Convergence of the point vortex method for the 2-D Euler equations. (English) Zbl 0694.76013
Summary: We prove consistency, stability and convergence of the point vortex approximation to the 2-D incompressible Euler equations with smooth solutions. We first show that the discretization error is second-order accurate. Then we show that the method is stable in \(\ell_ p\) norm. Consequently the method converges in \(\ell_ p\) norm for all time. The convergence is also illustrated by a numerical experiment.
MSC:
| 76B47 | Vortex flows for incompressible inviscid fluids |
| 35Q30 | Navier-Stokes equations |
| 65N15 | Error bounds for boundary value problems involving PDEs |
Keywords:
consistency; stability; convergence of the point vortex approximation; incompressible Euler equations with smooth solutions; discretization error; second-order accurateReferences:
| [1] | Anderson, SIAM J. Num. Anal. v. 22 pp 413– (1985) |
| [2] | Beak, Math. Comp. v. 46 pp 401– (1986) |
| [3] | Beale, Math. Comp. v. 32 pp 29– (1982) |
| [4] | Caflisch, SIAM J. Num. Anal. vol. 26 pp 1060– (1989) |
| [5] | Chorin, J. Fluid Mech. v. 57 pp 785– (1973) |
| [6] | and , A particle method to solve transport-diffusion equations–Part II: the Navier-Stokes equation, preprint. |
| [7] | and , Numerical Methods, Prentice Hall, 1974. |
| [8] | Introduction to Partial Differential Equations, Princeton University Press, 1976. · Zbl 0325.35001 |
| [9] | , and , Convergence of vortex blob methods with small blobs for the Euler equations, in preparation. |
| [10] | Hald, SIAM J. Numer. Anal. v. 16 pp 726– (1979) |
| [11] | and , Convergence of a point vortex method for the 3-D Euler equations, submitted to Comm. Pure Appl. Math. |
| [12] | Pearlman, J. Comp. Phys. v. 59 pp 200– (1985) |
| [13] | Leonard, J. Comput. Phys. v. 37 pp 289– (1980) |
| [14] | Singular Integrals and the Differentiability Properties of Functions, Princeton University Press, 1970. |
| [15] | Strang, Numerische Mathematik 6 pp 37– (1964) |
| [16] | Kato, Arch. Rational Mech. Anal. 25 pp 188– (1967) |
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