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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 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.

**1.**A. K. Aziz (ed.),*The mathematical foundations of the finite element method with applications to partial differential equations*, Academic Press, New York-London, 1972. MR**0347104****2.**J. Thomas Beale and Andrew Majda,*Rates of convergence for viscous splitting of the Navier-Stokes equations*, Math. Comp.**37**(1981), no. 156, 243–259. MR**628693**, 10.1090/S0025-5718-1981-0628693-0**3.**G. Benfatto and M. Pulvirenti,*Convergence of Chorin-Marsden product formula in the half-plane*, Comm. Math. Phys.**106**(1986), no. 3, 427–458. MR**859819****4.**Alexandre Joel Chorin,*Numerical study of slightly viscous flow*, J. Fluid Mech.**57**(1973), no. 4, 785–796. MR**0395483****5.**Alexandre J. Chorin, Marjorie F. McCracken, Thomas J. R. Hughes, and Jerrold E. Marsden,*Product formulas and numerical algorithms*, Comm. Pure Appl. Math.**31**(1978), no. 2, 205–256. MR**0488713****6.**A. Pazy,*Semigroups of linear operators and applications to partial differential equations*, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR**710486****7.**Roger Temam,*On the Euler equations of incompressible perfect fluids*, J. Functional Analysis**20**(1975), no. 1, 32–43. MR**0430568****8.**Roger Temam,*Navier-Stokes equations*, 3rd ed., Studies in Mathematics and its Applications, vol. 2, North-Holland Publishing Co., Amsterdam, 1984. Theory and numerical analysis; With an appendix by F. Thomasset. MR**769654****9.**Lung-an Ying,*Convergence of Chorin-Marsden formula for the Navier-Stokes equations on convex domains*, J. Comput. Math.**17**(1999), no. 1, 73–88. MR**1681882****10.**Lung-an Ying and Pingwen Zhang,*Vortex methods*, Mathematics and its Applications, vol. 381, Kluwer Academic Publishers, Dordrecht; Science Press Beijing, Beijing, 1997. MR**1705273**

Retrieve articles in *Mathematics of Computation*
with MSC (2000):
65M99,
35Q30,
76D05,
76M25

Retrieve articles in all journals with MSC (2000): 65M99, 35Q30, 76D05, 76M25

Additional Information

**Lung-An Ying**

Affiliation:
School of Mathematical Sciences, Peking University 100871, People’s Republic of China

Email:
yingla@pku.edu.cn

DOI:
https://doi.org/10.1090/S0025-5718-02-01423-0

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