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Mathematics of Smoothed Particle Hydrodynamics: A Study via Nonlocal Stokes Equations

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Abstract

Smoothed particle hydrodynamics (SPH) is a popular numerical technique developed for simulating complex fluid flows. Among its key ingredients is the use of nonlocal integral relaxations to local differentiations. Mathematical analysis of the corresponding nonlocal models on the continuum level can provide further theoretical understanding of SPH. We present, in this part of a series of works on the mathematics of SPH, a nonlocal relaxation to the conventional linear steady-state Stokes system for incompressible viscous flows. The nonlocal continuum model is characterized by a smoothing length \(\delta \) which measures the range of nonlocal interactions. It serves as a bridge between the discrete approximation schemes that involve a nonlocal integral relaxation and the local continuum models. We show that for a class of carefully chosen nonlocal operators, the resulting nonlocal Stokes equation is well-posed and recovers the original Stokes equation in the local limit when \(\delta \) approaches zero. For some other commonly used smooth kernels, there are risks in getting ill-posed continuum models that could lead to computational difficulties in practice. This leads us to discuss the implications of our finding on the design of numerical methods.

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Acknowledgements

The authors would like to thank R. Lehoucq, J. Foster and A. Tartakovsky for discussions on related subjects.

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  1. Xiaochuan Tian

    Present address: Department of Mathematics, University of Texas, Austin, TX, 78712, USA

Authors and Affiliations

  1. Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY, 10027, USA

    Qiang Du & Xiaochuan Tian

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  1. Qiang Du
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Correspondence to Qiang Du.

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Eitan Tadmor.

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This work is supported in part by the U.S. NSF Grants DMS-1719699 and DMS-1819233, AFOSR MURI center for material failure prediction through peridynamics, and ARO MURI Grant W911NF-15-1-0562.

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Du, Q., Tian, X. Mathematics of Smoothed Particle Hydrodynamics: A Study via Nonlocal Stokes Equations. Found Comput Math 20, 801–826 (2020). https://doi.org/10.1007/s10208-019-09432-0

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  • DOI: https://doi.org/10.1007/s10208-019-09432-0

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