Vortex methods. II. Higher order accuracy in two and three dimensions
HTML articles powered by AMS MathViewer
- by J. Thomas Beale and Andrew Majda PDF
- Math. Comp. 39 (1982), 29-52 Request permission
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.References
- J. Thomas Beale and Andrew Majda, Vortex methods. I. Convergence in three dimensions, Math. Comp. 39 (1982), no. 159, 1–27. MR 658212, DOI 10.1090/S0025-5718-1982-0658212-5 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.
- 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
- Lipman Bers, Fritz John, and Martin Schechter, Partial differential equations, Lectures in Applied Mathematics, Vol. III, Interscience Publishers, a division of John Wiley & Sons, Inc., New York-London-Sydney, 1964. With special lectures by Lars Garding and A. N. Milgram. MR 0163043
- Alexandre Joel Chorin, Numerical study of slightly viscous flow, J. Fluid Mech. 57 (1973), no. 4, 785–796. MR 395483, DOI 10.1017/S0022112073002016
- Gerald B. Folland, Introduction to partial differential equations, 2nd ed., Princeton University Press, Princeton, NJ, 1995. MR 1357411
- 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, DOI 10.1090/S0025-5718-1978-0492039-1
- 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, DOI 10.1090/S0025-5718-1978-0492039-1
- Jack K. Hale, Ordinary differential equations, Pure and Applied Mathematics, Vol. XXI, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1969. MR 0419901 L. Hörmander, "Fourier integral operators. I," Acta Math., v. 127, 1971, pp. 79-183. 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.
- A. Leonard, Vortex methods for flow simulation, J. Comput. Phys. 37 (1980), no. 3, 289–335. MR 588256, DOI 10.1016/0021-9991(80)90040-6
- F. J. McGrath, Nonstationary plane flow of viscous and ideal fluids, Arch. Rational Mech. Anal. 27 (1967), 329–348. MR 221818, DOI 10.1007/BF00251436
- Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972
- Michael Taylor, Pseudo differential operators, Lecture Notes in Mathematics, Vol. 416, Springer-Verlag, Berlin-New York, 1974. MR 0442523, DOI 10.1007/BFb0101246
Additional Information
- © Copyright 1982 American Mathematical Society
- 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