A Sweeping Positivity-Preserving High Order Finite Difference WENO Scheme for Euler Equations

TL;DR

The paper tackles the challenge of maintaining physical positivity in high-order Euler solver computations, focusing on density and nonlinear pressure. It introduces a conservative, positivity-preserving sweeping post-processing that combines a density sweep with a pressure sweep, enhanced by the Zhang–Shu scaling limiter, and extends Liu et al.'s scalar sweeping to concave pressure functions, ensuring $\rho>\epsilon$ and $p>\epsilon$ while preserving conservation and high-order accuracy. The method is demonstrated within a fifth-order finite-difference WENO framework and is applicable to FD, FV, and DG methods, with a two-dimensional extension that preserves the domain average and supports parallelization. Numerical tests across one- and two-dimensional problems—including double rarefaction, Sedov blast, isentropic vortex, Mach 2000 robustness, shock diffraction, and reactive Euler detonation—show that typically only one or two sweeps per Runge–Kutta stage are needed, maintaining accuracy and preventing blow-ups under challenging CFL conditions. The work suggests paths for extending positivity-preserving sweeping to Navier–Stokes and MHD applications, highlighting practical impact for robust, high-order simulations in CFD, plasma physics, and related fields.

Abstract

We develop a simple, high-order, conservative and robust positivity-preserving sweeping procedure for the density and the nonlinear pressure function in the compressible Euler equations. Using the scaling limiter in Zhang and Shu (2010), we obtain a nontrivial extension of the scalar sweeping technique in Liu, Cheng, and Shu (2016) for the positivity of pressure. The sweeping procedure developed in this paper is a post-processing technique, which can be applied to any concave functions of the conserved variables in hyperbolic conservation law systems. Thus, it has applications beyond the Euler equations. This procedure preserves positivity and conservation of physical quantities without destroying the accuracy of the underlying scheme. The algorithm works for general schemes including finite difference, finite-volume and discontinuous Galerkin methods; however, in this paper we focus on finite-difference weighted essentially non-oscillatory (WENO) methods. We provide numerical tests of the fifth order finite difference WENO scheme to demonstrate the accuracy and robustness of the technique.

Figures (7)

• Figure 1: Example \ref{['ex: 1D_rarefaction']}: Approximate solution of the 1D double rarefaction problem using fifth order WENO with the sweeping limiter. Green symbols are approximate results.
• Figure 2: Example \ref{['ex: 1D_Sedov']}: Numerical results 1D Sedov blast test case using fifth order WENO with sweeping limiter. Green symbols are approximate results. Red line is exact solution.
• Figure 3: Example \ref{['ex: 2D_vortex']}: Numerical results for 2D vortex evolution test case using fifth order WENO with sweeping limiter. Green symbols are approximate results. Red line is exact solution.
• Figure 4: Example \ref{['ex: 2D_Sedov']}: Numerical results for the 2D Sedov blast wave using fifth order WENO with sweeping limiter.
• Figure 5: Example \ref{['ex: Mach_2000']}: Color contour of results on a logarithmic scale for Mach 2000 test case using fifth order WENO with sweeping limiter.

Theorems & Definitions (8)

• Remark 1
• Example 1: 1D double rarefaction
• Example 2: 1D Sedov blast
• Example 3: 2D vortex evolution accuracy
• Example 4: 2D Sedov blast
• Example 5: Mach 2000 robustness test
• Example 6: Shock diffraction: High CFL
• Example 7: Reactive Euler detonation: High CFL