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

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