Information geometric regularization of the barotropic Euler equation

Abstract

Shock waves in gas dynamics feature jump discontinuities that hinder numerical simulations. Viscous regularizations are prone to excessive dissipation of fine-scale structures. In this work, we propose the first inviscid regularization of the multidimensional Euler equation based on ideas from semidefinite programming, information geometry, geometric hydrodynamics, and nonlinear elasticity. The Lagrangian flow maps of Euler solutions are a dynamical system on the manifold of diffeomorphisms. We observe that shock formation arises from the manifold's geodesic incompleteness. Our regularization embeds it into an ambient space equipped with the information geometry of the logarithmic barrier function. Thus, the diffeomorphism manifold inherits a geodesically complete geometry. The resulting regularized conservation law replaces shocks with smooth profiles without affecting oscillatory structures. One and two-dimensional numerical experiments show its practical potential to enable higher-order methods without explicit shock capturing. While we focus on the barotropic Euler equations for concreteness and simplicity of exposition, our regularization easily extends to more general Euler and Navier-Stokes-type equations. Our approach regularizes the Wasserstein geometry of the mass density with its information geometry. The former captures the natural trajectories of physical particles and the latter that of statistical estimators. Information geometric regularization accounts for the mass density's dual nature as a statistical/computational tool summarizing the motion of physical particles. Thus, our work is a starting point for information geometric mechanics that views solutions of continuum mechanical PDEs as parameters of statistical models for unresolved scales and uses their information geometry to evolve them in time.

Figures (19)

• Figure 1: Vanishing viscosity solutions. Solutions with shocks are defined by vanishing viscosity limits. For finite viscosity, the solution smoothes out over time.
• Figure 1: Geometric regularization. Regularizing the geometry of flow maps with $\alpha(-\log(\partial_x \Phi))$ avoids singularities and preserves the long-time behavior.
• Figure 1: The barrier function $\psi$ defines a dual exponential map on $\mathcal{M}$ through the Euclidean exponential map in the coordinate system given by $\nabla \psi$.
• Figure 1: Parallel transport on submanifolds. Parallel transport on a submanifold is parallel transport in the ambient manifold, followed by projection on the tangent space of the submanifold. For $S^1 \hookrightarrow \mathbb{R}^2$, the difference between parallel transport on ambient and submanifold arises from the nonlinearity of the embedding $\xi$. In unidimensional IGR, $\xi$ is affine but the parallel transport on the ambient space does not respect its linear structure, causing parallel transport on $\mathcal{M}$ and $\mathcal{E}$ to differ. In multidimensional IGR, $\xi$ is nonlinear and both of the above effects contribute.
• Figure 1: IGR on Euler: Lax-Wendroff (LW) with information geometric regularization (IGR) avoids oscillations in LW and the oversmoothing of Lax-Friedrichs (LF).

Theorems & Definitions (3)

• Remark 2.1
• Remark 4.1
• Remark 6.1: Boundary conditions