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Numerical solution of traffic flow models
Vacek, Lukáš ; Kučera, Václav (advisor) ; Janovský, Vladimír (referee)
Our work describes the simulation of traffic flows on networks. These are described by partial differential equations. For the numerical solution of our models, we use the discontinuous Galerkin method in space and a multistep method in time. This combination of the two methods on networks is unique and leads to a robust numerical scheme. We use several different approaches to model the traffic flow. Thus, our program must solve both scalar problems as well as systems of equations described by first and second order partial differential equations. The output of our programs is, among other things, the evolution of traffic density in time and 1D space. Since this is a physical quantity, we introduce limiters which keep the density in an admissible interval. Moreover, limiters prevent spurious oscillations in the numerical solution. All the above is performed on networks. Thus, we must deal with the situation at the junctions, which is not standard. The main task is to ensure that the law of conservation of the total amount of cars passing through the junction is still satisfied. This is achieved by modifying the numerical flux for junctions. The result of this work is the comparison of all the models, the demonstration of the benefits of the discontinuous Galerkin method and the influence of limiters.
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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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Periodic solutions of ordinary differential equations
Mitro, Erik ; Janovský, Vladimír (advisor) ; Felcman, Jiří (referee)
The thesis deals with periodic solutions of ordinary differential equations and examining of their stability. We are mainly limited to scalar differential equations. The first chapter is devoted to the stability of periodic solutions that is related to the Poincaré map. The aim is to decide on the asymptotic stability/instability of the fixed point of this map. To this end we need to compute derivatives of the Poincaré map of the first order or, possibly, of the higher orders. In the second chapter we introduce the concept of bifurcation and we examine the population model. In the third chapter we briefly mention the Van der Pol oscillator i.e the system of two equations. We illustrate the theory by examples.
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Filippov dynamical systems with applications
Šimonová, Dorota ; Janovský, Vladimír (advisor) ; Ratschan, Stefan (referee)
The thesis is motivated by problems of contact mechanics with friction. At the beginning we describe a class of piecewise smooth systems with discontinuous vector field called Filippov systems. We also show how to solve them. The rest of this thesis is focused on applications, especially dry friction model and finite element model of Coulomb friction with one contact point. We propose a technique for simulation of the second mentioned model which combines sovling methods for Filippov systems and impact oscillators. Powered by TCPDF (www.tcpdf.org)
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Numerická optimalizace
Márová, Kateřina ; Janovský, Vladimír (advisor) ; Lukšan, Ladislav (referee)
This thesis addresses the topic of unconstrained optimization. It describes seven derivative-free optimization methods for objective functions of multiple variables. Three groups of methods are distinguished. The Alternating Variable method and the method of Hooke and Jeeves represent the pattern search methods. Then there are two simplex algorithms: one by Spendley, Hext and Himsworth and the amoeba algorithm of Nelder and Mead. The family of methods with adaptive sets of search directions consists of Rosenbrock's method, the method of Davies, Swann and Campey, and Powell's method. All algorithms are implemented in MATLAB and tested on three functions of two variables. Their progression is illustrated by multiple figures and their comparative analysis is given. Powered by TCPDF (www.tcpdf.org)
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On a model of corruption in a democratic society
Splítek, Martin ; Janovský, Vladimír (advisor) ; Mlčoch, Lubomír (referee)
The aim of this work is to study the behavior of serious social pheno- menon - corruption, and we do this through a mathematical model of corruption in a democratic society, published in [1]. The model is a dynamical system of three differential equations, specified by three variables and ten parameters. The model is studied by means of numerical analysis, namely, the method of nume- rical integration of ordinary differential equations and the method of numerical continuation. We used toolbox Matcont [2], which works in the environment of program MATLAB [3]. The result is commented parametric study of the pheno- menon of corruption. Keywords: ordinary diferential equations, dynamic systems, bifurcation ana- lysis 1
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Steady states of dynamical systems
Šerý, David ; Janovský, Vladimír (advisor) ; Vlasák, Miloslav (referee)
In the thesis we analyse qualitative properties of dynamical systems near equilibria. We mainly deal with planar equations. The key notion is the stability of steady state. The stability analysis is closely connected to linearisation, which in many cases doesn't suffice. In that case Lyapunov function may help. We define stable and unstable manifold, basin of attraction, topological equivalence of equations and demonstrate their significance in qualitative analysis. The theory will be illustrated on examples. In the third chapter we briefly mention numerical continuation of steady states with respect to a parameter. 1
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Mathematical models of ecosystems
Scholle, David ; Janovský, Vladimír (advisor) ; Kofroň, Josef (referee)
This work is about models of population growth in different situations. At first, we will examine amount of spiders and their prey in the region of Langa Astigiana, based on models of dynamical systems. We will also consider the usage of spraying of near vineyards and effect of this on the ecosystem. The aim of this work is also to check the possibility of periodical cycles, and thus also of the Hopf Bifurcation, appearing. Next part talks about the model of a beehive and examines the influence of insecticides on the population of bee drones and worker bees. The aim of the last chapter is to examine the effectivity and possible impact of human intervention in the region of Šumava forest. The model will check the necessity of such action against parasites. The software used for these tasks will be mainly the continuation toolbox MatCont, which is a part of the program MatLab.
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Numerical solution of equations describing the dynamics of flocking
Živčáková, Andrea ; Kučera, Václav (advisor) ; Janovský, Vladimír (referee)
This work is devoted to the numerical solution of equations describing the dynamics of flocks of birds. Specifically, we pay attention to the Euler equations for compressible flow with a right-hand side correction. This model is based on the work Fornasier et al. (2010). Due to the complexity of the model, we focus only on the one-dimensional case. For the numerical solution we use a semi-implicit discontinuous Galerkin method. Discretization of the right-hand side is chosen so that we preserve the structure of the semi-implicit scheme for the Euler equations presented in the work Feistauer, Kučera (2007). The proposed numerical scheme was implemented and numerical experiments showing the robustness of the scheme were carried out. Powered by TCPDF (www.tcpdf.org)
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Numerical solution of ordinary differential equations
Monhartová, Petra ; Feistauer, Miloslav (advisor) ; Janovský, Vladimír (referee)
In the present work we study numerical methods for the nu- merical solution of initial value problems for ordinary differential equations. With the aid of the Taylor formula we derive several one-step methods. We compare numerical solution computed with explicit and implicit Eu- ler methods. Moreove, we are concerned with second-order and fourth-order Runge-Kutta methods. We find how accurately the numerical methods obta- ined with the aid of these methods approximate the exact solution. Further we estimate the error of these method by the half-step method. 1
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