Vortex methods. II. Higher order accuracy in two and three dimensions

Authors:
J. Thomas Beale and Andrew Majda

Journal:
Math. Comp. **39** (1982), 29-52

MSC:
Primary 65M15; Secondary 76C05

DOI:
https://doi.org/10.1090/S0025-5718-1982-0658213-7

MathSciNet review:
658213

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Abstract | References | Similar Articles | Additional Information

Abstract: In an earlier paper the authors introduced a new version of the vortex method for three-dimensional, incompressible flows and proved that it converges to arbitrarily high order accuracy, provided we assume the consistency of a discrete approximation to the Biot-Savart Law. We prove this consistency statement here, and also derive substantially sharper results for two-dimensional flows. A complete, simplified proof of convergence in two dimensions is included.

**[1]**J. Thomas Beale and Andrew Majda,*Vortex methods. I. Convergence in three dimensions*, Math. Comp.**39**(1982), no. 159, 1–27. MR**658212**, https://doi.org/10.1090/S0025-5718-1982-0658212-5**[2]**J. T. Beale & A. Majda,*The Design and Analysis of Vortex Methods*, Proc. Conf. on Transonic, Shock, and Multi-Dimensional Flows, Madison, Wisc., May 1981.**[3]**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**, https://doi.org/10.1090/S0025-5718-1981-0628693-0**[4]**Lipman Bers, Fritz John, and Martin Schechter,*Partial differential equations*, Lectures in Applied Mathematics, Vol. III, Interscience Publishers John Wiley & Sons, Inc. New York-London-Sydney, 1964. MR**0163043****[5]**Alexandre Joel Chorin,*Numerical study of slightly viscous flow*, J. Fluid Mech.**57**(1973), no. 4, 785–796. MR**0395483**, https://doi.org/10.1017/S0022112073002016**[6]**Gerald B. Folland,*Introduction to partial differential equations*, 2nd ed., Princeton University Press, Princeton, NJ, 1995. MR**1357411****[7]**Ole H. Hald,*Convergence of vortex methods for Euler’s equations. II*, SIAM J. Numer. Anal.**16**(1979), no. 5, 726–755. MR**543965**, https://doi.org/10.1137/0716055**[8]**Ole Hald and Vincenza Mauceri del Prete,*Convergence of vortex methods for Euler’s equations*, Math. Comp.**32**(1978), no. 143, 791–809. MR**492039**, https://doi.org/10.1090/S0025-5718-1978-0492039-1**[9]**Jack K. Hale,*Ordinary differential equations*, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1969. Pure and Applied Mathematics, Vol. XXI. MR**0419901****[10]**L. Hörmander, "Fourier integral operators. I,"*Acta Math.*, v. 127, 1971, pp. 79-183.**[11]**K. Kuwahara & H. Takami, "Numerical studies of a two-dimensional vortex motion by a system of point vortices,"*J. Phys. Soc. Japan*, v. 34, 1973, pp. 247-253.**[12]**A. Leonard,*Vortex methods for flow simulation*, J. Comput. Phys.**37**(1980), no. 3, 289–335. MR**588256**, https://doi.org/10.1016/0021-9991(80)90040-6**[13]**F. J. McGrath,*Nonstationary plane flow of viscous and ideal fluids*, Arch. Rational Mech. Anal.**27**(1967), 329–348. MR**0221818**, https://doi.org/10.1007/BF00251436**[14]**Elias M. Stein and Guido Weiss,*Introduction to Fourier analysis on Euclidean spaces*, Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical Series, No. 32. MR**0304972****[15]**Michael Taylor,*Pseudo differential operators*, Lecture Notes in Mathematics, Vol. 416, Springer-Verlag, Berlin-New York, 1974. MR**0442523**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1982-0658213-7

Keywords:
Vortex method,
incompressible flow

Article copyright:
© Copyright 1982
American Mathematical Society