Abstract
In this paper, we design and study two modifications of the first order standard pressure increment projection scheme for the Stokes system. The first scheme improves the existing schemes in the case of open boundary condition by modifying the pressure increment boundary condition, thereby minimizing the pressure boundary layer and recovering the optimal first order decay. The second scheme allows for variable time stepping. It turns out that the straightforward modification to variable time stepping leads to unstable schemes. The proposed scheme is not only stable but also exhibits the optimal first order decay. Numerical computations illustrating the theoretical estimates are provided for both new schemes.
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References
A.J. Chorin, On the convergence of discrete approximations to the Navier-Stokes equations. Math. Comput. 23, 341–353 (1969)
K. Goda, A multistep technique with implicit difference schemes for calculating two- or three-dimensional cavity flows. J. Comput. Phys. 30, 76–95 (1979)
J.L. Guermond, P. Minev, J. Shen, Error analysis of pressure-correction schemes for the time-dependent stokes equations with open boundary conditions. SIAM J. Numer. Anal. 43(1), 239–258 (2005). (electronic)
J.L. Guermond, P. Minev, J. Shen, An overview of projection methods for incompressible flows. Comput. Methods Appl. Mech. Eng. 195(44–47), 6011–6045 (2006)
J.-L. Guermond, L. Quartapelle, On the approximation of the unsteady Navier-Stokes equations by finite element projection methods. Numer. Math. 80(2), 207–238 (1998)
J.L. Guermond, J. Shen, Velocity-correction projection methods for incompressible flows. SIAM J. Numer. Anal. 41(1), 112–134 (2003). (electronic)
J.L. Guermond, J. Shen, On the error estimates of rotational pressure-correction projection methods. Math. Comput. 73, 1719–1737 (2004)
J.-L. Guermond, A.J. Salgado, A splitting method for incompressible flows with variable density based on a pressure poisson equation. J. Comput. Phys. 228(8), 2834–2846 (2009)
R. Témam, Sur l’approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires. II. Arch. Rational Mech. Anal. 33, 377–385 (1969)
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Bonito, A., Guermond, JL., Lee, S. (2015). Modified Pressure-Correction Projection Methods: Open Boundary and Variable Time Stepping. In: Abdulle, A., Deparis, S., Kressner, D., Nobile, F., Picasso, M. (eds) Numerical Mathematics and Advanced Applications - ENUMATH 2013. Lecture Notes in Computational Science and Engineering, vol 103. Springer, Cham. https://doi.org/10.1007/978-3-319-10705-9_61
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DOI: https://doi.org/10.1007/978-3-319-10705-9_61
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