Adaptivní hp nespojitá Galerkinova metoda pro nestacionární stlačitelné Eulerovy rovnice

• Název práce v češtině: Adaptivní hp nespojitá Galerkinova metoda; pro nestacionární stlačitelné Eulerovy rovnice
• Název v anglickém jazyce: Adaptive hp Discontinuous Galerkin Method; for Nonstationary Compressible Euler Equations
• Klíčová slova: numerické simulace, metoda konečných prvků, Eulerovy rovnice, hp-adaptivita, nespojitá Galerkinova metoda
• Klíčová slova anglicky: numerical simulation, finite element method, Euler equations, hp-adaptivity, discontinuous Galerkin method
• Akademický rok vypsání: 2009/2010
• Typ práce: diplomová práce
• Jazyk práce: angličtina
• Ústav: Katedra numerické matematiky (32-KNM)
• Vedoucí / školitel: prof. RNDr. Miloslav Feistauer, DrSc., dr. h. c.
• Řešitel: skrytý - zadáno a potvrzeno stud. odd.
• Datum přihlášení: 15.11.2009
• Datum zadání: 15.11.2009
• Datum a čas obhajoby: 08.02.2012 00:00
• Datum odevzdání elektronické podoby: 02.12.2011
• Datum odevzdání tištěné podoby: 09.12.2011
• Datum proběhlé obhajoby: 08.02.2012
• Oponenti: prof. RNDr. Vít Dolejší, Ph.D., DSc.
• Konzultanti: RNDr. Mgr. Pavel Šolín, Ph.D.

Zásady pro vypracování

The compressible Euler equations describe the motion of compressible inviscid fluids such as gases or the air. They are used in many areas ranging from aerospace, automotive, and nuclear engineering to chemistry, ecology, climatology, and others. Mathematically, the compressible Euler equations are 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 continuous FEM is not the optimal tool for the discretization of first-order equations. The most promissing approach to the approximate solution of compressible Euler equations is the discontinuous Galerkin method that combines the stability of FVM [4] with excellent approximation properties of higher-order FEM [6].

The objective of this Master's Thesis is to develop, implement and test new adaptive algorithms for nonstationary compressible Euler equations based on higher-order discontinuous Galerkin (hp-DG) methods. The basis for the new methods will be the discontinuous Galerkin methods [1,2,4] and space-time adaptive hp-FEM algorithms on dynamical meshes for nonstationary second-order problems [3,5,6]. The new algorithms will be implemented and tested in the framework of the open source library Hermes [7], and they will be applied to selected problems of transonic flow and low Mach number flows in climatology.

Seznam odborné literatury

[1] V. Dolejsi: On the Discontinuous Galerkin Method for the Numerical Solution of the Navier-Stokes Equations, Int. J. Numer. Methods Fluids, 45:1083-1106, 2004.
[2] V. Dolejsi, M. Feistauer: A Semi-Implicit Discontinuous Galerkin Finite Element Method for the Numerical Solution of Inviscid Compressible Flow, J. Comp. Phys., 198(2): 727-746, 2004.
[3] L. Dubcova, P. Solin, J. Cerveny, P. Kus: Space and Time Adaptive Two-Mesh hp-FEM for Transient Microwave Heating Problems, Electromagnetics, accepted March 2009, in press,
[4] M. Feistauer, J. Felcman, J. Straskraba: Mathematical And Computational Methods For Compressible Flow, Oxford Science Publications, 2003.
[5] P. Solin, L. Dubcova, J. Kruis: Adaptive hp-FEM with Dynamical Meshes for Transient Heat and Moisture Transfer Problems, J. Comput. Appl. Math, doi 10.1016/j.cam.2009.07.025, 2009.
[6] P. Solin, K. Segeth, I Dolezel: Higher-Order Finite Element Methods, Chapman & Hall / CRC Press, 2003.
[7] HERMES: Open Source Library for Rapid Prototyping of Adaptive hp-FEM Solvers, http://hpfem.org/.