- 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
- ECTS Credits: 4.5
- Semester: 1
- Year: 2
- Campus: Ivan Franko National University of Lviv
- Language: English
- Unit Coordinator: dr hab. inż. Edyta Hetmaniok, prof. PŚ
- ECTS Credits: 5
- Year: 2
- Campus: Silesian University of Technology
- Language: English
- Aims:
The aim of the course is to get knowledge on constructing mathematical models of selected technical and economic problems, verifying them and simulating by using Mathematica software.
- Content:
1. Types of models, types of variables, selection of variables to the model.
2. Construction of the model.
3. Linear model - assumptions, estimation of parameters, verification procedures.
4. Analysis of the selected nonlinear models.
5. Analysis of the selected models described by means of differential equations.
6. Computer simulations of selected models.
- Code: DT0627
- Unit Coordinator: Donatella Donatelli
- ECTS Credits: 3
- Semester: 1
- Year: 1
- Campus: University of L'Aquila
- Language: English
- Aims:
Learning Objectives:
The aim of the course is to give an overview of fluid dynamics from a mathematical viewpoint, and to introduce students to the mathematical modeling of fluid dynamic type. At the end of the course students will be able to perform a qualitative and quantitative analysis of solutions for particular fluid dynamics problems and to use concepts and mathematical techniques learned from this course for the analysis of other partial differential equations.Learning Outcomes:
On successful completion of this course, the student should:- understand the basic principles governing the dynamics of non-viscous fluids;
- be able to derive and deduce the consequences of the equation of conservation of mass;
- be able to apply Bernoulli's theorem and the momentum integral to simple problems including river flows;
- understand the concept of vorticity and the conditions in which it may be assumed to be zero;
- calculate velocity fields and forces on bodies for simple steady and unsteady flows derived from potentials;
- demonstrate skill in mathematical reasoning and ability to conceive proofs for fluid dynamics equations.
- demonstrate capacity for reading and understand other texts on related topics. - Content:
CONTENTS FOR: Modelling and analysis of fluids and biofluids (9 ECTS), Mathematical fluid dynamics (6 ECTS), Mathematical Modelling of Continuum Media (3 ECTS)
- Derivation of the governing equations: Euler and Navier-Stokes
- Eulerian and Lagrangian description of fluid motion; examples of fluid flows
- Fluidi di tipo Poiseulle e Couette
- Vorticity equation in 2D and 3D
CONTENTS FOR: Modelling and analysis of fluids and biofluids (9 ECTS), Mathematical fluid dynamics (6 ECTS), Mathematical fluid and biofluid dynamics (6 ECTS),- Dimensional analysis: Reynolds number, Mach Number, Frohde number.
- From compressible to incompressible models
- Existence of solutions for viscid and inviscid fluids
- Fluid dynamic modeling in various fields: mixture of fluids, combustion, astrophysics, geophysical fluids (atmosphere, ocean)CONTENTS FOR: Modelling and analysis of fluids and biofluids (9 ECTS), Mathematical fluid and biofluid dynamics (6 ECTS)
- Modeling for biofluids: hemodynamics, cerebrospinal fluids, cancer modelling, animal locomotion, bioconvection for swimming microorganisms.
- Pre-requisites:
PREREQUISITES for Mathematical Modelling of Continuum Media:
Basic notions of functional analysis, functions of complex values, standard properties of the heat equation, wave equation, Laplace and Poisson's equations.
PREREQUISITES for Mathematical fluid and biofluid dynamics, Mathematical fluid dynamics, Modelling and analysis of fluids and biofluids:
Basic notions of functional analysis, functions of complex values, standard properties of the heat equation, wave equation, Laplace and Poisson's equations, Sobolev spaces. - Reading list:
- Alexandre Chorin, Jerrold E. Marsden, A Mathematical Introduction to Fluid Mechanics. Springer.
- Roger M. Temam, Alain M. Miranville, Mathematical Modeling in Continum Mechanics. Cambridge University Press.
- Franck Boyer, Pierre Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models. Springer-Verlag Italia.
- Andrea Bertozzi, Andrew Majda, Vorticity and Incompressible Flow. Cambridge University Press.
