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Lecturer(s)
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Hozman Jiří, RNDr. Mgr. Ph.D.
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Course content
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1. Brief overview of equations describing the flow. Basic physical laws formulated in form of differential equations, constitutive and rheological relations, basic facts from the thermodynamics. 2. Mathematical theory of viscous incompressible flow. Function spaces, stationary Stokes problem, weak formulation, existence and uniqueness of a weak solution. 3. Stationary Navier-Stokes problem, existence and uniqueness of a weak solution, Oseen problem. 4. Nonstationary Navier-Stokes equations, functions with values in Banach space, weak formulation and solvability of the nonstationary problem. 5. Finite element method (FEM) for numerical solution of incompressible viscous flow. Continuous problem. Discrete Stokes problem. Finite element spaces. Babuška-Brezzi condition. Existence of an approximate solution. 6. Error estimates for the velocity and the pressure. Numerical realization of the discrete problem. Discrete Navier-Stokes problem. 7. Mathematical theory of compressible flow. Compressible Navier-Stokes equations, barotropic flow, Euler equations. 8. Finite volume method (FVM) for the Euler equations. Finite volume mesh and derivation of the finite volume scheme. Numerical flux, boundary conditions and the stability of FV scheme. 9. Numerical method for solution of compressible flow. Combined finite volume-finite element (FV-FE) method for viscous compressible flow. Triangulations, FV and FE spaces. 10. Space semidiscretization of the problem, time discretization, boundary conditions. 11. Discontinuous Galerkin method (DGFEM) for numerical solution of the Euler equations. Discretization, numerical solution. Limiting of the order of accuracy, approximation of the boundary. 12. DGFEM for convection-diffusion problems. Scalar problem, discretization, numerical solution. 13. DGFE discretization of the Navier-Stokes equations. Discrete problem, boundary conditions, numerical solution. 14. Reserve.
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Learning activities and teaching methods
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Monological explanation (lecture, presentation,briefing)
- Class attendance
- 56 hours per semester
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Learning outcomes
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Mathematical methods in fluid dynamics, Stokes and Navier-Stokes equations, their mathematical theory and numerical solution.
Student gains the basic knowledge of mathematical methods applied in fluid dynamics.
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Prerequisites
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Knowledge of subject Finite Element Method (KMD/MKP).
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Assessment methods and criteria
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Combined examination
Exam: Combined, according to the syllabus of subject.
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Recommended literature
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Feistauer M., Felcman J., Straškraba I.:. Mathematical and Computational Methods for Compressible Flow. Clarendon Press, Harlow, 2003.
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Feistauer M.:. Mathematical Methods in Fluid Dynamics. Longman Scientific-Technical, Harlow, 1993.
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