  A key numerical difficulty in compressible fluid dynamics is the formation of
shock waves. Shock waves feature jump discontinuities in the velocity and
density of the fluid and thus preclude the existence of classical solutions to
the compressible Euler equations. Weak entropy solutions are commonly defined
by viscous regularization, but even small amounts of viscosity can
substantially change the long-term behavior of the solution. 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. From a Lagrangian
perspective, shock formation in entropy solutions amounts to inelastic
collisions of fluid particles. Their trajectories are akin to that of projected
gradient descent on a feasible set of non-intersecting paths. We regularize
these trajectories by replacing them with solution paths of interior point
methods based on log determinantal barrier functions. These paths are geodesic
curves with respect to the information geometry induced by the barrier
function. Thus, our regularization replaces the Euclidean geometry of
trajectories with a suitable information geometry. We extend this idea to
infinite families of paths by viewing Euler's equations as a dynamical system
on a diffeomorphism manifold. Our regularization embeds this manifold into an
information geometric ambient space, equipping it with a geodesically complete
geometry. Expressing the resulting Lagrangian equations in Eulerian form, we
derive a regularized Euler equation in conservation form. Numerical experiments
on one and two-dimensional problems show its promise as a numerical tool. 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.
