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The Crank-Nicolson/Adams-Bashforth scheme for the time-dependent Navier-Stokes equations with nonsmooth initial data

Authors

  • Yinnian He

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    1. Faculty of Science, State Key Laboratory of Multiphase Flow in Power Engineering, Xi'an Jiaotong University, Xi'an 710049, People's Republic of China
    • Faculty of Science, State Key Laboratory of Multiphase Flow in Power Engineering, Xi'an Jiaotong University, Xi'an 710049, People's Republic of China
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Abstract

In this article, we study the stability and convergence of the Crank-Nicolson/Adams-Bashforth scheme for the two-dimensional nonstationary Navier-Stokes equations with a nonsmooth initial data. A finite element method is applied for the spatial approximation of the velocity and pressure. The time discretization is based on the implicit Crank-Nicolson scheme for the linear terms and the explicit Adams-Bashforth scheme for the nonlinear term. Moreover, we prove that the scheme is almost unconditionally stable for a nonsmooth initial data u0 with div u0 = 0, i.e., the time step τ satisfies: τ ≤ C0 if u0H1L; τ |log h| ≤ C0 if u0H1 for the mesh size h and some positive constant C0. Finally, we obtain some error estimates for the discrete velocity and pressure under the above stability condition. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 28: 155-187, 2012

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