Abstract
In this paper we review fourth-order approximations of the biharmonic operator in one, two and three dimensions. In addition, we describe recent developments on second and fourth order finite difference approximations of the two dimensional Navier-Stokes equations. The schemes are compact both for the biharmonic and the Laplacian operators. For the convective term the fourth order scheme invokes also a sixth order Pade approximation for the first order derivatives, using an approximation suggested by Carpenter-Gottlieb-Abarbanel (J. Comput. Phys. 108:272–295, 1993). We also introduce the derivation of a pure streamfunction formulation for the Navier-Stokes equations in three dimensions.
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Ben-Artzi, M.: Planar Navier-Stokes equations, vorticity approach. In: Fridlander, S.J., Serre, J. (eds.) Handbook of Mathematical Fluid Dynamics, vol. II. Elsevier, Amsterdam (2003). Chapter 5
Ben-Artzi, M., Croisille, J.-P., Fishelov, D., Trachtenberg, S.: A pure-compact scheme for the streamfunction formulation of Navier-Stokes equations. J. Comput. Phys. 205(2), 640–664 (2005)
Ben-Artzi, M., Croisille, J.-P., Fishelov, D.: Convergence of a compact scheme for the pure streamfunction formulation of the unsteady Navier-Stokes system. SIAM J. Numer. Anal. 44(5), 1997–2024 (2006)
Ben-Artzi, M., Croisille, J.-P., Fishelov, D.: A fast direct solver for the biharmonic problem in a rectangular grid. SIAM J. Sci. Comput. 31(1), 303–333 (2008)
Ben-Artzi, M., Chorev, I., Croisille, J.-P., Fishelov, D.: A compact difference scheme for the biharmonic equation in planar irregular domains. SIAM J. Numer. Anal. 47(4), 3087–3108 (2009)
Ben-Artzi, M., Croisille, J.-P., Fishelov, D.: A high order compact scheme for the pure-streamfunction formulation of the Navier-Stokes Equations. J. Sci. Comput. 42(2), 216–250 (2010)
Carpenter, M.H., Gottlieb, D., Abarbanel, S.: The stability of numerical boundary treatments for compact high-order schemes finite difference schemes. J. Comput. Phys. 108, 272–295 (1993)
E, W., Liu, J.-G.: Essentially compact schemes for unsteady viscous incompressible flows. J. Comput. Phys. 126, 122–138 (1996)
Ghia, U., Ghia, K.N., Shin, C.T.: High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method. J. Comput. Phys. 48, 387–411 (1982)
Landau, L.D., Lifshitz, E.M.: Fluid Mechanics. Pergamon, Elmsford (1959). Chapter II, Sect. 15
Ruas, V., Quartapelle, L.: Uncouples finite element solutions of biharmonic problems for vector potentials. Int. J. Numer. Methods Fluids 11, 811–822 (1990)
Rubel, A., Volpe, G.: Biharmonic vector stream function formulation and multigrid solutions for a three-dimensional driven-cavity stokes flow. In: AIAA Computational Fluid Dynamics Conference, 9th, Buffalo, NY, 13–15 June 1989, pp. 380–388. AIAA, Washington (1989)
Stephenson, J.W.: Single cell discretizations of order two and four for biharmonic problems. J. Comput. Phys. 55, 65–80 (1984)
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Dedicated to the memory of Professor David Gottlieb for his Wisdom and Generosity.
Partially supported by a French-Israeli scientific cooperation grant 3-1355.
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Fishelov, D., Ben-Artzi, M. & Croisille, JP. Recent Developments in the Pure Streamfunction Formulation of the Navier-Stokes System. J Sci Comput 45, 238–258 (2010). https://doi.org/10.1007/s10915-010-9374-1
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DOI: https://doi.org/10.1007/s10915-010-9374-1