Sacha Cardonna

Sacha Cardonna

Ph.D. Student in Mathematics

Documents

A selection of research internships, reports and course projects completed during my university training.

These documents reflect work in progress at the time they were written. They may contain typos or incomplete arguments, but remain available as a record of my academic development.

01

Research internships

01

Local monolithic DG/FV subcells convex property preserving scheme on nonlinear shallow water equations

Numerical analysis · Scientific computing

Building and implementing a new strategy for stabilizing discontinuous Galerkin numerical methods using a Finite-Volume subcells approach for the nonlinear shallow-water equations. The work considers a fully a priori monolithic DG/FV scheme designed to preserve convex properties.

Advisors · François Vilar and Fabien Marche

Slides
02

Asymptotic analysis of PDEs sequences and homogenization theory

Partial differential equations · Homogenization

Study of two problems, including a Dirichlet problem posed on a variable open set. The second part models the formation of ice fog—water vapour condensing into droplets and rapidly freezing—as a homogenization problem.

Advisor · Michel Bellieud

Report Slides
03

From differential geometry to mathematical billiards

Differential geometry · Dynamical systems

Study of mathematical billiards as an elementary dynamical system, with a characterization of periodic trajectories through their initial angle of incidence. Joint work with Samuel Raë.

Advisor · Daniel Massart

Report Slides
04

Proof of Dirichlet Prime Number theorem

Number theory · Complex analysis

A proof that, for coprime integers a and b, the arithmetic progression an + b contains infinitely many prime numbers, drawing on tools from complex analysis and group theory.

Advisor · Sylvain Brochard

Report Slides
02

Course projects

Projects and reports7 items · reports, notes and simulations

Numerical analysis

Hybrid High-Order method on Leray–Lions operators

Study of a nonconforming finite-element method and its main discrete functional-analysis results.

Report

Scientific computing

Müller's SPH C++ implementation for fluid dynamics

Construction and implementation of a Smoothed Particle Hydrodynamics method.

Report Simulation

Measure theory

Some results about measure theory

Radon measures and the Besicovitch, Vitali, Tietze and Lusin theorems.

Notes

Error estimation

Projects for a posteriori estimates course

Theoretical results and implementations in Fortran, Python and C++.

Reports

Analysis

Reports for analysis courses

Selected results on Hölder functions and distribution theory.

Reports

Algebra

Reports for algebra courses

Cube automorphisms, Euclidean rings and Diophantine equations.

Reports

Machine learning

Projects for machine learning and optimization course

Dataset analysis and implementation of regression methods in Python.

Reports