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Rates of convergence for viscous splitting of the Navier-Stokes equations

Authors: J. Thomas Beale and Andrew Majda
Journal: Math. Comp. 37 (1981), 243-259
MSC: Primary 65M15; Secondary 76D05
MathSciNet review: 628693
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Abstract: Viscous splitting algorithms are the underlying design principle for many numerical algorithms which solve the Navier-Stokes equations at high Reynolds number. In this work, error estimates for splitting algorithms are developed which are uniform in the viscosity $ \nu $ as it becomes small for either two- or three-dimensional fluid flow in all of space. In particular, it is proved that standard viscous splitting converges uniformly at the rate $ C\nu \Delta t$, Strang-type splitting converges at the rate $ C\nu {(\Delta t)^2}$, and also that solutions of the Navier-Stokes and Euler equations differ by $ C\nu $ in this case. Here C depends only on the time interval and the smoothness of the initial data. The subtlety in the analysis occurs in proving these estimates for fixed large time intervals for solutions of the Navier-Stokes equations in two space dimensions. The authors derive a new long-time estimate for the two-dimensional Navier-Stokes equations to achieve this. The results in three space dimensions are valid for appropriate short time intervals; this is consistent with the existing mathematical theory.

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  • [1] G. K. Batchelor, An Introduction to Fluid Mechanics, Cambridge Univ. Press, New York, 1970. MR 622931 (82g:76002)
  • [2] A. J. Chorin, "Numerical study of slightly viscous flow," J. Fluid Mech., v. 57, 1973, pp. 785 - 796. MR 0395483 (52:16280)
  • [3] A. J. Chorin, T. Hughes, M. McCracken & J. Marsden, "Product formulas and numerical algorithms," Comm. Pure Appl. Math., v. 31, 1978, pp. 205-256. MR 0488713 (58:8230)
  • [4] D. Ebin & J. Marsden, "Groups of diffeomorphisms and the motion of an incompressible fluid," Ann. of Math., v. 92, 1970, pp. 102-163. MR 0271984 (42:6865)
  • [5] O. Hald, "Convergence of vortex methods for Euler's equations. II," SIAM J. Numer. Anal., v. 16, 1979, pp. 726-755. MR 543965 (81b:76015b)
  • [6] T. Kato, "Nonstationary flows of viscous and ideal fluids in $ {{\text{R}}^3}$," J. Funct. Anal., v. 9, 1972, pp. 296-305. MR 0481652 (58:1753)
  • [7] T. Kato, "On classical solutions of the two-dimensional Euler equation," Arch. Rational Mech. Anal., v. 25, 1967, pp. 188-200. MR 0211057 (35:1939)
  • [8] F. J. McGrath, "Nonstationary plane flow of viscous and ideal fluids," Arch. Rational Mech. Anal., v. 27, 1968, pp. 329-348. MR 0221818 (36:4870)
  • [9] F. Milinazzo & P. G. Saffman, "The calculation of large Reynolds number two-dimensional flow using discrete vortices with random walk," J. Comput. Phys., v. 23, 1977, pp. 380-392. MR 0452145 (56:10426)
  • [10] G. Strang, "On the construction and comparison of difference schemes," SIAM J. Numer. Anal., v. 5, 1968, pp. 506-517. MR 0235754 (38:4057)
  • [11] G. Strang, "Accurate partial difference methods. II. Nonlinear problems," Numer. Math., v. 6, 1964, pp. 37-49. MR 0166942 (29:4215)
  • [12] R. Temam, The Navier-Stokes Equations, North-Holland, Amsterdam, 1977.

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Keywords: Splitting algorithms, Navier-Stokes equations, Euler's equations
Article copyright: © Copyright 1981 American Mathematical Society

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