Abstract
In many important scientific applications (e.g., incompressible fluids) the diivergence-free property is not preserved by the partial differential equations describing the flow. Accordingly projection of a vector field v onto its solenoidal (divergence-free) part plays a fundamental role and in some respects is one of the most difficult aspects in the numerical analysis of such problems.
We first survey and describe the schemes that have been devised to deal computationally with this difficulty. Relatively few have been implemented in three dimensions and even fewer for three-dimensional stationary flows.
We then present a new scheme for the direct computation of the projection of an arbitrary three-dimensional vector field v(x) onto its solenoidal (divergence-free) part. The algorithm combines finite differences before and after the calculation of a singular integral. We prove convergence for this algorithm and present illustrative numerical results for the cases tested. A number of applications are discussed.
Partially supported by a Computing resources Grant from the National Center of Atmospheric Research.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
P. Morse and H. Feshbach, Methods of Theoretical Physics, Parts I and II, McGraw-Hill, New York, 1953.
O. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1963.
D. Sattinger, Topics in Stability and Bifurcation Theory, Lec. Notes in Math. 309, Springer, Berlin, 1973.
R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, Elsevich-North Holland, New York, 1977.
P. Gresho, R. Lee, R. Sani, T. Stullich, On the time-dependend FEM solution of the incompressible Navier-Stokes Equations in two and three dimensions, Lawrence Livermore Lab. Rept. UCRL-81323 (1978).
A. Chorin, The numerical solution of the Navier-Stokes equation for an incompressible fluid, Bull. Amer. Math. Soc. 73 (1967), 928–931.
A. Chorin, A numerical method for solving incompressible viscous flow problems, J. Comp. Physics 2 (1967), 12–26.
C. Peskin, Flow patterns around heart valves: a numerical method, J. Comp. Physics 10 (1972), 252–271.
G. Prodi, Theoremi di Tipo Locale per il Sistema di Navier-Stokes e Stabilita delle Soluzione Stazionarei, Rend. Sem. Mat. Univ. Padova 32 (1962), 374–397.
D. Sattinger, The mathematical problem of hydrodynamical stability, J. Math. Mech. 19 (1970), 797–817.
M. Fortin, Approximation des Fonctions à Divergence Nulle par la Méthode des Eléments Finis, Lec. Notes in Physics 18, Springer, Berlin (1973), 99–103.
M. Crouzeix and P. Raviart, Conforming and Nonconforming Finite Element Methods for Solving the Stationary Stokes Equations (to appear).
F. Thomasset, Application d'une Méthode d'éléments finis d'ordre un à la résolution numérique des équations de Navier-Stokes, IRIA Rept. No. 150, Le Chesnay, France, 1975.
U. Schumann, Subgrid scale model for finite difference simulations of turbulent flows in plane channels and annuli, J. Comp. Physics 18 (1975), 376–404.
O. Ladyzhenskaya and V. Rivland, On the alternating direction method for the computation of a viscous incompressible fluid flow in cylindrical coordinates, Izv. Akad. Nank. 35 (1971), 259–268.
J. Lions, On the numerical approximation of some equations arising in hydrodynamics, A.M.S. Symposium, Durham, April, 1968.
R. Temam, Une méthode d'approximation de la solution des équations de Navier-Stokes, Bull. Soc. Math. France 98 (1968), 115–152.
R. Glowinski and O. Pironneau, "Numerical methods for the 2-dimensional Stokes Problem through the stream function-voticity formulation", 1st France-Japan Colloq. on Funct. Analysis and Num. Analysis, Tokyo, 1976.
H. Fujita and T. Kato, On the Navier-Stokes Initial Value Problem I, Tech. Rept. 121, Stanford University, 1963.
W. Ames, Some computation-steeples in fluid mechanics, SIAM Review 15 (1973), 524–552.
A. Amsden and F. Harlow, A simplified MAC technique for incompressible fluid flow calculations, J. Comp. Physics 6 (1970), 322–325.
R. Sweet, A cyclic reduction algorithm for solving block tridiagonal systems of arbitrary dimension, SIAM J. Num. Anal. 14 (1977), 706–720.
G. Williams, Numerical integration of the three-dimensional Navier-Stokes equations for incompressible flow, J. Fluid. Mech. 37 (1969), 727–750.
W. Ames, Numerical Methods for Partial Differential Equations, 2nd Ed., Academic Press, New York, 1977.
P. Eagles, On stability of Taylor vortices by fifth-order amplitude expansions, J. Fluid Mech. 49 (1971), 529–550.
K. Gustafson, Estimation of eigenvalue aggregates determining hydrodynamic stability, Notices Amer. Math. Soc. 23 (1976), A-682.
J. Marsden and M. McCracken, The Hopf Bifurcation and its applications, Springer, Berlin, 1976.
O. Ladyzhenskaya, Mathematical analysis of Navier-Stokes equations for incompressible liquids, in "Annual Review of Fluid Mechanics", Vol. 7, Annual Reviews Inc., Palo Alto, California, 1975.
D. Young, Nonlinear Diffusion with Traveling Waves and Numerical Solutions, Thesis, University of Colorado, 1979, to appear.
R. Richtmyer, Invariant manifolds and attractors in the Taylor Problem, preprint, 1978.
R. Beam and R. Warming, An implicit finite difference algorithm for hyperbolic systems in conservation-law form, J. Comp. Physics 22 (1976), 87–110.
F. Johnson and L. Erickson, A general panel method for the analysis and design of arbitrary configurations in incompressible flows, NASA report, NASA CR-3079 (1979).
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1982 Springer-Verlag
About this paper
Cite this paper
Gustafson, K.E., Young, D.P. (1982). Computation of solenoidal (divergence-free) vector fields. In: Hinze, J. (eds) Numerical Integration of Differential Equations and Large Linear Systems. Lecture Notes in Mathematics, vol 968. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0064884
Download citation
DOI: https://doi.org/10.1007/BFb0064884
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-11970-8
Online ISBN: 978-3-540-39374-0
eBook Packages: Springer Book Archive