Commun. Comput. Phys., 9 (2011), pp. 994-1019.


High Order Compact Schemes in Projection Methods for Incompressible Viscous Flows

Michel Fournie 1*, Alain Rigal 1

1 Institut de Mathematiques de Toulouse, Universite de Toulouse et CNRS (UMR 5219), France.

Received 23 July 2009; Accepted (in revised version) 8 July 2010
Available online 2 November 2010
doi:10.4208/cicp.230709.080710a

Abstract

Within the projection schemes for the incompressible Navier-Stokes equations (namely "pressure-correction" method), we consider the simplest method (of order one in time) which takes into account the pressure in both steps of the splitting scheme. For this scheme, we construct, analyze and implement a new high order compact spatial approximation on nonstaggered grids. This approach yields a fourth order accuracy in space with an optimal treatment of the boundary conditions (without error on the velocity) which could be extended to more general splitting. We prove the unconditional stability of the associated Cauchy problem via von Neumann analysis. Then we carry out a normal mode analysis so as to obtain more precise results about the behavior of the numerical solutions. Finally we present detailed numerical tests for the Stokes and the Navier-Stokes equations (including the driven cavity benchmark) to illustrate the theoretical results.

AMS subject classifications: 65M06, 76D05

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Key words: Incompressible Navier-Stokes, fractional step method, high order compact scheme, boundary conditions.

*Corresponding author.
Email: michel.fournie@math.univ-toulouse.fr (M. Fournie), alain.rigal@math.univ-toulouse.fr (A. Rigal)
 

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