Sacha Cardonna

Sacha Cardonna

Ph.D. Student in Mathematics

About me

I’m a Ph.D. student in Mathematics, under the supervision of François Vilar & Fabien Marche at Institut Montpelliérain Alexander Grothendieck (IMAG - UMR 5149).

I’m also a teaching assistant for the Department of Mathematics at Montpellier Faculty of Sciences & Polytech Montpellier Engineering School.

Research interests

During my thesis, I design and implement high-order methods on water waves models, especially nonlinear Shallow-Water equations (NSW), which represent a nonlinear hyperbolic system with source term:

$$ \begin{cases} \partial_t \eta + \nabla_{\mathbf{x}} \cdot \mathbf{q} = 0, \\ \partial_t \mathbf{q} + \nabla_{\mathbf{x}} \cdot \left( \mathbf{u} \otimes \mathbf{q} + \frac{g\eta}{2}(\eta - 2b)\mathbb{I}_2 \right) = -g\eta \nabla_{\mathbf{x}} b, \end{cases} \nonumber $$

where $\eta$ is water total elevation, $\mathbf{q}=(q_x,q_y)^T$ is the horizontal discharge, and $\mathbf{B} = (0, -g\eta \nabla_{\mathbf{x}} b)^T$ the topography source term.

More generally, I am interested in modeling and numerical analysis of partial differential equations (PDEs), and their applications to physics problems, mainly fluid mechanics.

  • Models. Conservation laws, non-linear hyperbolic systems of PDEs, models coupling.
  • Numerics. Discontinuous Galerkin & Finite-Volume schemes, well-balanced schemes, ALE approaches.
  • Applications. Fluid mechanics, nonlinear Shallow-Water equations, dispersive PDEs.
  • Scientific computing. Object oriented and generic programming, parallel computing.

Highlights

  • I started writing my first article, in collaboration with A. Haidar and my thesis directors, see Research for more details.
  • I introduced high-schoolers to mathematics from party games during MathC2+ program, see Diffusion for the materials.