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

2026-04-07

AI Summary

This work addresses the challenge of Gibbs–Runge oscillations induced by shock waves in high-Mach-number flows, which commonly plague conventional numerical methods. Existing shock-capturing techniques often excessively smear fine-scale features such as turbulence and acoustic structures due to artificial viscosity or slope limiters. To overcome this limitation, the authors propose a finite volume method based on information-geometric regularization (IGR), which replaces shock singularities with tunable smooth profiles. This approach achieves inviscid, non-dissipative shock regularization without relying on Riemann solvers or high-order WENO reconstructions. In standard benchmark tests, the method attains accuracy comparable to state-of-the-art schemes like WENO and LAD while significantly reducing memory access and computational cost per time step.

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 Schäfer 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.

Problem

• shock waves
• Gibbs-Runge oscillations
• finite volume methods
• Navier-Stokes equations
• numerical regularization

Innovation

• information geometric regularization
• finite volume methods
• shock waves
• inviscid regularization
• Navier-Stokes equations