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Introduction to numerical analysis and scientific computing with python

Syllabus 2023-2024

  • Basics on Scientific Computing
  • Vector spaces, vector norms, matrices, and matrix norms
  • Basic linear algebra: direct solution of linear systems
  • Not so basic linear algebra: iterative solution of linear systems
  • Polynomial interpolation
  • Interpolatory Quadrature rules
  • L2 projection / Least square approximation
  • Introduction to Finite Difference Methods
  • Introduction to Finite Element Methods

Python laboratories

Topics in high order accurate time integration methods

Ordinary differential equations (ODEs) describe many physical, biological and chemical phenomena. It is, thus, important to find approximations of ODEs which are highly accurate and, in order to obtain it within reasonable computational times, high order accurate time integration methods are very often chosen to proceed in time. In this course, we will revise ODEs and the theoretical results that guarantee their existence and uniqueness [1, 2].

Advanced programming

Students will acquire a comprehensive understanding of advanced programming concepts, specifically in C++ and Python. They will become familiar with object-oriented and generic programming paradigms, as well as proficient in utilizing common data structures, algorithms, and relevant libraries and frameworks for scientific computing. Furthermore, students will be introduced to fundamental software development tools in a Linux environment, encompassing essential aspects like software documentation, version control, testing, and project management.

Turbulent compressible fluid dynamics

Turbulence plays a fundamental role in many different applications varying from aeronautical to environmental simulations. This course aims at giving an overview of the phenomenology and mathematical modelling of compressible turbulent flows. We will start from a first part focused on gasdynamics (thermodynamics, compressible navier-stokes equations, speed of sound, shock waves). The second part of the course will instead be focused on turbulence phenomenology and modelling.

Models and applications in CFD

The course provides an introduction to the numerical simulation of laminar and turbulent incompressible flows by using a finite volume method. Each topic will be corroborated by a set of numerical examples to be performed within the open source C++ finite volume library OpenFOAM.

 

Module 0

  • Well posedness of an abstract problem
  • Numerical approximation of an abstract problem: consistency, convergence and stability
  • Lax–Richtmyer theorem

Module 1

Topics in computational fluid dynamics

Topics/Syllabus

  • Introduction to CFD, examples.
  • Constitutive laws
  • Incompressible flows.
  • Numerical methods for potential and thermal flows
  • Boundary layer theory
  • Thermodynamics effects, energy equation, enthalpy and entropy

Computational mechanics by reduced order methods

Mathematics Area, PhD in Mathematical Analysis, Modelling and Applications (AMMA)
Master in High Performance Computing (MHPC)
Lectures Prof Gianluigi Rozza, Tutorials coordinated by Dr Michele Girfoglio, Dr Federico Pichi and Dr Ivan Prusak.

Learning outcomes and objectives

Topics in continuum mechanics

This is a 60-hours introductory course on continuum mechanics and its applications. The aim is to provide first year students with a solid understanding of the fundamental principles of the subject.

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