Numerical solution of stiff and singularly perturbed boundary value problems with a segmentedadaptive formulation of the tau method
Authors:
P. Onumanyi and E. L. Ortiz
Journal:
Math. Comp. 43 (1984), 189203
MSC:
Primary 65L10
MathSciNet review:
744930
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Abstract: This paper concerns the application of Ortiz' recursive formulation of the Tau method to the construction of piecewise polynomial approximations to the solution of linear and nonlinear boundary value problems for ordinary differential equations. A practical error estimation technique, related to the concept of correction in Zadunaisky's sense, is considered and used in the design of an adaptive approach to the Tau method. It proves efficient in the numerical treatment of problems with rapid functional variations, stiff and singularly perturbed problems. A technique of increased accuracy at matching points of segmented Tau approximants is also discussed and successfully applied to several problems. Numerical examples show that, for a given degree of approximation, our segmented Tau approximant gives an accuracy comparable to that of the best segmented approximation of the exact solution by means of algebraic polynomials.
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 J. W. Barrett & K. W. Morton, Optimal Finite Element Solutions to DiffusionConvection Problems in One Dimension, University of Reading, Numerical Analysis Report 3/78, 1978.
 [2]
 I. Christie, D. F. Griffith, A. R. Mitchell & O. C. Zienkiewicz, "Finite element methods for second order differential equations with significant derivatives," Internat. J. Numer. Methods Engrg., v. 10, 1976, pp. 13891396. MR 0445844 (56:4178)
 [3]
 M. R. Crisci & E. Russo, "An extension of Ortiz' recursive formulation of the Tau method to certain linear systems of ordinary differential equations," Math. Comp., v. 41, 1983, pp. 2742. MR 701622 (85b:65061)
 [4]
 C. de Boor & B. Swartz, "Collocation at Gaussian points," SIAM J. Numer. Anal., v. 10, 1973, pp. 582606. MR 0373328 (51:9528)
 [5]
 F. R. de Hoog & R. Weiss, "Difference methods for boundary value problems with a singularity of the first kind," SIAM. J. Numer. Anal., v. 13, 1976, pp. 775813. MR 0440931 (55:13799)
 [6]
 J. H. Freilich & E. L. Ortiz, "Numerical solution of systems of ordinary differential equations with the Tau method: An error analysis," Math. Comp., v. 39, 1982, pp. 467479. MR 669640 (84k:65066)
 [7]
 J. H. Freilich & E. L. Ortiz, On the Error Analysis of a Vector Rational Approximation of the Solution of a System of Ordinary Differential Equations With the Tau Method, Imperial College Research Report NAS 75, 1975, pp. 114.
 [8]
 K. G. Guderley, "A unified view of some methods for stiff twopoint boundary value problems," SIAM Rev., v. 17, 1975, pp. 416442. MR 0366046 (51:2297)
 [9]
 G. Kedem, "A posteriori error bounds for twopoint boundary value problems," SIAM J. Numer. Anal., v. 18, 1981, pp. 431448. MR 615524 (82h:65062)
 [10]
 C. Lanczos, "Trigonometric interpolation of empirical and analytical functions," J. Math. Phys., v. 17, 1938, pp. 123199.
 [11]
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 C. Lanczos, "Legendre versus Chebyshev polynomials," Topics in Numerical Analysis (J. J. H. Miller, Ed.), Academic Press, New York, 1973. MR 0341880 (49:6626)
 [13]
 Y. L. Luke, The Special Functions and Their Approximations, Vols. I and II. Academic Press, New York, 1969. MR 0241700 (39:3039)
 [14]
 P. Llorente & E. L. Ortiz, "Sur quelques aspects algébriques d'une méthode de M. Lanczos," Math. Notae, v. 21, 1968, pp. 1721. MR 0246519 (39:7823)
 [15]
 G. Meinardus, Approximation of Functions: Theory and Numerical Methods, SpringerVerlag, Berlin, 1967. MR 0217482 (36:571)
 [16]
 S. Namasivayam & E. L. Ortiz, Perturbation Terms and Approximation Error in the Numerical Solution of Differential Equations With the Tau Method, Imperial College, Research Report NAS 050981, 1981, pp. 15.
 [17]
 S. Namasivayam & E. L. Ortiz, Approximate coefficients and approximation error in the numerical solution of differential equations, with an application to the Tau method, Research Report NAS 040882.
 [18]
 F. A. Oliveira, "Collocation and residual correction," Numer. Math., v. 36, 1980, pp. 2731. MR 595804 (82a:65060)
 [19]
 P. Onumanyi, Numerical Experiments With Some Nonlinear Ordinary Differential Equations Using the Tau Method, M. Sc. Thesis, Imperial College, 1978.
 [20]
 P. Onumanyi, Numerical Solution of Nonlinear Boundary Value Problems With the Tau Method, Ph.D. Thesis, Imperial College, 1981.
 [21]
 P. Onumanyi & E. L. Ortiz, "Numerical solution of high order boundary value problems for ordinary differential equations with an estimation of the error," Internat. J. Numer. Methods Engrg., v. 18, 1982, pp. 775781. MR 664672 (83f:76013)
 [22]
 P. Onumanyi, E. L. Ortiz & H. Samara, "Software for a method of finite approximations for the numerical solution of differential equations," Appl. Math. Modelling, v. 5, 1981, pp. 282286.
 [23]
 E. L. Ortiz, On the Generation of the Canonical Polynomials Associated With Certain Linear Differential Operators, Imperial College Research Report NAS 64, 1964, pp. 122.
 [24]
 E. L. Ortiz, "The Tau method," SIAM J. Numer. Anal., v. 6, 1969, pp. 480492. MR 0258287 (41:2934)
 [25]
 E. L. Ortiz, "Canonical polynomials in the Lanczos Tau method," Studies in Numerical Analysis (B. Scaife, Ed.), Academic Press, New York, 1974, pp. 7393. MR 0474847 (57:14478)
 [26]
 E. L. Ortiz, "Step by step Tau method, Piecewise polynomial approximations," Comput. Math. Appl., v. 1, 1975, pp. 381392. MR 0464550 (57:4480)
 [27]
 E. L. Ortiz, Sur Quelques Nouvelles Applications de la Méthode Tau, Séminaire Lions, Analyse et Contrôle de Systèmes, IRIA, Paris, 1975, pp. 247257.
 [28]
 E. L. Ortiz, "On the numerical solution of nonlinear and functional differential equations with the Tau method," in Numerical Treatment of Differential Equations in Applications (R. Ansorge and W. Törnig, Eds.), Lecture Notes in Math., Vol. 679, SpringerVerlag, Berlin, 1978, pp. 127139. MR 515576 (80c:65180)
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 E. L. Ortiz, "Polynomial and rational approximation of boundary layer problems with the Tau method," in Boundary and Interior LayersComputational and Asymptotic Methods (J. J. H. Miller, Ed.), Boole Press, Dublin, 1980, pp. 387391. MR 589393 (81m:65138)
 [30]
 E. L. Ortiz & A. Pham Ngoc Dinh, "On the convergence of the Tau method for nonlinear differential equations of Riccati's type," Nonlinear Analysis, 1984. (In press.)
 [31]
 E. L. Ortiz & A. Pham Ngoc Dinh, "An error analysis of the Tau method for a class of singularly perturbed problems for differential equations," Math. Methods Appl. Sci., 1984. (In press.) MR 771805 (86d:65101)
 [32]
 E. L. Ortiz, F. J. Rodriguez Cañizares & W. F. C. Purser, Automation of the Tau Method, Imperial College Research Report NAS 0172, 1972, pp. 154. (Presented to the Conference on Numerical Analysis organized by the Royal Irish Academy, Dublin, 1972.)
 [33]
 E. L. Ortiz & H. Samara, "An operational approach to the Tau method for the numerical solution of nonlinear differential equations," Computing, v. 27, 1981, pp. 1525. MR 623173 (83b:65079)
 [34]
 R. D. Russell & L. F. Shampine, "A collocation method for boundaryvalue problems," Numer. Math., v. 19, 1972, pp. 128. MR 0305607 (46:4737)
 [35]
 M. R. Scott, "On the conversion of boundaryvalue problems into stable initialvalue problems via several invariant imbedding algorithms," in Numerical Solution of Boundary Value Problems for Ordinary Differential Equations (A. K. Aziz, Ed.), Academic Press, New York, 1975, pp. 89146. MR 0416044 (54:4120)
 [36]
 H. J. Stetter, "Economical global error estimation," in Stiff Differential Equations (R. A. Willoughby, Ed.), Plenum Press, New York, 1974, pp. 245258. MR 0405863 (53:9655)
 [37]
 P. Zadunaisky, "On the estimation of errors in the numerical integration of ordinary differential equations," Numer. Math., v. 27, 1976, pp. 2139. MR 0431696 (55:4691)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198407449309
PII:
S 00255718(1984)07449309
Keywords:
Initial value problems,
boundary value problems,
systems of ordinary differential equations,
stiff problems,
singularly perturbed problems,
stepbystep approximation,
adaptive methods,
Tau method
Article copyright:
© Copyright 1984
American Mathematical Society
