Information Geometric Regularization: Shock capturing without viscosity

Shock waves in high-speed gas dynamics cause near-discontinuities in the momentum, density, and energy of the gas. To mitigate the resulting Gibbs-Runge type oscillations, most contemporary work uses artificial viscosity or limiters. These approaches face a delicate balance in achieving sufficiently regular solutions without damping out fine-scale features such as turbulence. Information geometric regularization (IGR) is the first inviscid regularization of gas dynamics. It replaces shock singularities with smooth profiles of adjustable width, without dissipating fine-scale features. It achieves this goal by changing the geometry of characteristic curves such that instead of crossing, they asymptotically approach each other. This allows omitting conventional limiters and Riemann solvers, and thus enables the use of lightweight and highly performant linear discretizations. Most recently, this approach has enabled the first compressible CFD simulation of mesh sizes exceeding 100 trillion points.