- ECTS Credits: 6
- Semester: 1
- Year: 2
- Campus: University of Aveiro
- Language: English
- Unit Coordinator: Libor Čermák
- ECTS Credits: 4
- Semester: 1
- Year: 2
- Campus: Brno University of Technology
- Language: English
- Aims:
The course is intended as an introduction to the computational fluid dynamics. Considerable emphasis will be placed on the inviscid compressible low: namely, the derivation of Euler equations, properties of hyperbolic systems and an introduction of several methods based on the finite volumes. Methods for computations of viscous lows will be also studied, namely the pressure-correction method and the spectral element method. Students ought to realize that only the knowledge of substantial physical and mathematical aspects of particular types of lows enables them to choose an effective numerical method and an appropriate software product. The development of individual semester assignement constitutes an important experience enabling to verify how the subject matter was managed.
- Content:
1. Material derivative, transport theorem, mass, momentum and energy conservation laws.
2. Constitutive relations, thermodynamic state equations, Navier-Stokes and Euler equations, initial and boundary conditions.
3. Trauc low equation, acoustic equations, shallow water equations.
4. Hyperbolic system, classical and week solution, discontinuities.
5. The Riemann problem in linear and nonlinear case, wave types.
6. Finite volume method in one and two dimensions, numerical lux.
7. Local error, stability, convergence.
8. The Godunov's method, lux vector splitting methods: the Vijayasundaram, the Steger- Warming, the Van Leer.
9. Viscous incompressible low: finite volume method for orthogonal staggered grids, pressure correction method SIMPLE.
10. Pressure correction method for colocated variable arrangements, non-orthogonal and unstructured meshes.
11. Stokes problem, spectral element method.
12. Steady Navier-Stokes problem, spectral element method.
13. Unsteady Navier-Stokes problem.
- Pre-requisites:
Evolution partial differential equations, functional analysis, numerical methods for partial differential equations.
- Reading list:
R.J. LeVeque: Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge, 2002.
E.F. Toro: Riemann Solvers and Numerical Methods for Fluid Dynamics, A Practical Introduction, Springer, Berlin, 1999.
S.V. Patankar: Numerical Heat Transfer and Fluid Flow, McGraw-Hill, New York, 1980.
J.H. Ferziger, M. Peric: Computational Methods for Fluid Dynamics, Springer-Verlag, New York, 2002.
M.O. Deville, P.F. Fischer, E.H. Mund: High-Order Methods for Incompressible Fluid Flow. Cambridge University Press, Cambdrige, 2002.
A. Quarteroni, A. Valli: Numerical Approximatipon of Partial Differential Equations. Springer- Verlag, Berlin, 1994
- Additional info:
Basic physical laws of continuum mechanics: laws of conservation of mass, momentum and energy. Theoretical study of hyperbolic conservation laws, particularly of Euler equations that describe the motion of inviscid compressible luids. Numerical modelling based on the finite volume method. Numerical modelling of incompressible lows: Navier-Stokes equations, pressure-correction method, spectral element method.
- Unit Coordinator: Jerzy Dajka
- ECTS Credits: 6
- Semester: 1
- Year: 2
- Campus: University of Silesia in Katowice
- Language: English
- Aims:
My aim is to present mathematical methods for quantum information processing. As in most applications it is enough to work with qubits and systems of qubits, mathematical methods originate from linear algebra, which is usually one of first curses taught. It makes quantum information accessible for very 'fresh' students. I would like to convince students that quantum information processing is useful, interesting, counter-intuitive, sometimes seemingly as mysterious as the Schroedinger cat.
- Content:
Content:
Mathematical formalism of quantum mechanics. Postulates of quantum mechanics.
Quantum information: quantum gates, no-go theorems, measurement. Quantum entanglement: mathematical basis.
Selected applications: teleportation, dense coding. Quantum cloning and applications.
Basic protocols for quantum cryptography: BB84, B92.
Quantum nonlocality: Bell and Leggett-Garg inequalities, contextuality.
Dynamics of quantum systems, open quantum systems.
Quantum error correction.
- Pre-requisites:
Pre-requisites:
Basic linear algebra is enough. A bit of number theory can be useful but not necessary. - Reading list:
Reading list:
Quantum Computation and Quantum Information by Michael A. Nielsen & Isaac L. ChuangLecture notes by John Preskill
- Unit Coordinator: Donato Pera
- Programme: InterMaths
- ECTS Credits: 6
- Semester: 1
- Year: 2
- Campus: University of L'Aquila
- Language: English
- Delivery: In-class
- Aims:
The aim of the course is the study of analytical, numerical and computational methods (on parallel computing structures), for the solution of partial differential equations considered as basic elements for the construction of mathematical models for natural disasters. During the course will be introduced basic concepts related to the analytical and numerical solution for wave equations, elastodynamics equations and advection-reaction-diffusion systems.
In addition to the classical analytical and numerical approach, some basic elements for the simulation of the studied problems on parallel computing structures will be introduced, with reference to the programming of Shared Memory, Distributed Memory and/or GPU computing architectures.
The course activities are consistent with the professional profiles proposed by the Mathematical Engineering master course in relation to the acquisition of programming skills for complex computing structures and the solution of theoretical models.Learning outcomes
At the end of the course the student should be able to:
1) Know the basic aspects related to the analytical and numerical solutions of the proposed models.
2) Use parallel computing codes related to the models proposed during the course.
3) Propose solutions related to problems similar to the models proposed during the course.
4) Understand technical-scientific texts on related topics. - Content:
Wave equations analytical and numerical methods, elastodynamic equations analytical and numerical methods.
Diffusion equations analytical and numerical methods.
Advection-reaction-diffusion systems analytical and numerical methods.
Introduction to parallel computing architectures:
Shared memory systems, distributed memory systems and GPU computing.
Models for performance evaluation Speedup, Efficiency and Amdahl's law.
Introduction to Linux/Unix operating systems and scheduling for HPC applications.
Message passing interface (MPI) programming
MPI basic notions, point-to-point communications, collective communications - Pre-requisites:
Basic knowledge of mathematical analysis, numerical analysis and scientific programming
