Numerical Evaluation of the Accuracy and Stability Properties of High-order Direct Stokes Solvers with or without Temporal Splitting

The temporal stability and effective order of two different direct high-order Stokes solvers are examined. Both solvers start from the primitive variables formulation of the Stokes problem, but are distinct by the numerical uncoupling they apply on the Stokes operator. One of these solvers introduces an intermediate divergence free velocity for performing a temporal splitting (J. Comput. Phys. [1991] 97, 414–443) while the other treats the whole Stokes problem through the evaluation of a divergence free acceleration field (C.R. Acad. Sci. Paris [1994] 319 Serie I, 1455–1461; SIAM J. Scient. Comput. [2000] 22(4), 1386–1410). The second uncoupling is known to be consistent with the harmonicity of the pressure field (SIAM J. Scient. Comput. [2000] 22(4), 1386–1410). Both solvers proceed by two steps, a pressure evaluation based on an extrapolated in time (of theoretical order Je) Neumann condition, and a time implicit (of theoretical order Ji) diffusion step for the final velocity. These solvers are implemented with a Chebyshev mono-domain and a Legendre spectral element collocation schemes. The numerical stability of these four options is investigated for the sixteen combinations of (Je,Ji), 1 ≤ Je, Ji ≤ 4.

[1]  T. A. Zang,et al.  Spectral methods for fluid dynamics , 1987 .

[2]  Michel O. Deville,et al.  An Object-Oriented Toolbox for Spectral Element Analysis , 1999, J. Sci. Comput..

[3]  S. Orszag,et al.  Boundary conditions for incompressible flows , 1986 .

[4]  P. Fischer,et al.  High-Order Methods for Incompressible Fluid Flow , 2002 .

[5]  G. Dahlquist.  A special stability problem for linear multistep methods , 1963 .

[6]  Jie Shen,et al.  Velocity-Correction Projection Methods for Incompressible Flows , 2003, SIAM J. Numer. Anal..

[7]  Chao Yang,et al.  ARPACK users' guide - solution of large-scale eigenvalue problems with implicitly restarted Arnoldi methods , 1998, Software, environments, tools.

[8]  C. Canuto.  Spectral methods in fluid dynamics , 1991 .

[9]  D. Gottlieb,et al.  Numerical analysis of spectral methods : theory and applications , 1977 .

[10]  G. Labrosse,et al.  Compatibility conditions for the Stokes system discretized in 2-D Cartesian domains , 1993 .

[11]  S. Orszag,et al.  High-order splitting methods for the incompressible Navier-Stokes equations , 1991 .

[12]  N. Parolini,et al.  Role of the LBB condition in weak spectral projection methods: 405 , 2001 .

[13]  Jie Shen,et al.  Quelques résultats nouveaux sur les méthodes de projection , 2001 .

[14]  Joel Ferziger,et al.  Higher Order Methods for Incompressible Fluid Flow: by Deville, Fischer and Mund, Cambridge University Press, 499 pp. , 2003 .

[15]  A. Chorin.  Numerical Solution of the Navier-Stokes Equations* , 1989 .

[16]  Emmanuel Leriche,et al.  High-Order Direct Stokes Solvers with or Without Temporal Splitting: Numerical Investigations of Their Comparative Properties , 2000, SIAM J. Sci. Comput..

[17]  E. Leriche,et al.  Direct numerical simulation of the flow in a lid-driven cubical cavity , 2000 .

[18]  R. Temam.  Sur l'approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires (II) , 1969 .

[19]  A. Batoul,et al.  Une méthode de résolution directe (pseudo-spectrale) du problème de Stokes 2D/3D instationnaire. Application à la cavité entraînée carrée , 1994 .

[20]  Jie Shen,et al.  A new class of truly consistent splitting schemes for incompressible flows , 2003 .