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A posteriori bounds in the numerical solution of mildly nonlinear parabolic equations

Author: Alfred Carasso
Journal: Math. Comp. 24 (1970), 785-792
MSC: Primary 65.68
MathSciNet review: 0281374
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Abstract: We derive a posteriori bounds for $ (V - \hat V)$ and its difference quotient $ {(V - \hat V)_x}$, where $ V$ and $ \hat V$ are, respectively, the exact and computed solution of a difference approximation to a mildly nonlinear parabolic initial boundary problem, with a known steadystate solution. It is assumed that the computation is over a long interval of time. The estimates are valid for a class of difference approximations, which includes the CrankNicolson method, and are of the same magnitude for both $ (V - \hat V)$ and $ (V - \hat V)x$.

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  • [1] A. Carasso, "Finite-difference methods and the eigenvalue problem for nonselfadjoint Sturm-Liouville operators," Math. Comp., v. 23, 1969, pp. 717-729. MR 0258291 (41:2938)
  • [2] A. Carasso & S. V. Parter, "An analysis of 'boundary-value techniques' for parabolic problems," Math. Comp., v. 24, 1970, pp. 315-340. MR 0284019 (44:1249)
  • [3] A. Carasso, "Long range numerical solution of mildly non-linear parabolic equations," Numer. Math. (To appear.) MR 0286301 (44:3514)
  • [4] J. Douglas, Jr., A Survey of Numerical Methods for Parabolic Differential Equations, Advances in Computers, vol. 2, Academic Press, New York, 1961, pp. 1-54. MR 25 #5604. MR 0142211 (25:5604)
  • [5] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, N. J., 1964. MR 31 #6062. MR 0181836 (31:6062)
  • [6] F. John, "On integration of parabolic equations by difference methods. I: Linear and quasi-linear equations for the infinite interval," Comm. Pure Appl. Math., v. 5, 1952, pp. 155-211. MR 13, 947. MR 0047885 (13:947b)
  • [7] H. O. Kreiss & O. B. Widlund, Difference Approximations for Initial Value Problems for Partial Differential Equations, Department of Computer Sciences, Report NR 7, Upsala University, 1967.
  • [8] M. Lees, "Approximate solutions of parabolic equations," J. Soc. Indust. Appl. Math., v. 7, 1959, pp. 167-183. MR 22 #1092. MR 0110212 (22:1092)
  • [9] R. D. Richtmyer & K. W. Morton, Difference Methods for Initial-Value Problems, 2nd ed., Interscience Tracts in Pure and Appl. Math., no. 4, Interscience, New York, 1967. MR 36 #3515. MR 0220455 (36:3515)
  • [10] O. B. Widlund, "On difference methods for parabolic equations and alternating direction implicit methods for elliptic equations," IBM J. Res. Develop., v. 11, 1967, pp. 239-243. MR 36 #7356. MR 0224312 (36:7356)

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Keywords: Parabolic equations, Crank-Nicolson method, limiting steady state, computations over long times
Article copyright: © Copyright 1970 American Mathematical Society

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