A Conservative, Positivity-Preserving Sweeping Framework for Hyperbolic Systems

High-order discretizations of hyperbolic conservation laws generally lack inherent mechanisms to preserve the positivity of physical quantities such as density and pressure. Violations near strong shocks or near-vacuum states can cause loss of hyperbolicity and numerical blow-up. In this talk, we present a conservative sweeping framework that enforces positivity as a standalone post-processing step, without modifying the underlying spatial discretization. The framework generalizes a scalar mass-redistribution procedure to nonlinear constraints, most notably pressure in the compressible Euler equations. It extends to any concave function of the conserved variables, and it uses the Zhang–Shu scaling limiter as a projection technique for nonlinear constraints. We demonstrate the method with a fifth-order finite difference WENO scheme on extreme compressible flows. Examples include high-Mach astrophysical jets and strong blast waves. The nonlinear extension may in principle require repeated passes, but even in these demanding regimes, one or two sweeps are sufficient. We then show how its non-intrusive design carries over to fully implicit continuous Galerkin discretizations, where it corrects large positivity violations while maintaining conservation. This is joint work with Chi-Wang Shu and Eirik Endeve