M2 - Well-balanced semi-implicit scheme for the Euler equations with gravity

[oussama-floor9]

Student: Oussama BOUHENNICHE
Academic supervisors: Andrea Thomann, Victor Michel-Dansac
Host institution: Inria ( Macaron team, Nancy–Grand Est, hosted in Strasbourg), funded by ITI IRMIA++
Period: March – July 2025

Context

The internship focused on the numerical modeling of compressible flows under gravity, a key challenge in geophysical and astrophysical simulations.
Explicit numerical schemes face severe limitations:

• Acoustic CFL restriction, which enforces prohibitively small time steps.
• Failure to preserve hydrostatic equilibria, leading to spurious oscillations and instabilities.

The goal was to develop a semi-implicit well-balanced scheme for the Euler equations with gravity, stable in the low-Mach regime and capable of preserving hydrostatic equilibrium exactly.

Methodology

• Tools: Python (NumPy, SciPy), GitLab, Gaya HPC cluster, iterative solvers (BiCGStab) with ILU preconditioning.
• Approach:
  • Reformulated Euler equations with a flux-splitting strategy (explicit transport, implicit pressure–gravity).
  • Designed a semi-implicit IMEX Runge–Kutta finite-volume scheme with centered div-grad operators.
  • Implemented two closures: energy-based and pressure-based.
  • Modular Python package with mesh handling, discretization, fluxes, IMEX integration, and post-processing.
  • Benchmarks: isothermal & polytropic atmospheres, Graf–Gresho vortex, Sod shock tube, Kelvin–Helmholtz, Rayleigh–Taylor, rising bubble, and shock–bubble interaction.

Results

• Well-balanced property validated: hydrostatic equilibria preserved up to round-off error.
• Asymptotic-preserving stability: scheme remains robust as Mach number decreases (low-Mach regime).
• Accuracy: second-order convergence confirmed on Gresho vortex test with gravity.
• Robustness: Successfully handled instability-driven problems (Rayleigh–Taylor, rising bubble, shock–bubble interaction).
• Delivered a modular and extensible Python solver.

Shock–bubble interaction

Rising bubble

Rayleigh–Taylor instability

Useful links

• 📄 Internship report
• 🎤 Presentation slides
• 👔 My CV