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A Simple Bound-Preserving Sweeping Technique for Conservative Numerical Approximations

Abstract

In this paper, we propose a simple bound-preserving sweeping procedure for conservative numerical approximations. Conservative schemes are of importance in many applications, yet for high order methods, the numerical solutions do not necessarily satisfy maximum principle. This paper constructs a simple sweeping algorithm to enforce the bound of the solutions. It has a very general framework acting as a postprocessing step accommodating many point-based or cell average-based discretizations. The method is proven to preserve the bounds of the numerical solution while conserving a prescribed quantity designated as a weighted average of values from all points. The technique is demonstrated to work well with a spectral method, high order finite difference and finite volume methods for scalar conservation laws and incompressible flows. Extensive numerical tests in 1D and 2D are provided to verify the accuracy of the sweeping procedure.

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Author information

Correspondence to Yingda Cheng.

Additional information

Yuan Liu: Research is supported by Mississippi State University startup grant and a grant from the Simons Foundation (426993). Yingda Cheng: Research is supported by NSF Grants DMS-1318186 and DMS-1453661. Chi-Wang Shu: Research is supported by ARO Grant W911NF-15-1-0226 and NSF Grant DMS-1418750.

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Liu, Y., Cheng, Y. & Shu, C. A Simple Bound-Preserving Sweeping Technique for Conservative Numerical Approximations. J Sci Comput 73, 1028–1071 (2017). https://doi.org/10.1007/s10915-017-0395-x

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Keywords

  • Bound-preserving
  • Conservative schemes
  • Fourier spectral methods
  • High order finite difference and finite volume methods