CERAMATHS - DMATHS seminar: presentation by Robin Colombier

The CÉRAMATHS mathematics department seminar will welcome Robin Colombier (UPHF, CÉRAMATHS) at 2pm on Thursday, July 3, 2025, for the following talk:

Numerical schemes for solving Quantum Navier-Stokes equations

The path of electrons in a semiconductor can, in certain situations, be modeled by the hydrodynamic equations of fluid mechanics. It is in this context that we focus on the Quantum Navier-Stokes (QNS) equations. These are a system of partial differential equations of order 3, which makes them difficult to solve numerically, especially if certain physical properties of the solutions are to be preserved. One approach regularly proposed in the literature is to transform the system in order to reduce its order. In this presentation, I will introduce a physical explanation of the NSQ equations, and then present two resolution schemes on a Cartesian grid. The first ensures discrete entropy decay. To achieve this, the system is reformulated in a way that allows a time splitting scheme to be implemented. This consists of solving a hyperbolic part using a time-explicit finite volume method, followed by solving a parabolic part using a time-implicit finite difference method. A second fully implicit scheme allows an additional electric potential to be taken into account while respecting energy decay, enabling the simulation of physical cases such as that of a resonant tunnel diode.