Convergence of gauge method for incompressible flow
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- by Cheng Wang and Jian-Guo Liu PDF
- Math. Comp. 69 (2000), 1385-1407 Request permission
Abstract:
A new formulation, a gauge formulation of the incompressible Navier-Stokes equations in terms of an auxiliary field $\mathbf {a}$ and a gauge variable $\phi$, $\mathbf {u} =\mathbf {a}+\nabla \phi$, was proposed recently by E and Liu. This paper provides a theoretical analysis of their formulation and verifies the computational advantages. We discuss the implicit gauge method, which uses backward Euler or Crank-Nicolson in time discretization. However, the boundary conditions for the auxiliary field $\mathbf {a}$ are implemented explicitly (vertical extrapolation). The resulting momentum equation is decoupled from the kinematic equation, and the computational cost is reduced to solving a standard heat and Poisson equation. Moreover, such explicit boundary conditions for the auxiliary field $\mathbf {a}$ will be shown to be unconditionally stable for Stokes equations. For the full nonlinear Navier-Stokes equations the time stepping constraint is reduced to the standard CFL constraint ${\triangle t} / {\triangle x} \le C$. We also prove first order convergence of the gauge method when we use MAC grids as our spatial discretization. The optimal error estimate for the velocity field is also obtained.References
- Ann S. Almgren, John B. Bell, and William G. Szymczak, A numerical method for the incompressible Navier-Stokes equations based on an approximate projection, SIAM J. Sci. Comput. 17 (1996), no. 2, 358–369. MR 1374285, DOI 10.1137/S1064827593244213
- Alexandre Joel Chorin, Numerical solution of the Navier-Stokes equations, Math. Comp. 22 (1968), 745–762. MR 242392, DOI 10.1090/S0025-5718-1968-0242392-2
- John B. Bell, Phillip Colella, and Harland M. Glaz, A second-order projection method for the incompressible Navier-Stokes equations, J. Comput. Phys. 85 (1989), no. 2, 257–283. MR 1029192, DOI 10.1016/0021-9991(89)90151-4
- Hidegorô Nakano, Über Abelsche Ringe von Projektionsoperatoren, Proc. Phys.-Math. Soc. Japan (3) 21 (1939), 357–375 (German). MR 94
- Thomas F. Buttke and Alexandre J. Chorin, Turbulence calculations in magnetization variables, Appl. Numer. Math. 12 (1993), no. 1-3, 47–54. Special issue to honor Professor Saul Abarbanel on his sixtieth birthday (Neveh, 1992). MR 1227179, DOI 10.1016/0168-9274(93)90111-4
- Weinan E and J.-G. Liu, Gauge method for viscous incompressible flows, submitted to J. Comp. Phys. (1996)
- Weinan E and J.-G. Liu, Gauge finite element method for incompressible flows, to appear in Int. J. Num. Meth. Fluids (2000).
- Weinan E and Jian-Guo Liu, Projection method. I. Convergence and numerical boundary layers, SIAM J. Numer. Anal. 32 (1995), no. 4, 1017–1057. MR 1342281, DOI 10.1137/0732047
- Weinan E and Jian-Guo Liu, Finite difference schemes for incompressible flows in the velocity-impulse density formulation, J. Comput. Phys. 130 (1997), no. 1, 67–76. MR 1427404, DOI 10.1006/jcph.1996.5537
- P.M. Gresho and R.L. Sani, On pressure boundary conditions for the incompressible Navier-Stokes equations, Int. J. Numer. Methods Fluids 7 (1987), 1111–1145.
- G. E. Karniadakis and S. J. Sherwin, Spectral/hp Element Methods for CFD, Oxford University Press, New York (1999)
- J. Kim and P. Moin, Application of a fractional-step method to incompressible Navier-Stokes equations, J. Comput. Phys. 59 (1985), no. 2, 308–323. MR 796611, DOI 10.1016/0021-9991(85)90148-2
- John H. Maddocks and Robert L. Pego, An unconstrained Hamiltonian formulation for incompressible fluid flow, Comm. Math. Phys. 170 (1995), no. 1, 207–217. MR 1331698
- S.A. Orszag, M. Israeli, and M.O. Deville, Boundary conditions for incompressible flows, J. Scientific Computing 1 (1986), 75–111.
- V. I. Oseledets, A new form of writing out the Navier-Stokes equation. Hamiltonian formalism, Uspekhi Mat. Nauk 44 (1989), no. 3(267), 169–170 (Russian); English transl., Russian Math. Surveys 44 (1989), no. 3, 210–211. MR 1024057, DOI 10.1070/RM1989v044n03ABEH002122
- G. Russo and P. Smereka, Impulse formulation of the Euler equations: general properties and numerical methods, J. Fluid Mech. 391 (1999), 189–209.
- Jie Shen, On error estimates of projection methods for Navier-Stokes equations: first-order schemes, SIAM J. Numer. Anal. 29 (1992), no. 1, 57–77. MR 1149084, DOI 10.1137/0729004
- Jie Shen, On error estimates of some higher order projection and penalty-projection methods for Navier-Stokes equations, Numer. Math. 62 (1992), no. 1, 49–73. MR 1159045, DOI 10.1007/BF01396220
- Gilbert Strang, Accurate partial difference methods. II. Non-linear problems, Numer. Math. 6 (1964), 37–46. MR 166942, DOI 10.1007/BF01386051
- R. Témam, Sur l’approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires. II, Arch. Rational Mech. Anal. 33 (1969), 377–385 (French). MR 244654, DOI 10.1007/BF00247696
- J. van Kan, A second-order accurate pressure-correction scheme for viscous incompressible flow, SIAM J. Sci. Statist. Comput. 7 (1986), no. 3, 870–891. MR 848569, DOI 10.1137/0907059
- Ai Hui Zhou and Qun Lin, Optimal and superconvergence estimates of the finite element method for a scalar hyperbolic equation, Acta Math. Sci. (English Ed.) 14 (1994), no. 1, 90–94. MR 1280088, DOI 10.1016/S0252-9602(18)30094-8
Additional Information
- Cheng Wang
- Affiliation: Institute for Physical Science and Technology and Department of Mathematics, University of Maryland, College Park, Maryland 20742
- Address at time of publication: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
- Email: cwang@math.umd.edu
- Jian-Guo Liu
- Affiliation: Institute for Physical Science and Technology and Department of Mathematics, University of Maryland, College Park, Maryland 20742
- MR Author ID: 233036
- ORCID: 0000-0002-9911-4045
- Email: jliu@math.umd.edu
- Received by editor(s): November 10, 1997
- Received by editor(s) in revised form: December 7, 1998
- Published electronically: March 24, 2000
- Additional Notes: The research was supported by NSF grant DMS-9805621 and Navy ONR grant N00014-96-1013
- © Copyright 2000 American Mathematical Society
- Journal: Math. Comp. 69 (2000), 1385-1407
- MSC (1991): Primary 65M12, 76M20
- DOI: https://doi.org/10.1090/S0025-5718-00-01248-5
- MathSciNet review: 1710695