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

DOI:
https://doi.org/10.1090/S0025-5718-1981-0628693-0

MathSciNet review:
628693

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 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 , Strang-type splitting converges at the rate , and also that solutions of the Navier-Stokes and Euler equations differ by 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.

**[1]**G. K. Batchelor and H. K. Moffatt (eds.),*25th anniversary issue: editorial reflections on the development of fluid mechanics*, Cambridge University Press, Cambridge-New York, 1981. J. Fluid Mech. 106 (1981). MR**622931****[2]**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**[3]**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. MR**0488713**, https://doi.org/10.1002/cpa.3160310205**[4]**David G. Ebin and Jerrold Marsden,*Groups of diffeomorphisms and the motion of an incompressible fluid.*, Ann. of Math. (2)**92**(1970), 102–163. MR**0271984**, https://doi.org/10.2307/1970699**[5]**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**[6]**Tosio Kato,*Nonstationary flows of viscous and ideal fluids in 𝑅³*, J. Functional Analysis**9**(1972), 296–305. MR**0481652****[7]**Tosio Kato,*On classical solutions of the two-dimensional nonstationary Euler equation*, Arch. Rational Mech. Anal.**25**(1967), 188–200. MR**0211057**, https://doi.org/10.1007/BF00251588**[8]**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**[9]**F. Milinazzo and P. G. Saffman,*The calculation of large Reynolds number two-dimensional flow using discrete vortices with random walk*, J. Computational Phys.**23**(1977), no. 4, 380–392. MR**0452145****[10]**Gilbert Strang,*On the construction and comparison of difference schemes*, SIAM J. Numer. Anal.**5**(1968), 506–517. MR**0235754**, https://doi.org/10.1137/0705041**[11]**Gilbert Strang,*Accurate partial difference methods. II. Non-linear problems*, Numer. Math.**6**(1964), 37–46. MR**0166942**, https://doi.org/10.1007/BF01386051**[12]**R. Temam,*The Navier-Stokes Equations*, North-Holland, Amsterdam, 1977.

Retrieve articles in *Mathematics of Computation*
with MSC:
65M15,
76D05

Retrieve articles in all journals with MSC: 65M15, 76D05

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1981-0628693-0

Keywords:
Splitting algorithms,
Navier-Stokes equations,
Euler's equations

Article copyright:
© Copyright 1981
American Mathematical Society