An explicit high-order single-stage single-step positivity-preserving finite difference WENO method for the compressible Euler equations

TL;DR

This paper introduces a high-order, positivity-preserving finite difference WENO method for the compressible Euler equations, combining a novel time discretization with flux optimization to ensure stability and accuracy.

Contribution

It presents a new single-stage, single-step method that guarantees positivity preservation using a flux blending approach and a specialized time discretization based on the Picard integral formulation.

Findings

Method achieves high-order accuracy in 1D and 2D simulations.

Ensures positivity preservation during the numerical solution.

Demonstrates stability and effectiveness through numerical tests.

Abstract

In this work we construct a high-order, single-stage, single-step positivity-preserving method for the compressible Euler equations. Space is discretized with the finite difference weighted essentially non-oscillatory (WENO) method. Time is discretized through a Lax-Wendroff procedure that is constructed from the Picard integral formulation (PIF) of the partial differential equation. The method can be viewed as a modified flux approach, where a linear combination of a low- and high-order flux defines the numerical flux used for a single-step update. The coefficients of the linear combination are constructed by solving a simple optimization problem at each time step. The high-order flux itself is constructed through the use of Taylor series and the Cauchy-Kowalewski procedure that incorporates higher-order terms. Numerical results in one- and two-dimensions are presented.