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Convergence of a step-doubling Galerkin method for parabolic problems

Authors: Bruce P. Ayati and Todd F. Dupont
Journal: Math. Comp. 74 (2005), 1053-1065
MSC (2000): Primary 65M06, 65M12, 65M60; Secondary 35K15, 35K20, 65M15
Published electronically: September 10, 2004
MathSciNet review: 2136993
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Abstract | References | Similar Articles | Additional Information

Abstract: We analyze a single step method for solving second-order parabolic initial-boundary value problems. The method uses a step-doubling extrapolation scheme in time based on backward Euler and a Galerkin approximation in space. The technique is shown to be a second-order correct approximation in time. Since step-doubling can be used as a mechanism for step-size control, the analysis is done for variable time steps. The stability properties of step-doubling are contrasted with those of Crank-Nicolson, as well as those of more general extrapolated theta-weighted schemes. We provide an example computation that illustrates both the use of step-doubling for adaptive time step control and the application of step-doubling to a nonlinear system.

References [Enhancements On Off] (What's this?)

  • 1. Bruce P. Ayati, $BuGS$ 1.0 user guide, Tech. Report CS-96-18, University of Chicago, 1996.
  • 2. Bruce P. Ayati and Todd F. Dupont, Convergence of a step-doubling Galerkin method for parabolic problems, Tech. Report CS-99-02, University of Chicago, 1999.
  • 3. James H. Bramble and Peter H. Sammon, Efficient higher order single step methods for parabolic problems. $I$, Math. Comp. 35 (1980), 655-677. MR 81h:65110
  • 4. M. P. Brenner, X. D. Shi, and S. R. Nagel, Iterated instabilities during droplet fission, Phys. Rev. Letters 73 (1994), no. 25, 3391-3394.
  • 5. Todd F. Dupont and A. E. Hosoi, Modeling and computation for applications in science and engineering, ch. Some reduced-dimension models based on numerical methods, pp. 59-80, Oxford University Press, 1998. MR 2000i:76048
  • 6. Jens Eggers, Nonlinear dynamics and breakup of free-surface flows, Rev. of Modern Phys. 69 (1997), no. 3, 865-929.
  • 7. Jens Eggers and Todd F. Dupont, Drop formation in a one-dimensional approximation of the Navier-Stokes equation, J. Fluid Mech. 262 (1994), 205-221. MR 94m:76029
  • 8. C. William Gear, Numerical initial value problems in ordinary differential equations, Prentice-Hall, New Jersey, 1971. MR 47:4447
  • 9. Gene H. Golub and Charles F. Van Loan, Matrix computations, third ed., The Johns Hopkins University Press, Baltimore, 1996. MR 97g:65006
  • 10. A. E. Hosoi and John W. M. Bush, Evaporative instabilities in climbing films, J. Fluid Mech. 442 (2001), 217-239. MR
  • 11. A. E. Hosoi and L. Mahadevan, Axial instability of a free-surface in a partially filled horizontal rotating cylinder, Physics of Fluids 11 (1999), 97-106. MR 99k:76055
  • 12. L. F. Shampine, Local error estimation by doubling, Computing 34 (1985), 179-190. MR 87b:65093
  • 13. Vidar Thomée, Galerkin finite element methods for parabolic problems, Springer Series in Computational Mathematics, vol. 25, Springer-Verlag, Berlin, 1997. MR 98m:65007
  • 14. Mary F. Wheeler, A priori $L_2$ error estimates for Galerkin approximations to parabolic partial differential equations, SIAM J. Numer. Anal. 10 (1973), 723-759. MR 50:3613

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Additional Information

Bruce P. Ayati
Affiliation: Department of Mathematics, Southern Methodist University, Dallas, Texas 75275

Todd F. Dupont
Affiliation: Departments of Computer Science and Mathematics, The University of Chicago, Chicago, Illinois 60637

Keywords: Variable time steps, step-size control, parabolic partial differential equation.
Received by editor(s): October 22, 2003
Received by editor(s) in revised form: February 27, 2004
Published electronically: September 10, 2004
Additional Notes: The second author was supported by the ASCI Flash Center at the University of Chicago under DOE contract B532820, and by the MRSEC Program of the National Science Foundation under award DMR-0213745.
Article copyright: © Copyright 2004 American Mathematical Society

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