Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



On the numerical computation of parabolic problems for preceding times

Authors: B. L. Buzbee and Alfred Carasso
Journal: Math. Comp. 27 (1973), 237-266
MSC: Primary 65M30
MathSciNet review: 0368448
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Abstract: We develop and analyze a general procedure for computing selfadjoint parabolic problems backwards in time, given an a priori bound on the solutions. The method is applicable to mixed problems with variable coefficients which may depend on time. We obtain error bounds which are naturally related to certain convexity inequalities in parabolic equations. In the time-dependent case, our difference scheme discerns three classes of problems. In the most severe case, we recover a convexity result of Agmon and Nirenberg. We illustrate the method with a numerical experiment.

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  • [1] R. S. Anderssen, A Review of Numerical Methods for Certain Improperly Posed Parabolic Partial Differential Equations, Technical Report No. 36, Computer Centre, Australian National University, August 1970.
  • [2] B. L. Buzbee, G. H. Golub, and C. W. Nielson, On direct methods for solving Poisson’s equations, SIAM J. Numer. Anal. 7 (1970), 627–656. MR 0287717
  • [3] J. R. Cannon, Some numerical results for the solution of the heat equation backwards in time, Numerical Solutions of Nonlinear Differential Equations (Proc. Adv. Sympos., Madison, Wis., 1966) John Wiley & Sons, Inc., New York, 1966, pp. 21–54. MR 0207221
  • [4] J. R. Cannon and Jim Douglas Jr., The approximation of harmonic and parabolic functions on half-spaces from interior data, Numerical Analysis of Partial Differential Equations (C.I.M.E. 2 Ciclo, Ispra, 1967) Edizioni Cremonese, Rome, 1968, pp. 193–230. MR 0243755
  • [5] Alfred Carasso, The abstract backward beam equation, SIAM J. Math. Anal. 2 (1971), 193–212. MR 0288438
  • [6] Alfred Carasso, The backward beam equation: Two 𝐴-stable schemes for parabolic problems, SIAM J. Numer. Anal. 9 (1972), 406–434. MR 0315910
  • [7] R. Courant, Methods of Mathematical Physics. Vol. II: Partial Differential Equations, Interscience, New York, 1962. MR 25 #4216.
  • [8] J. Douglas, Jr., Approximate Solution of Physically Unstable Problems, École CEA-EDF, Paris, 1965.
  • [9] Jim Douglas Jr., The approximate solution of an unstable physical problem subject to constraints, Functional Analysis and Optimization, Academic Press, New York, 1966, pp. 65–66. MR 0216127
  • [10] Jim Douglas Jr., Approximate continuation of harmonic and parabolic functions, Numerical Solution of Partial Differential Equations (Proc. Sympos. Univ. Maryland, 1965) Academic Press, New York, 1966, pp. 353–364. MR 0202333
  • [11] Avner Friedman, Partial differential equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969. MR 0445088
  • [12] Fritz John, Numerical solution of the equation of heat conduction for preceding times, Ann. Mat. Pura Appl. (4) 40 (1955), 129–142. MR 0087224
  • [13] Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
  • [14] R. Lattès and J.-L. Lions, Méthode de quasi-réversibilité et applications, Travaux et Recherches Mathématiques, No. 15, Dunod, Paris, 1967 (French). MR 0232549
  • [15] M. M. Lavrentiev, Some Improperly Posed Problems of Mathematical Physics, Springer Tracts in Natural Philosophy, vol. II, Springer-Verlag, New York, 1967.
  • [16] M. M. Lavrentiev, Numerical solution of conditionally properly posed problems, Numerical solution of partial differential equations, II (SYNSPADE, 1970) (Proc. Sympos., Univ. Maryland, College Park, Md., 1970) Academic Press, New York, 1971, pp. 417–432. MR 0309325
  • [17] M. Lees and M. H. Protter, Unique continuation for parabolic differential equations and inequalities, Duke Math. J. 28 (1961), 369–382. MR 0140840
  • [18] J.-L. Lions and B. Malgrange, Sur l’unicité rétrograde dans les problèmes mixtes paraboliques, Math. Scand. 8 (1960), 277–286 (French). MR 0140855
  • [19] J.-L. Lions, Équations différentielles opérationnelles et problèmes aux limites, Die Grundlehren der mathematischen Wissenschaften, Bd. 111, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961 (French). MR 0153974
  • [20] Y. L. Luke, The Special Functions and Their Approximations. Vols. 1, 2, Math. in Sci. and Engineering, vol. 53, Academic Press, New York, 1969. MR 39 #3039; MR 40 #2909.
  • [21] Keith Miller, Three circle theorems in partial differential equations and applications to improperly posed problems, Arch. Rational Mech. Anal. 16 (1964), 126–154. MR 0164136
  • [22] L. E. Payne, On some non well posed problems for partial differential equations, Numerical Solutions of Nonlinear Differential Equations (Proc. Adv. Sympos., Madison, Wis., 1966) John Wiley & Sons, Inc., New York, 1966, pp. 239–263. MR 0213749
  • [23] Carlo Pucci, Sui problemi di Cauchy non “ben posti”, Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. (8) 18 (1955), 473–477 (Italian). MR 0087873
  • [24] Robert D. Richtmyer and K. W. Morton, Difference methods for initial-value problems, Second edition. Interscience Tracts in Pure and Applied Mathematics, No. 4, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1967. MR 0220455
  • [25] Richard Saylor, Time reversal in abstract Cauchy problems, SIAM J. Math. Anal. 2 (1971), 454–457. MR 0288628
  • [26] R. S. Varga, "Some results in approximation theory with applications to numerical analysis," Numerical Solution of Partial Differential Equations. II (SYNSPADE 1970), B. Hubbard (Editor), Academic Press, New York, 1971.
  • [27] F. John, Partial differential equations, Mathematics Applied to Physics, Springer, New York, 1970, pp. 229–315. MR 0261133
  • [28] Joel N. Franklin, Well-posed stochastic extensions of ill-posed linear problems, J. Math. Anal. Appl. 31 (1970), 682–716. MR 0267654
  • [29] W. L. Miranker, A well posed problem for the backward heat equation, Proc. Amer. Math. Soc. 12 (1961), 243–247. MR 0120462, 10.1090/S0002-9939-1961-0120462-2

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Keywords: Improperly posed problems, backward parabolic equations, time-dependent coefficients, finite-difference scheme, jury-procedure, the backward beam equation, block Gaussian elimination, method of lines, variable domain operator, smoothing by growing diffusion coefficient, convexity inequalities for parabolic equations, long-time backward computation, matrix decomposition code
Article copyright: © Copyright 1973 American Mathematical Society