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An Explicit High-Order Single-Stage Single-Step Positivity-Preserving Finite Difference WENO Method for the Compressible Euler Equations

Journal of Scientific Computing volume 68, pages 171–190 (2016)Cite this article

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 method. Time is discretized through a Lax–Wendroff procedure that is constructed from the Picard integral formulation 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.

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Acknowledgments

The authors would like to thank the anonymous reviewer for the helpful suggestions to further improve this work. This work has been supported in part by Air Force Office of Scientific Research Grants FA9550-11-1-0281, FA9550-12-1-0343 and FA9550-12-1-0455, and by National Science Foundation Grant Numbers DMS-1115709 and DMS-1316662.

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Authors and Affiliations

  1. Department of Mathematics, U.S. Naval Academy, 121 Blake Road, Annapolis, MD, 21402, USA

    David C. Seal

  2. Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY, 12180, USA

    Qi Tang

  3. Department of Mathematical Sciences, Michigan Technological University, Houghton, MI, 49931, USA

    Zhengfu Xu

  4. Department of Computational Mathematics, Science and Engineering, East Lansing, MI, 48824, USA

    Andrew J. Christlieb

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  1. David C. Seal
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  2. Qi Tang
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  3. Zhengfu Xu
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  4. Andrew J. Christlieb
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Correspondence to David C. Seal.

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Seal, D.C., Tang, Q., Xu, Z. et al. An Explicit High-Order Single-Stage Single-Step Positivity-Preserving Finite Difference WENO Method for the Compressible Euler Equations. J Sci Comput 68, 171–190 (2016). https://doi.org/10.1007/s10915-015-0134-0

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  • DOI: https://doi.org/10.1007/s10915-015-0134-0

Keywords

  • Hyperbolic conservation laws
  • Lax–Wendroff
  • Weighted essentially non-oscillatory
  • Positivity-preserving

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