Calcolo Scientifico

Repository for the course Calcolo Scientifico for Scienze Matematiche per l'Intelligenza Artificiale A.A. 2025/2026

OPIS

Codice OPIS: J5K9SA1T

Rooms

• On Tuesdays 8:30-10:00 in Laboratorio 1, Castelnuovo
• On Wednesdays 13:15-15:00 in Aula Picone, Castelnuovo

Exceptions (extra classes on Thursday!)

• 23/04/26: 10:15-12:00 in Laboratorio 1, Castelnuovo
• 5-6/05/26: no lesson

Form for project day and exams

Please fill in this form form link

Useful links

• Course page on Sapienza Website
• Moodle page for communications

Zoom link

Zoom link for lectures

Slides

Number	Topic	PDF slides	HTML slides	Markdown slides
1	Introduction
1.0	Introduction to the course	PDF	html	md
1.1	Introduction to git(hub)	PDF	html	md
1.2	Introduction to PDEs	PDF	html	md
1.3	Introduction to functional analysis	PDF	html	md
2	Elliptic problems
2.1	Elliptic problems	PDF	html	md
2.1.1	(Extra) Poisson with dirac source	PDF	html	md
2.2	Finite differences for elliptic problems	PDF	html	md
2.3	Finite elements for elliptic problems	PDF	html	md
2.4	Reduced order methods for elliptic problems	PDF Notebook	html	md
3	Parabolic problems
3.1	Parabolic problems and their discretization	PDF	html	md
4	Hyperbolic equations
4.1	Linear Transport equation problems and finite difference	PDF	html	md
4.2	Nonlinear conservation laws	PDF	html	md
5	PINN	PDF	html	md

Logbook, Notes and Recordings

Date	Time	Topics	Notes	Recording link	Password
03/03/26	08:30	Introduction to the course, derivatives notation	Notes	AI summary	Forgot recording
04/03/26	13:15	Classification of PDEs, first order linear PDEs, canonical form, second order linear PDEs	Notes	Zoom link	31MB@a2L
10/03/26	08:30	Canonical form of second order linear PDE	Notes	Zoom link	SKn8.v=n
11/03/26	13:15	Solution of elliptic equation (with Fourier), Cauchy-Kovaleskaya, well posedness	Notes	Zoom link	jzS0V.zn
17/03/26	08:30	Functional spaces, linear functional, bilinear functionals, Hilbert spaces, distributions	Notes	Zoom link	XX.A!0*8
18/03/26	13:15	Derivatives of distributions, Sobolev spaces, Poincare, intro to elliptic	Notes	Zoom link	!x$t1R9q
24/03/26	08:30	Weak formulation of elliptic problems, Lax-Milgram	Notes	Zoom link	bR1..x*5
25/03/26	13:15	Divided differences, finite differences for 1D poisson, error analysis	Notes	Zoom link	yQJ85&d.
31/03/26	08:30	Coding finite difference 1D for Poisson	Notebook	Audio, Zoom no audio	i$@hzM4a
01/04/26	13:15	Coding finite difference 2D for Poisson	Notes	Zoom link	!6r*@RwB
14/04/26	08:30	Finite Element in 1D for Poisson	Notes	Zoom link	L1i*8^Yx
15/04/26	13:15	Finite Element in 1D for Poisson	Notes	Zoom link	$1c?U1X^
16/04/26	10:15	Coding Finite Element 1D for Poisson	Notebook
21/04/26	08:30	Finite element for multi-D	Notes	Zoom link	zi*0q.e9
22/04/26	13:15	Coding finite element 2D for Poisson with FEniCS	Notebook
23/04/26	10:15	Model order reduction	Notes Notebook	Zoom link	8JT%*t#1
28/04/26	08:30	Parabolic equations	Notes	Zoom link no audio Audio	@L6F=wCb
29/04/26	13:15	FD and FEM for parabolic equations	Notes	Zoom link (last year)	6ZC@D#?+
12/05/26	08:30	Advection equation	Notes	Zoom Link	eGp$c^@7
13/05/26	13:15	FD for advection equation	See above	Zoom link	%zF$0hX1
19/05/26	08:30	Coding FD for advection and nonlinear laws	Notebook	Zoom link	UU@5EV#6
20/05/26	13:15	Scalar hyperbolic conservation laws (nonlinear)	Notes	Zoom link	Hhe&?4C#
26/05/26	08:30	Coding hyperbolic conservation laws	Notes	Zoom link	*9BGW0ni
27/05/26	13:15	Physics Informed Neural Networks	Zoom Lavagna Pass: wtoTU+3Y	Zoom notebook	8UN!#?x9
10/06/26	13:15	Projects presentation

Notebooks

Date	Topic	Notebook	Solutions	Last save
31/03/26	Finite difference for Poisson	Notebook	Solutions
16/04/26	Finite element 1D for Poisson	Notebook	Solutions
22/04/26	Finite element 2D for Poisson with FEniCS	Notebook	Solutions
23/04/26	Reduced order models for parametric problems with FEniCS	Notebook	-
29/04/26	Finite difference for Heat equation	Notebook	Solutions
19/05/26	Finite difference for transport equation	Notebook	Solutions
26/05/26	Finite volume for conservation laws	Notebook	Solutions
27/05/26	PINN	Notebook	Solutions

Projects ideas (contact me for more details)

1. You personalised project that you came up with studying the course

If you have not many ideas I can suggest you something, probably more complicated:

1. Finite element with naive basis functions (e.g. trigonometric functions) (in 1D/2D)
2. Spectral element method (Chapter of Quarteroni)
3. Nonlinear parabolic problems (e.g. Allen-Cahn) with finite elements (and FEniCS?)
4. WENO reconstruction for finite difference for transport equation in 1D
5. Dirac delta source terms for Poisson in dimension 1 problems with their Riesz representative, do the simulations work as expected?
6. Navier-Stokes with FEniCS on a complex geometry (Chapter of Quarteroni)
7. Reduced basis for elastic block Problem from this RBniCS test
8. Error control for reduced order models (a posteriori error estimator) (Hesthaven book)
9. SUPG for advection-diffusion time dependent problem (1D or 2D with FEniCS), with energy stability analysis (Chapter of Quarteroni)
10. Saddle point problems: Stokes problem for incompressible fluids (Chapter of Quarteroni)
11. Von Neumann stability analysis for a Finite Difference discretization of the wave equation
12. Von Neumann stability analysis for FEM $\mathbb P^p$ for parabolic equations
13. Wave equations in 2D (with compatible Finite Difference discretization)
14. High order FD discretization of Burgers' equations in 1D with WENO
15. Hyperbolic system of conservation laws (Euler equations) in 1D with finite difference methods
16. 2D Euler equations solved on a Cartesian grid for a DMR test
17. Comparison of PINN with classical solvers (for various problems)

Literature

• Quarteroni, Alfio. Modellistica Numerica per Problemi Differenziali. Springer Science & Business Media, 2016. [Intro alle PDE, Metodi agli elementi finiti, Metodi alle differenze finite, Riduzione del Modello]
• Evans, Lawrence C. Partial differential equations. Vol. 19. American Mathematical Society, 2010. [Introduzione alle PDE]
• Cangiani, Andrea. Note del corso Numerical Solution of Partial Differential Equations in SISSA, 2025 [Sezioni 1-5].
• LeVeque, Randall J. Finite difference methods for ordinary and partial differential equations: steady-state and time-dependent problems. Society for Industrial and Applied Mathematics, 2007. [Metodi alle differenze finite]
• LeVeque, Randall J. Finite volume methods for hyperbolic problems. Vol. 31. Cambridge university press, 2002. [Metodi ai volumi finiti]
• Langtangen, Hans Petter, and Anders Logg. Solving PDEs in python: the FEniCS tutorial I. Springer Nature, 2017. [Manuale per usare FEniCS]
• Hesthaven, J., Rozza G. and Stamm B. Certified Reduced Basis Methods for Parametrized Partial Differential Equations. Springer, 2016. [Riduzione del Modello] https://link.springer.com/book/10.1007/978-3-319-22470-1

Program

The course studies partial differential equations (PDEs) and some numerical methods for approximating their solutions.

• Introduction to PDEs and functional analysis [12h];
• Methods for elliptic equations (finite difference and finite element) [16h = 8 class + 8 lab];
• ODE review [2h] (if necessary);
• Methods for parabolic equations (finite difference and finite element) [4h = 2h classes + 2h lab];
• Methods for hyperbolic equations (finite difference and finite volume) [10h = 6h classes + 4h lab];
• Physics informed neural networks [2h = 1h class + 1h lab];
• Model order reduction [2h = 1h class + 1h lab].

At first, we will introduce PDEs with some examples from various physical problems, the concept of weak derivatives and Sobolev spaces. For some classes of PDEs, we will verify the existence, uniqueness and/or regularity of PDEs' solution (transport equation, Poisson equation, heat equation, Stokes problem for incompressible fluids, conservation laws, Burgers equations, Euler for fluid dynamics).

We will see different types of discretization of PDEs starting from finite differences (FD). We will define finite difference discretization, their consistency, their accuracy, and we will apply it to 1D, 2D, time-independent, and time-dependent problems. We will see how to implement and solve these problems in Python by defining sparse matrix structures. We will study the stability of these methods and derive the CFL conditions.

Subsequently, we will study finite element methods (FEM) starting from the definition of the approximation spaces and their properties, the quadratures and a priori estimates for some problems. We will study the convergence of methods for coercive linear problems and for linear problems in saddle point formulations. Finally, we will see the onset of instabilities due to advection-dominated problems and some stabilization techniques. We will implement finite elements for 1D and 2D problems. Moreover, we will use the FEniCS library for problems with more complicated geometries.

Then, we will study the finite volume method for conservation laws, introducing the discretization and the concept of consistency for numerical fluxes. We will study the stability of the method and see different types of reconstructions and numerical fluxes. Moreover, we will implement the method on nonlinear problems.

The final classes will be devoted to less standard methods as physics informed neural networks and model order reduction techniques for parametrised PDEs.

Tools

• git (github)
• python
• FEniCS Finite Element python library
• PINA Physics Informed Neural Network library based on PyTorch