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Exact and approximate Riemann solvers for the Euler equations
Živčáková, Andrea ; Kučera, Václav (advisor) ; Felcman, Jiří (referee)
In this work we deal with the solution and implementation of the problem of solving a partial differential equation with a piecewise constant initial condition, the so-called Riemann's problem. Specifically, we study the equations of conservation laws describing inviscid adiabatic flow of an ideal gas - the Euler equations. After some investigation, we show that these equations can be transformed to a quasilinear hyperbolic partial differential equation of first order. We are especially interested in the one-dimensional Euler equations for which we want to get an analytically exact Riemann's solver. The solution is found by investigation of properties of waves, namely rarefaction waves, shock waves and contact discontinuities were treated. The output of this work is a program in C for finding the exact Riemann's solver for one-dimensional Euler equations. The program is based on a theoretical analysis summarized in the first two chapters, and is tested on standard test data. The theory is based on the books [1] and [2].
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Adaptivní hp nespojitá Galerkinova metoda pro nestacionární stlačitelné Eulerovy rovnice
Korous, Lukáš ; Feistauer, Miloslav (advisor) ; Dolejší, Vít (referee)
The compressible Euler equations describe the motion of compressible inviscid fluids. They are used in many areas ranging from aerospace, automotive, and nuclear engineering to chemistry, ecology, climatology, and others. Mathematically, the compressible Euler equations represent a hyperbolic system consisting of several nonlinear partial differential equations (conservation laws). These equations are solved most frequently by means of Finite Volume Methods (FVM) and low-order Finite Element Methods (FEM). However, both these approaches are lacking higher order accuracy and moreover, it is well known that conforming FEM is not the optimal tool for the discretization of first-order equations. The most promissing approach to the approximate solution of the compressible Euler equations is the discontinuous Galerkin method that combines the stability of FVM, with excellent approximation properties of higher-order FEM. The objective of this Master Thesis was to develop, implement and test new adaptive algorithms for the nonstationary compressible Euler equations based on higher-order discontinuous Galerkin (hp-DG) methods. The basis for the new methods were the discontinuous Galerkin methods and space-time adaptive hp-FEM algorithms on dynamical meshes for nonstationary second-order problems. The new algorithms...
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Mathematical models in hydromechanics (and aerodynamics)
Ježková, Jitka ; Zatočilová, Jitka (referee) ; Nechvátal, Luděk (advisor)
Bachelor thesis is a summarizing text which deals with the state and the motion of ideal liquid and gas. The main goal is to derive Euler equations describing the flow of fluids. From these equations we can obtain Bernoulli equation that is directly used to solve problems of fluid flow. The next step is to derive the continuity equation expressing the fact that the mass is preserved in the system. In the case of ideal gas the state equation of ideal gas is added and therefore solutions of various types of tasks of hydrodynamics and aerodynamics can be achieved.
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Numerické simulace stlačitelného proudění v kanále s pohyblivou stěnou
Prokopova, J. ; Feistauer, M. ; Kučera, V. ; Horáček, Jaromír
This work is concerned with the simulation of inviscid compressible flow in time dependent domains. The special treatment is given to the ALE (Arbitrary Lagrangian-Eulerian) method. We present an ALE formulation of the Euler equations describing the compressible flow and discretize them. The space discretize them. The space discretization is carried out by the discontinuous Galerkin method and for the time discretization a semi-implicit scheme is used. In the end of the paper some computational results are presented.
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Numerické řešení stacionárního a nestacionárního proudění kolem profilu v kanálu
Furmánek, P. ; Horáček, Jaromír ; Kozel, Karel
The work deals with steady and unsteady solution of subsonic flow over a profile DCA 18%in a channel. For the computation the predictor-correstor MacCormack scheme with modified TVD Cousonś artificial dissipation is used. Firstly, the steady state solution compared to the experimental results is presented. Than a simple unsteady model based on pressure change at the outlet area of the computational domain and finaly an unsteady model obtained with the use of ALE method (moving mesh) are presented.
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