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A traffic flow with a bottelneck
Kovařík, Adam ; Janovský, Vladimír (advisor) ; Vejchodský, Tomáš (referee)
Title: A traffic flow with a bottelneck Author: Adam Kovařík Department: Department of Numerical Mathematics Supervisor: prof. RNDr. Vladimír Janovský, DrSc. Supervisor's e-mail address: janovsky@karlin.mff.cuni.cz Abstract: In this paper we study a microscopic follow-the-leader traffic model on a circu- lar road with a bottleneck. We assume that all drivers are identical and overtaking is not permitted. We sketch a small part of the rich dynamics of the model including Hopf and Neimark-Sacker bifurcations. We introduce so called POM and quasi-POM solutions and an algorithm how to search them. The main goal of this work is to investigate how the optimal velocity model with a bottleneck deals with so called aggressive behavior of dri- vers. The effect of variable reaction time and a combination of both named factors is also tested. Using numerical simulations we'll find out that aggressiveness and faster reactions have positive effect on traffic flow. In the end we discuss models with two bottlenecks and with one extraordinary driver. Keywords: dynamical systems, ODEs, traffic flow, bottleneck, aggressiveness. 1
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Computational comparison of hp-adaptive approaches
Kubásek, Petr ; Vejchodský, Tomáš (advisor) ; Feistauer, Miloslav (referee)
Cílem této práce je porovnat řízení hp-adaptivního procesu pomocí referenčního řešení a různých aposteriorních odhadů chyby. Tyto přístupy jsou porovnávány z hlediska globální diskretizační chyby a potřebného počtu stupňů volnosti. Konkrétně se zabýváme explicitními residuálními odhady, implicitními residuálními odhady Dirichletova a Neumannova typu a hierarchickými odhady. Všechny odhady jsou v práci podrobně odvozeny včetně jejich nejvýznamnějších vlastností. Jednotlivé přístupy jsou srovnávány pomocí numerických experimentů. Na jejich základě lze ríci, že nejlepších výsledků dosahuje adaptivita řízená pomocí referenčního řešení společně s implicitním Dirichletovým odhadem. Referenční řešení se zdá být nejspolehlivější metodou zatímco implicitní Dirichletův odhad je, s výjimkou některých případů, nejrychlejší.
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Stochastické modelování reakčně-difuzních procesů v biologii
Lipková, Jana ; Maslowski, Bohdan (advisor) ; Vejchodský, Tomáš (referee)
Many biological processes can be described in terms of chemical reactions and diffusion. In this thesis, reaction-diusion mechanisms related to the formation of Turing patterns are studied. Necessary and sufficient conditions under which Turing instability occur is presented. Behaviour of Turing patterns is investigated with a use of deterministic approach, compartment-based stochastic simulation algorithm and molecular-based stochastic simulation algorithm.
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Computation of an anisotropic and nonlinear magnetic field by the finite element method
Kunický, Zdeněk ; Vejchodský, Tomáš (advisor) ; Křížek, Michal (referee)
In the present work we study the modelling of stationary magnetic fields in nonlinear anisotropic media by FEM. The magnetic characteristics of such materials are thoroughly examined and eventually applied to the construction of a full 2D model of an anisotropic steel sheet. Some improvements in the construction in comparison with the ones previously published are achieved. We also present an extension of a 3D model of steel and dielectric laminations for anisotropic sheets. We point out that the standard formulations and the subsequent theorems for the boundary value problems in fact do not correspond with the physical situation. Instead, we propose new formulations that reflect the real physical properties of matter. General existence and uniqueness theorems for the obtained boundary value problems are proved as well as the convergence theorems for the discrete solutions. Finally, the conventional and full 2D model of an anisotropic steel sheet are compared in two transformer core models using the adaptive Newton-Raphson iterative scheme and the obtained results are presented and analysed.
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Automatic hp-adaptivity on Meshes with Arbitrary-Level Hanging Nodes in 3D
Kůs, Pavel ; Vejchodský, Tomáš (advisor) ; Segeth, Karel (referee) ; Dolejší, Vít (referee)
The thesis is concerned with theoretical and practical aspects of the hp- adaptive finite element method for solving elliptic and electromagnetic prob- lems described by partial differential equations in three spatial dimensions. Besides the standard element refinements, the hp-adaptivity allows indepen- dent adaptation of degrees of the polynomial approximation as well. This leads to exponentially fast convergence even for problems with singularities. The efficiency of the hp-adaptivity is enhanced even more by the ability of the algorithm to work with meshes with arbitrary-level hanging nodes. This generality, however, leads to great complexity of the implementation. There- fore, the thesis concentrates on the mathematical analysis of algorithms that have led to successful implementation of the method. In addition, the the- sis discusses the numerical integration in 3D and the implementation of the method itself. Finally, numerical results obtained by this new implemen- tation are presented. They confirm advantages of hp-adaptivity on meshes with arbitrary-level hanging nodes. 1
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Use of the hp discontinuous Galerkin method for a simulation of compressible flows
Tarčák, Karol ; Dolejší, Vít (advisor) ; Vejchodský, Tomáš (referee)
Title: Application of hp-adaptive discontinuous Galerkin method to com- pressible flow simulation Author: Karol Tarčák Department: Department of Numerical Mathematics Supervisor: prof. RNDr. Vít Dolejší, Ph.D., DSc. Abstract: In the present work we study an residuum estimate of disconti- nuous Galerkin method for the solution of Navier-Stokes equations. Firstly we summarize the construction of the viscous compressible flow model via Navier-Stokes partial differential equation and discontinuous Galerkin met- hod. Then we propose an extension of an already known residuum estimate for stationary problems to non-stationary problems. We observe the beha- vior of the proposed estimate and modify an existing hp-adaptive algorithm to use our estimate. Finally we apply the modified algorithm on test cases and present adapted meshes from the numerical experiments. Keywords: discontinuous Galerkin method, adaptivity, error estimate 4
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Programs and Algorithms of Numerical Mathematics 18 : Janov nad Nisou, June 19-24, 2016 : proceedings of seminar
Chleboun, J. ; Kůs, Pavel ; Přikryl, Petr ; Segeth, Karel ; Šístek, Jakub ; Vejchodský, Tomáš
This book comprises papers that originated from the invited lectures, survey lectures, short communications, and posters presented at the 18th seminar Programs and Algorithms of Numerical Mathematics (PANM) held in Janov nad Nisou, Czech Republic, June 19-24, 2016. All the papers have been peer-reviewed. The seminar was organized by the Institute of Mathematics of the Czech Academy of Sciences under the auspices of EU-MATHS-IN.cz, Czech Network for Mathematics in Industry. It continued the previous seminars on mathematical software and numerical methods held (biennially, with only one exception) in Al šovice, Bratří kov, Janov nad Nisou, Ko řenov, L ázně Libverda, Dolní Maxov, and Prague in the period 1983-2014. The objective of this series of seminars is to provide a forum for presenting and discussing advanced theoretical as well as practical topics in numerical analysis, computer implementation of algorithms, new approaches to mathematical modeling, and single- or multi-processor applications of computational methods.
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Programs and Algorithms of Numerical Mathematics 19 : Hejnice, June 24-29, 2018 : proceedings of seminar
Chleboun, J. ; Kůs, Pavel ; Přikryl, Petr ; Rozložník, Miroslav ; Segeth, Karel ; Šístek, Jakub ; Vejchodský, Tomáš
These proceedings contain peer-reviewed papers that are based on the invited lectures, survey lectures, short communications, and posters presented at the 19th seminar Programs and Algorithms of Numerical Mathematics (PANM) held in the International Center for Spiritual Rehabilitation, Hejnice, Czech Republic, June 24-29, 2018. The seminar was organized by the Institute of Mathematics of the Czech Academy of Sciences under the auspices of EU-MATHS-IN.cz, Czech Network for Mathematics in Industry, and with the financial support provided by the RSJ Foundation. It continued the previous seminars on mathematical software and numerical methods held (biennially, with only one exception) in Alšovice, Bratříkov, Janov nad Nisou, Kořenov, Lázně Libverda, Dolní Maxov, and Prague in the period 1983-2016. The objective of this series of seminars is to provide a forum for presenting and discussing advanced topics in numerical analysis, computer implementation of numerical algorithms, new approaches to mathematical modeling, and single- or multi-processor applications of computational methods.
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Numerical solution of convection-diffusion problems by discontinuous Galerkin method
Vlasák, Miloslav ; Dolejší, Vít (advisor) ; Janovský, Vladimír (referee) ; Vejchodský, Tomáš (referee)
This work is concerned with the theoretical analysis of the discontinuous Galerkin finite element method. We use a discontinuous Galerkin formulation for a scalar convection-diffusion equation with nonlinear convective term. The resulting semidiscretized equations with symmetric (SIPG) or nonsymmetric (NIPG) diffusive term are then discretized in time by Backward Differential formulae (BDF), implicit Runge-Kutta methods and Time discontinuous Galerkin. All of these schemes are linearized by a suitable explicit extrapolations to avoid nonlinearity in the convective term. These final schemes are theoretically analyzed and error estimates are derived. We also present some superconvergence result for Time discontinuous Galerkin for nonsymmetric operator. Numerical experiments verify the theoretical results.
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Adaptive methods for singularly perturbed partial differential equations
Lamač, Jan ; Knobloch, Petr (advisor) ; Franz, Sebastian (referee) ; Vejchodský, Tomáš (referee)
This thesis deals with solving singularly perturbed convection- diffusion equations. Firstly, we construct a matched asymptotic expansion of the solution of the singularly perturbed convection-diffusion equation in 1D and derive a formula for the zeroth-order asymptotic expansion in several two- dimensional polygonal domains. Further, we present a set of stabilization meth- ods for solving singularly perturbed problems and prove the uniform convergence of the Il'in-Allen-Southwell scheme in 1D. Finally, we introduce a modification of the streamline upwind Petrov/Galerkin (SUPG) method on convection-oriented meshes. This new method enjoys several profitable properties such as the ful- filment of the discrete maximum principle. Besides the analysis of the method and derivation of a priori error estimates in respective energy norms we also carry out several numerical experiments verifying the theoretical results.
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