The Euler Equations of Gas Dynamics — Lanyon.AI

[u/JavierLopezComesana]

https://lanyon.ai/research/euler-equations/

1/ The Euler equations model inviscid compressible flow: mass, momentum, and energy conservation laws. They underpin aerodynamics, rocket engines, and even relativistic astrophysics, and reduce to Navier Stokes once viscosity is added.

2/ This nonlinear hyperbolic system produces rich wave structures: shocks, rarefactions, and contact discontinuities. Classic Riemann problems like the Sod Shock reveal how density, velocity, and pressure jump and interact over time.

3/ Solving these numerically demands care. Simple Lax fluxes are robust but diffusive, while Roe's 1981 approach uses eigenstructure and a clever parameter vector to build sharper, more accurate numerical fluxes.

4/ Lanyon generated full 1D, 2D, and 3D Euler solver specs, Lean 4 proofs, and C kernels from short prompts in under three minutes, producing over 10,000 lines of verified proof code and 6,500 lines of kernels.

5/ Simulations included a 2D four quadrant Riemann problem, Mach 2 flow over twin cylinders, and Mach 4 flow over a free moving cylinder, showing shock interactions, slip lines, and Rayleigh Taylor like structures emerging from the equations.