Convergence study of the ChorinMarsden formula
Author:
LungAn Ying
Journal:
Math. Comp. 72 (2003), 307333
MSC (2000):
Primary 65M99; Secondary 35Q30, 76D05, 76M25
Published electronically:
May 3, 2002
MathSciNet review:
1933823
Fulltext PDF Free Access
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Abstract: Using the fundamental solution of the heat equation, we give an expression of the solutions to twodimensional initialboundary value problems of the NavierStokes 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 for the NavierStokes equations. The result is in the direction of justifying the ChorinMarsden 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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A.J.Chorin, T.J.R.Hughes, M.F.McCracken, and J.E.Marsden, Product formulas and numerical algorithms, Comm. Pure Appl. Math., 31, 205256, 1978. MR 58:8230
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A.Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, SpringerVerlag, 1983. MR 85g:47061
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R.Temam, NavierStokes Equations, Theory and Numerical Analysis, 3rd ed., North Holland, 1984. MR 86m:76003
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L.A.Ying, Convergence of ChorinMarsden formula for the NavierStokes equations on convex domains, J. Comp. Math., 17, 7388, 1999. MR 2000d:65162
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L.A.Ying, and P.Zhang, Vortex Methods, Science Press, Beijing/New York, and Kluwer Academic Publishers, Dordrecht/Boston/London, 1997. MR 2000f:76093
 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 NavierStokes equations, Math. Comp., 37, 243259, 1981. MR 82i:65056
 3.
 G.Benfatto, and M.Pulvirenti, Convergence of ChorinMarsden product formula in the halfplane, Comm. Math. Phys., 106, 427458, 1986. MR 88a:35186
 4.
 A.J.Chorin, Numerical study of slightly viscous flow, J. Fluid Mech., 57, 785796, 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, 205256, 1978. MR 58:8230
 6.
 A.Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, SpringerVerlag, 1983. MR 85g:47061
 7.
 R.Temam, On the Euler equations of incompressible perfect fluids, J. Funct. Anal., 20, 3243, 1975. MR 55:3573
 8.
 R.Temam, NavierStokes Equations, Theory and Numerical Analysis, 3rd ed., North Holland, 1984. MR 86m:76003
 9.
 L.A.Ying, Convergence of ChorinMarsden formula for the NavierStokes equations on convex domains, J. Comp. Math., 17, 7388, 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
LungAn Ying
Affiliation:
School of Mathematical Sciences, Peking University 100871, People’s Republic of China
Email:
yingla@pku.edu.cn
DOI:
http://dx.doi.org/10.1090/S0025571802014230
PII:
S 00255718(02)014230
Keywords:
Vortex method,
NavierStokes 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
