full Boltzmann equation. Several test cases involving a large range of Knudsen numbers are numerically investigated.

Application of Moment Realizability Criteria for Coupling of the Boltzmann and Euler Equations
(1998)

The moment realizability criteria have been used to test the domains of validity of the Boltzmann and Euler Equations. With the help of this criteria teh coupling of the Boltzmann and Euler equations have been performed in two dimensional spatial space. The time evolution of domain decompositions for such equations have been presented in different time steps. The numerical resulta obtained from the coupling code have been compared with those from the pure Boltzmann one.

This paper presents a case study of duty rostering for physicians at a department of orthopedics and trauma surgery. We provide a detailed description of the rostering problem faced and present an integer programming model that has been used in practice for creating duty rosters at the department for more than a year. Using real world data, we compare the model output to a manually generated roster as used previously by the department and analyze the quality of the rosters generated by the model over a longer time span. Moreover, we demonstrate how unforeseen events such as absences of scheduled physicians are handled.

This papers deals with the minimization of seminorms \(\|L\cdot\|\) on \(\mathbb R^n\) under the constraint of a bounded I-divergence \(D(b,H\cdot)\). The I-divergence is also known as Kullback-Leibler divergence and appears in many models in imaging science, in particular when dealing with Poisson data. Typically, \(H\) represents here, e.g., a linear blur operator and \(L\) is some discrete derivative operator. Our preference for the constrained approach over
the corresponding penalized version is based on the fact that the I-divergence of data
corrupted, e.g., by Poisson noise or multiplicative Gamma noise can be estimated by statistical methods. Our minimization technique rests upon relations between constrained and penalized convex problems and resembles the idea of Morozov's discrepancy principle.
More precisely, we propose first-order primal-dual algorithms which reduce the problem to the solution of certain proximal minimization problems in each iteration step. The most interesting of these proximal minimization problems is an I-divergence constrained least squares problem. We solve this problem by connecting it to the corresponding I-divergence
penalized least squares problem with an appropriately chosen regularization parameter. Therefore, our algorithm produces not only a sequence of vectors which converges to a minimizer of the constrained problem but also a sequence of parameters which convergences to a regularization parameter so that the penalized problem has the same solution as our constrained one. In other words, the solution of this penalized problem fulfills the I-divergence constraint. We provide the proofs which are necessary to understand
our approach and demonstrate the performance of our algorithms for different
image restoration examples.

A standard approach for deducing a variational denoising method is the maximum a posteriori strategy. Here, the denoising result is chosen in such a way that it maximizes the conditional density function of the reconstruction given its observed noisy version. Unfortunately, this approach does not imply that the empirical distribution of the reconstructed noise components follows the statistics of the assumed noise model. In this paper, we propose to overcome this drawback by applying an additional transformation to the random vector modeling the noise. This transformation is then incorporated into the standard denoising approach and leads to a more sophisticated data fidelity term, which forces the removed noise components to have the desired statistical properties. The good properties of our new approach are demonstrated for additive Gaussian noise by numerical examples. Our method shows to be especially well suited for data containing high frequency structures, where other denoising methods which assume a certain smoothness of the signal cannot restore the small structures.

This paper presents a new similarity measure and nonlocal filters for images corrupted by multiplicative noise. The considered filters are generalizations of the nonlocal means filter of Buades et al., which is known to be well suited for removing additive Gaussian noise. To adapt to different noise models, the patch comparison involved in this filter has first of all to be performed by a suitable noise dependent similarity measure. To this purpose, we start by studying a probabilistic measure recently proposed for general noise models by Deledalle et al. We analyze this measure in the context of conditional density functions and examine its properties for images corrupted by additive and multiplicative noise. Since it turns out to have unfavorable properties for multiplicative noise we deduce a new similarity measure consisting of a probability density function specially chosen for this type of noise. The properties of our new measure are studied theoretically as well as by numerical experiments. To obtain the final nonlocal filters we apply a weighted maximum likelihood estimation framework, which also incorporates the noise statistics. Moreover, we define the weights occurring in these filters using our new similarity measure and propose different adaptations to further improve the results. Finally, restoration results for images corrupted by multiplicative Gamma and Rayleigh noise are presented to demonstrate the very good performance of our nonlocal filters.

In this paper we develop a data-driven mixture of vector autoregressive models with exogenous components. The process is assumed to change regimes according to an underlying Markov process. In contrast to the hidden Markov setup, we allow the transition probabilities of the underlying Markov process to depend on past time series values and exogenous variables. Such processes have potential applications to modeling brain signals. For example, brain activity at time t (measured by electroencephalograms) will can be modeled as a function of both its past values as well as exogenous variables (such as visual or somatosensory stimuli). Furthermore, we establish stationarity, geometric ergodicity and the existence of moments for these processes under suitable conditions on the parameters of the model. Such properties are important for understanding the stability properties of the model as well as deriving the asymptotic behavior of various statistics and model parameter estimators.

In this paper we consider a multivariate switching model, with constant states means
and covariances. In this model, the switching mechanism between the basic states of
the observed time series is controlled by a hidden Markov chain. As illustration, under
Gaussian assumption on the innovations and some rather simple conditions, we prove
the consistency and asymptotic normality of the maximum likelihood estimates of the model parameters.

The mathematical modelling of problems in science and engineering leads often to partial differential equations in time and space with boundary and initial conditions.The boundary value problems can be written as extremal problems(principle of minimal potential energy), as variational equations (principle of virtual power) or as classical boundary value problems.There are connections concerning existence and uniqueness results between these formulations, which will be investigated using the powerful tools of functional analysis.The first part of the lecture is devoted to the analysis of linear elliptic boundary value problems given in a variational form.The second part deals with the numerical approximation of the solutions of the variational problems.Galerkin methods as FEM and BEM are the main tools. The h-version will be discussed, and an error analysis will be done.Examples, especially from the elasticity theory, demonstrate the methods.

We propose a model for acid-mediated tumor invasion involving two different scales: the microscopic one, for the dynamics of intracellular protons and their exchange with their extracellular counterparts, and the macroscopic scale of interactions between tumor cell and normal cell populations, along with the evolution of extracellular protons. We also account for the tactic behavior of cancer cells, the latter being assumed to biase their motion according to a gradient of extracellular protons (following [2,31] we call this pH taxis). A time dependent (and also time delayed) carrying capacity for the tumor cells in response to the effects of acidity is considered as well. The global well posedness of the resulting multiscale model is proved with a regularization and fixed point argument. Numerical simulations are performed in order to illustrate the behavior of the model.

Diese Arbeit beschäftigt sich mit dem Algorithmus von Kalman zur Schätzung von gegenwärtigen und zukünftigen Zuständen in zeitdiskreten dynamischen Systemen. In der Literatur ist dieser Algorithmus allgemein als Kalman-Filter bekannt. Im Vordergrund der Betrachtungen stehen dabei die Schätzfehler des Kalman-Filters, insbesondere für den Fall, daß das benutzte Modell nicht mit dem realen System übereinstimmt. Es wird der Frage nachgegangen, welche Einflüsse die Modellfehler auf die Schätzfehler des Kalman-Filters haben. Dies ist ein wichtiger Gesichtspunkt, den man bei der Anwendung des Kalman-Filters beachten sollte, da man i.a. nicht davon ausgehen kann, daß Modell und reales System übereinstimmen.; Um diese Fragestellung stärker zu motivieren, werden im nächsten Abschnitt ein paar allgemeine Überlegungen zur Modellbildung angestellt. Danach werden einige Modelle zur Behandlung von Zeitreihen angesprochen. Zur Hinführung auf den Kalman-Filter wird dann in Kapital 2 das Problem des Schätzens etwas allgemeiner behandelt. In Kapitel 3 erfolgt dann eine Herleitung des Kalman-Filters und die Untersuchung der Fehlerprozesse für den Fall, daß Modell und reales System übereinstimmen. Da für die zeitliche Entwicklung der Fehlerprozesse die Stabilität des Kalman-Filters von Bedeutung ist, wird auch diese besprochen. In Kapitel 4 werden schließlich die Fehlerprozesse für den Fall behandelt, daß Modell und reales System nicht übereinstimmen.

The asymptotic behaviour of a singular-perturbed two-phase Stefan problem due to slow diffusion in one of the two phases is investigated. In the limit the model equations reduce to a one-phase Stefan problem. A boundary layer at the moving interface makes it necessary to use a corrected interface condition obtained from matched asymptotic expansions. The approach is validated by numerical experiments using a front-tracking method.

The asymptotic analysis of IBVPs for the singularly perturbed parabolic PDE ... in the limit epsilon to zero motivate investigations of certain recursively defined approximative series ("ping-pong expansions"). The recursion formulae rely on operators assigning to a boundary condition at the left or the right boundary a solution of the parabolic PDE. Sufficient conditions for uniform convergence of ping-pong expansions are derived and a detailed analysis for the model problem ... is given.

Second Order Scheme for the Spatially Homogeneous Boltzmann Equation with Maxwellian Molecules
(1995)

In the standard approach, particle methods for the Boltzmann equation are obtained using an explicit time discretization of the spatially homogeneous Boltzmann equation. This kind of discretization leads to a restriction of the discretization parameter as well as on the differential cross section in the case of the general Boltzmann equation. Recently, it was shown, how to construct an implicit particle scheme for the Boltzmann equation with Maxwellian molecules. The present paper combines both approaches using a linear combination of explicit and implicit discretizations. It is shown that the new method leads to a second order particle method, when using an equiweighting of explicit and implicit discretization.

Simulation methods like DSMC are an efficient tool to compute rarefied gas flows. Using supercomputers it is possible to include various real gas effects like vibrational energies or chemical reactions in a gas mixture. Nevertheless it is still necessary to improve the accuracy of the current simulation methods in order to reduce the computational effort. To support this task the paper presents a comparison of the classical DSMC method with the so called finite Pointset Method. This new approach was developed during several years in the framework of the European space project HERMES. The comparison given in the paper is based on two different testcases: a spatially homogeneous relaxation problem and a 2-dimensional axisymmetric flow problem at high Mach numbers.

The paper presents theoretical and numerical investigations on simulation methods for the Boltzmann equation with axisymmetric geometry. The main task is to reduce the computational effort by taking advantage of the symmetry in the solution of the Boltzmann equation.; The reduction automatically leads to the concept of weighting functions for the radial space coordinate and therefore to a modified Boltzmann equation. Consequently the classical simulation methods have to be modified according to the new equation.; The numerical results shown in this paper - rarefied gas flows around a body with axisymmetric geometry - were done in the framework of the European space project HERMES.