Course Unit Page


This teaching activity contributes to the achievement of the Sustainable Development Goals of the UN 2030 Agenda.

Quality education Affordable and clean energy Industry, innovation and infrastructure

Academic Year 2021/2022

Learning outcomes

The course is designed with the main objective of introducing students to modeling techniques and numerical simulations. During the course, the student is introduced to the main aspects that lead to mathematical modeling, numerical discretization, solution of the equations and scientific visualization. Fundamental methods of numerical approximations are discussed with particular emphasis on finite element, finite volume and finite difference methods. Students will use open-source computational software to solve and visualize the results of the modeled problems. Part of the course will be dedicated to a project that allows them to verify the direct application of theoretical concepts previously learned.

Course contents

The course is designed with the objective to introduce students to the modeling techniques  and numerical simulation. The course covers aspects that lead to mathematical and numerical modeling, discretization and numerical solution of the main equations important for the study of energy  systems. Part of the course is dedicated to a project that allows to verify the direct application of theoretical concepts learned previously. The course consists of three main activities:

Modeling and numerical simulation:
• Modeling of energy systems. Multiscale and multiphysics modeling. Weak solution and classical models of differential equations. Weak formulation and variational form of the main conservation equations. Functional spaces. Modeling of the main relevant engineering equations.
• Numerical approximation. Approximation and interpolation of solution spaces. Finite difference, finite volume  and finite element method.  Approximation of some relevant equations in the main engineering fields.
• Solution of discrete numerical models (this section  alternates lectures and training). Installation and use of libraries for linear algebra solvers. Open-source libraries for solution of discrete systems corresponding to finite volumes and finite element approximations.

Specific topics:
- Scientific visualization: software Paraview; Gnuplot.
- Multigriglia methods for solving algebraic systems: the principle of a multigriglia solver; cycle  V and W.
- Numerical models for advection and turbulence

Training and use of scientific software:
Using open-source libraries LibMesh for solving systems discretized using finite volume and finite element method.


The course is based on teacher's notes.
Numerical Models for Differential Problems, A. Quarteroni, Springer (2009)

Teaching methods

Lectures and tutorials

Assessment methods

Homework (30%)
Discussion of a project assigned by the instructor (70%)

Teaching tools

Personal computer and workstation

Office hours

See the website of Sandro Manservisi