Computational Shock Development: accurately resolving cusps and discontinuities in the Euler equations

April 18, 2025

TIME: 3:30 PM

LOCATION: GMCS 314

SPEAKER: Steve Shkoller, University of California, Davis

ABSTRACT: We present a novel Computational Shock Development (CSD) algorithm for the 1D Euler equations, based on an Arbitrary Lagrangian-Eulerian mesh-movement scheme, that is designed to capture the fine cusp structures which emerge when sound waves first steepen to form a gradient singularity, and subsequently when the discontinuous shock wave develops simultaneously with the so-called weak contact and weak rarefaction waves. Using recently established theorems for solutions to the Euler equations as new accuracy metrics, we demonstrate the ability of the CSD algorithm to accurately compute (a) the spacetime location of the first gradient singularity, (b) the precise H\”{o}lder $C^{\frac{1}{3}}$-cusp structure of the Euler solution at the time of first gradient singularity, and (c) the H\”{o}lder $C^{\frac{1}{2}}$-cusps formed by derivatives of the Euler solution at the weak contact and weak rarefaction waves for times past the first gradient singularity. Then, adapting our methodology to evolve discontinuous Riemann data, we use our adapted CSD-Riemann algorithm to accurately simulate notoriously challenging shock tube experiments for which other state-of-the-art algorithms fail. This talk is based on joint work with Raag Ramani.

HOST: Stathis Charalampidis