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Convergence study of the Chorin-Marsden formula

Author: Lung-An Ying
Journal: Math. Comp. 72 (2003), 307-333
MSC (2000): Primary 65M99; Secondary 35Q30, 76D05, 76M25
Published electronically: May 3, 2002
MathSciNet review: 1933823
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Abstract: Using the fundamental solution of the heat equation, we give an expression of the solutions to two-dimensional initial-boundary value problems of the Navier-Stokes equations, where the vorticity is expressed in terms of a Poisson integral, a Newtonian potential, and a single layer potential. The density of the single layer potential is the solution to an integral equation of Volterra type along the boundary. We prove there is a unique solution to the integral equation. One fractional time step approximation is given, based on this expression. Error estimates are obtained for linear and nonlinear problems. The order of convergence is $\frac 14$ for the Navier-Stokes equations. The result is in the direction of justifying the Chorin-Marsden formula for vortex methods. It is shown that the density of the vortex sheet is twice the tangential velocity for the half plane, while in general the density differs from it by one additional term.

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  • 1. A.K.Aziz (ed.), The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, Academic Press, New York and London, 1972. MR 49:11824
  • 2. J.T.Beale, and A.Majda, Rates of convergence for viscous splitting of the Navier-Stokes equations, Math. Comp., 37, 243-259, 1981. MR 82i:65056
  • 3. G.Benfatto, and M.Pulvirenti, Convergence of Chorin-Marsden product formula in the half-plane, Comm. Math. Phys., 106, 427-458, 1986. MR 88a:35186
  • 4. A.J.Chorin, Numerical study of slightly viscous flow, J. Fluid Mech., 57, 785-796, 1973. MR 52:16280
  • 5. A.J.Chorin, T.J.R.Hughes, M.F.McCracken, and J.E.Marsden, Product formulas and numerical algorithms, Comm. Pure Appl. Math., 31, 205-256, 1978. MR 58:8230
  • 6. A.Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, 1983. MR 85g:47061
  • 7. R.Temam, On the Euler equations of incompressible perfect fluids, J. Funct. Anal., 20, 32-43, 1975. MR 55:3573
  • 8. R.Temam, Navier-Stokes Equations, Theory and Numerical Analysis, 3rd ed., North Holland, 1984. MR 86m:76003
  • 9. L.-A.Ying, Convergence of Chorin-Marsden formula for the Navier-Stokes equations on convex domains, J. Comp. Math., 17, 73-88, 1999. MR 2000d:65162
  • 10. L.-A.Ying, and P.Zhang, Vortex Methods, Science Press, Beijing/New York, and Kluwer Academic Publishers, Dordrecht/Boston/London, 1997. MR 2000f:76093

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Additional Information

Lung-An Ying
Affiliation: School of Mathematical Sciences, Peking University 100871, People’s Republic of China

Keywords: Vortex method, Navier-Stokes equation, fractional step method
Received by editor(s): June 5, 2000
Received by editor(s) in revised form: January 3, 2001
Published electronically: May 3, 2002
Article copyright: © Copyright 2002 American Mathematical Society

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