Non-Intrusive Positivity Preservation: A Conservative Sweeping Framework for High-Order Discretizations of Hyperbolic Systems
Online
https://www.cscdr.umassd.edu/seminars
Abstract:
High-order numerical simulations of hyperbolic conservation laws frequently lack inherent mechanisms to preserve physical positivity, leading to computational failures near strong shocks or near-vacuum states. To address this, we introduce a conservative sweeping framework designed to enforce positivity of density and pressure as a standalone post-processing procedure. The approach generalizes a scalar mass-redistribution technique using the Zhang-Shu scaling limiter as a projection strategy to treat nonlinear pressure in the Euler equations. Because the positivity-preserving mechanism operates independently from the spatial discretization, it provides highly modular and non-intrusive stabilization for existing solvers.
We first demonstrate the framework's robust performance in explicit fifth-order finite difference WENO schemes for extreme compressible Euler flows such as high-Mach jets and strong blast waves. Crucially, as a post-processing step, this framework easily extends to implicit time-stepping schemes, where guaranteeing positivity and conservation traditionally presents a significant algorithmic hurdle. We present results highlighting its ability to successfully correct large positivity violations in fully implicit continuous Galerkin applications. The general framework is applicable to finite volume and discontinuous Galerkin discretizations, and may be used to treat any concave function of the conserved variables. Thus, it is an ideal candidate for internal energy treatment in Navier-Stokes and MHD simulations.