Convergence study of the Chorin-Marsden formula

Ying, Lung-An

doi:10.1090/S0025-5718-02-01423-0

Convergence study of the Chorin-Marsden formula
HTML articles powered by AMS MathViewer

by Lung-An Ying PDF

Math. Comp. 72 (2003), 307-333 Request permission

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.

References

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
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, DOI 10.1090/S0025-5718-1981-0628693-0
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
Alexandre Joel Chorin, Numerical study of slightly viscous flow, J. Fluid Mech. 57 (1973), no. 4, 785–796. MR 395483, DOI 10.1017/S0022112073002016
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 488713, DOI 10.1002/cpa.3160310205
A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486, DOI 10.1007/978-1-4612-5561-1
Roger Temam, On the Euler equations of incompressible perfect fluids, J. Functional Analysis 20 (1975), no. 1, 32–43. MR 0430568, DOI 10.1016/0022-1236(75)90052-x
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
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
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