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Spectral Method for Navier–Stokes Equations with Slip Boundary Conditions

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Abstract

In this paper, we propose a spectral method for the \(n\)-dimensional Navier–Stokes equations with slip boundary conditions by using divergence-free base functions. The numerical solutions fulfill the incompressibility and the physical boundary conditions automatically. Therefore, we need neither the artificial compressibility method nor the projection method. Moreover, we only have to evaluate the unknown coefficients of expansions of \(n-1\) components of the velocity. These facts simplify actual computation and numerical analysis essentially, and also save computational time. As the mathematical foundation of this new approach, we establish some approximation results, with which we prove the spectral accuracy in space of the proposed algorithm. Numerical results demonstrate its high efficiency and coincide the analysis very well. The main idea, the approximation results and the techniques developed in this paper are also applicable to numerical simulations of other problems with divergence-free solutions, such as certain partial differential equations describing electro-magnetic fields.

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Authors and Affiliations

  1. Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China

    Ben-yu Guo & Yu-jian Jiao

  2. Scientific Computing Key Laboratory of Shanghai Universities, Shanghai Normal University, Shanghai, 200234, China

    Ben-yu Guo & Yu-jian Jiao

  3. E-Institute for Computational Science of Shanghai Universities, Shanghai Normal University, Shanghai, 200234, China

    Ben-yu Guo & Yu-jian Jiao

Authors
  1. Ben-yu Guo
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  2. Yu-jian Jiao
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Correspondence to Ben-yu Guo.

Additional information

This work is supported in part by NSF of China N.11171227, Fund for Doctoral Authority of China N.20123127110001, Fund for E-institute of Shanghai Universities N.E03004, and Leading Academic Discipline Project of Shanghai Municipal Education Commission N.J50101.

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Guo, By., Jiao, Yj. Spectral Method for Navier–Stokes Equations with Slip Boundary Conditions. J Sci Comput 58, 249–274 (2014). https://doi.org/10.1007/s10915-013-9729-5

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  • DOI: https://doi.org/10.1007/s10915-013-9729-5

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