Discontinuous Galerkin Semidiscretization of the Information Geometric Regularized Compressible Euler Equations

Abstract

Shock stabilization in compressible Euler flows remains a central challenge for high-order numerical methods. Existing shock-capturing approaches, including limiters, artificial viscosity, and reconstruction-based methods, involve tradeoffs between robustness, accuracy, preservation of fine-scale flow features, and computational complexity. In this work, we develop a discontinuous Galerkin (DG) discretization of the information geometric regularization (IGR) framework introduced by Cao and Sch\"afer for the compressible Euler equations. The method regularizes shocks at the PDE level by augmenting the Euler equations with the entropic pressure $\Sigma$, obtained from an auxiliary elliptic equation. Within the DG formulation, the regularization enters only through the augmented pressure $P+\Sigma$ in the Euler fluxes, preserving the conservative structure of the discretization while using a common approximation space for both the hyperbolic and elliptic equations. Numerical experiments spanning one and two-dimensional benchmark problems show the proposed formulation stabilizes shocks without shock-capturing limiters or artificial viscosity, although positivity-preserving methods may still be required when the density or pressure approaches zero. Compared with a characteristic TVB-limited DG formulation, the IGR-DG method resolves increasingly finer-scale flow features as the polynomial order is increased while maintaining stable shock resolution. The entropic pressure remains localized to regions of strong compression with minimal activation in smooth regions of the flow, providing selective PDE-level regularization while preserving the underlying solution elsewhere.