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Numerical solution of equations describing the dynamics of flocking
Živčáková, Andrea ; Kučera, Václav (advisor)
This work is devoted to the numerical solution of equations describing the dynamics of flocks of birds. Specifically, we pay attention to the Euler equati- ons 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 numeri- cal scheme was implemented and numerical experiments showing the robustness of the scheme were carried out. 1
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Goal-oriented a posteriori error estimates and adaptivity for the numerical solution of partial differential equations
Roskovec, Filip ; Dolejší, Vít (advisor) ; Kanschat, Guido (referee) ; Zeman, Jan (referee)
A posteriori error estimation is an inseparable component of any reliable numerical method for solving partial differential equations. The aim of the goal-oriented a posteriori error estimates is to control the computational error directly with respect to some quantity of interest, which makes the method very convenient for many engineering applications. The resulting error estimates may be employed for mesh adaptation which enables to find a numerical approximation of the quantity of interest under some given tolerance in a very efficient manner. In this thesis, the goal-oriented error estimates are derived for discontinuous Galerkin discretizations of the linear scalar model problems, as well as of the Euler equations describing inviscid compressible flows. It focuses on several aspects of the goal-oriented error estimation method, in particular, higher order reconstructions, adjoint consistency of the discretizations, control of the algebraic errors arising from iterative solutions of both algebraic systems, and linking the estimates with the hp-anisotropic mesh adaptation. The computational performance is demonstrated by numerical experiments.
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Numerical solution of equations describing the dynamics of flocking
Živčáková, Andrea ; Kučera, Václav (advisor)
This work is devoted to the numerical solution of equations describing the dynamics of flocks of birds. Specifically, we pay attention to the Euler equati- ons 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 numeri- cal scheme was implemented and numerical experiments showing the robustness of the scheme were carried out. 1
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The Gibbs phenomenon in the discontinuous Galerkin method
Stará, Lenka ; Kučera, Václav (advisor) ; Sváček, Petr (referee)
The solution of the Burgers' equation computed by the standard finite element method is degraded by oscillations, which are the manifestation of the Gibbs phenomenon. In this work we study the following numerical me- thods: Discontinuous Galerkin method, stable low order schemes and the flux corrected technique method in order to prevent the undesired Gibbs phenomenon. The focus is on the reduction of severe overshoots and under- shoots and the preservation of the smoothness of the solution. We consider a simple 1D problem on the interval (0, 1) with different initial conditions to demonstrate the properties of the presented methods. The numerical results of individual methods are provided. 1
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Numerical solution of the shallow water equations
Šerý, David ; Dolejší, Vít (advisor) ; Felcman, Jiří (referee)
The thesis deals with the numerical solution of partial differential equati- ons describing the flow of the so-called shallow water neglecting the flow in the vertical direction. These equations are of hyperbolical type of the first or- der with a reactive term representing the bottom topology. We discretize the resulting system of equations by the implicit space-time discontinuous Ga- lerkin method (STDGM). In the literature, the explicit techniques are used most of the time. The implicit approach is suitable especially for adaptive methods, because it allows the usage of different meshes for different time niveaus. In the thesis we derive the corresponding method and an adaptive algorithm. Finally, we present usage of the method in several examples. 1
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Discontinuous Galerkin method for the solution of boundary-value problems in non-smooth domains
Bartoš, Ondřej ; Feistauer, Miloslav (advisor) ; Dolejší, Vít (referee)
This thesis is concerned with the analysis of the finite element method and the discontinuous Galerkin method for the numerical solution of an elliptic boundary value problem with a nonlinear Newton boundary condition in a two-dimensional polygonal domain. The weak solution loses regularity in a neighbourhood of boundary singularities, which may be at corners or at roots of the weak solution on edges. The main attention is paid to the study of error estimates. It turns out that the order of convergence is not dampened by the nonlinearity, if the weak solution is nonzero on a large part of the boundary. If the weak solution is zero on the whole boundary, the nonlinearity only slows down the convergence of the function values but not the convergence of the gradient. The same analysis is carried out for approximate solutions obtained with numerical integration. The theoretical results are verified by numerical experiments. 1
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Incompressible and compressible viscous flow with low Mach numbers
Balázsová, M. ; Feistauer, M. ; Sváček, Petr ; Horáček, Jaromír
In this paper we compare incompressible flow and low Mach number compressible viscous flow. Incompressible Navier-Stokes equations were treated with the aid of discontinuous Galerkin method in space and backward difference method in time. We present numerical results for a flow in a channel which represents a simplified model of the human vocal tract. Presented numerical results give a good correspondence between the incompressible flow and the compressible flow with low Mach numbers.
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Adaptive space-time discontinuous Galerkin method for the solution of non-stationary problems
Vu Pham, Quynh Lan ; Dolejší, Vít (advisor) ; Feistauer, Miloslav (referee)
This thesis studies the numerical solution of non-linear convection-diffusion problems using the space- time discontinuous Galerkin method, which perfectly suits the space as well as time local adaptation. We aim to develop a posteriori error estimates reflecting the spatial, temporal, and algebraic errors. These estimates are based on the measurement of the residuals in dual norms. We derive these estimates and numerically verify their properties. Finally, we derive an adaptive algorithm and apply it to the numerical simulation of non-stationary viscous compressible flows. Powered by TCPDF (www.tcpdf.org)
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Numerical solution of nonlinear transport problems
Bezchlebová, Eva ; Feistauer, Miloslav (advisor) ; Vlasák, Miloslav (referee)
Práce je zaměřená na numerickou simulaci dvoufázového proudění. Je studován matematický model a numerická aproximace toku dvou nemísitelných nestlačitelných tekutin. Rozhraní mezi tekutinami je popsáno pomocí pomocí tzv. level set metody. Představena je diskretizace problému v prostoru a v čase. Metoda konečných prvk· se zpětnou Eulerovou metodou je aplikována na Navierovy-Stokesovy rovnice a časoprostorová nespojitá Galerkinova metoda je použita k řešení transportního problému. D·raz je kladen na analýzu chyby nespojité Galerkinovy metody přímek a časoprostorové nespojité Galerkinovy metody pro transportní problém. Jsou prezentovány numerické výsledky. 1
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