Post-processing of solutions of incompressible Navier-Stokes equations on rotating spheres

Mahadevan Ganesh, Quoc Thong Le Gia

Abstract


We describe a post-processing technique (requiring only solutions of linear stationary problems) to improve the resolution of Galerkin solutions of the time dependent nonlinear incompressible Navier--Stokes equations on the rotating unit sphere. Numerical experiments illustrate the advantage of this more efficient method to simulate higher modes to approximate the divergence-free velocity field.

References
  • B. G. Archilla, J. Novo, and E. S. Titi. Postprocessing the Galerkin method: a novel approach to approximate inertial manifolds. SIAM J. Numer. Anal., 35:941--972, 1998. doi:10.1137/S0036142995296096
  • A. Debussche, T. Dubois, and R. Temam. The nonlinear Galerkin method: a multiscale method applied to the simulation of homogeneous turbulent flows. Theoret. Comput. Fluid Dynamics, 7:279--315, 1995. doi:10.1007/BF00312446
  • M. J. Fengler and W. Freeden. A nonlinear Galerkin scheme involving vector and tensor spherical harmonics for solving the incompressible Navier--Stokes equation on the sphere. SIAM J. Sci. Comput., 27:967--994, 2004. doi:10.1137/040612567
  • C. Foias, O. Manley, and R. Temam. Modelling of the interaction of small and large eddies in two dimensional turbulence flows, RAIRO Model. Math. Anal. Numer., 22:93--118, 1988.
  • M. Ganesh, Q. T. Le Gia, and I. H. Sloan. A pseudospectral quadrature method for Navier--Stokes equations on rotating spheres. Preprint. http://www.mines.edu/mganesh/NSE-08-pap.pdf
  • A. A. Il'in. The Navier-Stokes and Euler equations on two dimensional closed manifolds, Math. USSR Sbornik, 69:559--579, 1991. doi:10.1070/SM1991v069n02ABEH002116
  • A. A. Il'in and A. N. Filatov. On unique solvability of the Navier--Stokes equations on the two dimensional sphere. Soviet Math. Dokl., 38:9--13, 1989.
  • M. Marion and R. Temam. Nonlinear Galerkin methods, SIAM J. Numer. Anal., 26:1139--1157, 1989. doi:10.1137/0726063
  • R. Temam and S. Wang, Inertial forms of Navier--Stokes equations on the sphere. J. Funct. Anal., 117:215--24, 1993. doi:10.1006/jfan.1993.1126

Full Text:

PDF BibTeX


DOI: http://dx.doi.org/10.21914/anziamj.v50i0.1436



Remember, for most actions you have to record/upload into this online system
and then inform the editor/author via clicking on an email icon or Completion button.
ANZIAM Journal, ISSN 1446-8735, copyright Australian Mathematical Society.