Convergence study of the Chorin-Marsden formula. (English) Zbl 1093.76049
Summary: Using the fundamental solution of heat equation, we give an expression of the solutions to two-dimensional initial-boundary value problems for Navier-Stokes equations, where the vorticity is expressed in terms of Poisson integral, Newtonian potential, and 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 that there is a unique solution to the integral equation. A fractional time step approximation is given, based on this expression. Error estimates are obtained for linear and nonlinear problems. The order of convergence is \(\frac 1 4\) 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.
MSC:
| 76M23 | Vortex methods applied to problems in fluid mechanics |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
References:
| [1] | A. K. Aziz , The mathematical foundations of the finite element method with applications to partial differential equations, Academic Press, New York-London, 1972. · Zbl 0259.00014 |
| [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. · Zbl 0518.76027 |
| [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. · Zbl 0615.76032 |
| [4] | Alexandre Joel Chorin, Numerical study of slightly viscous flow, J. Fluid Mech. 57 (1973), no. 4, 785 – 796. · doi:10.1017/S0022112073002016 |
| [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. · Zbl 0358.65082 · doi:10.1002/cpa.3160310205 |
| [6] | A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. · Zbl 0516.47023 |
| [7] | Roger Temam, On the Euler equations of incompressible perfect fluids, J. Functional Analysis 20 (1975), no. 1, 32 – 43. · Zbl 0309.35061 |
| [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. · Zbl 0568.35002 |
| [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. · Zbl 0915.76070 |
| [10] | Lung-an Ying and Pingwen Zhang, Vortex methods, Mathematics and its Applications, vol. 381, Kluwer Academic Publishers, Dordrecht; Science Press Beijing, Beijing, 1997. · Zbl 0926.76002 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
