Shocks without shock capturing: Information geometric regularization of finite volume methods for Navier--Stokes-like problems

TL;DR

This paper introduces an information geometric regularization (IGR) method for finite volume simulations of Navier--Stokes-like problems, effectively handling shock waves without traditional dissipation or limiters.

Contribution

It provides a practical finite-volume implementation of IGR, demonstrating competitive accuracy and efficiency compared to established shock-capturing schemes.

Findings

IGR recovers expected solutions across benchmarks.

IGR achieves accuracy comparable to WENO and LAD schemes.

IGR is computationally efficient with fewer memory accesses.

Abstract

Shock waves in high-speed fluid dynamics produce near-discontinuities in the fluid momentum, density, and energy. Most contemporary works use artificial viscosity or limiters as numerical mitigation of the Gibbs--Runge oscillations that result from traditional numerics. These approaches face a delicate balance in achieving sufficiently regular solutions without dissipating fine-scale features, such as turbulence or acoustics. Recent work by Cao and Sch\"afer introduces information geometric regularization (IGR), the first inviscid regularization method for fluid dynamics. IGR replaces shock singularities with smooth profiles of adjustable width, without dissipating fine-scale features. This work provides a strategy for the practical use of IGR in finite-volume-based numerical methods. We illustrate its performance on canonical test problems and compare it against established approaches based on limiters and Riemann solvers. Results show that the finite volume IGR approach recovers the expected solutions in all cases. Across canonical benchmarks, IGR achieves accuracy competitive with WENO and LAD shock-capturing schemes in both smooth and discontinuous flow regimes. The IGR approach is computationally light, with meaningfully fewer memory accesses and arithmetic operations per time step.