Numerical solution of stiff and singularly perturbed boundary value problems with a segmented-adaptive formulation of the tau method

Authors:
P. Onumanyi and E. L. Ortiz

Journal:
Math. Comp. **43** (1984), 189-203

MSC:
Primary 65L10

MathSciNet review:
744930

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**[1]**J. W. Barrett & K. W. Morton,*Optimal Finite Element Solutions to Diffusion-Convection Problems in One Dimension*, University of Reading, Numerical Analysis Report 3/78, 1978.**[2]**I. Christie, D. F. Griffiths, A. R. Mitchell, and O. C. Zienkiewicz,*Finite element methods for second order differential equations with significant first derivatives*, Internat. J. Numer. Methods Engrg.**10**(1976), no. 6, 1389–1396. MR**0445844****[3]**M. R. Crisci and E. Russo,*An extension of Ortiz’ recursive formulation of the tau method to certain linear systems of ordinary differential equations*, Math. Comp.**41**(1983), no. 163, 27–42. MR**701622**, 10.1090/S0025-5718-1983-0701622-9**[4]**Carl de Boor and Blâir Swartz,*Collocation at Gaussian points*, SIAM J. Numer. Anal.**10**(1973), 582–606. MR**0373328****[5]**Frank R. de Hoog and Richard Weiss,*Difference methods for boundary value problems with a singularity of the first kind*, SIAM J. Numer. Anal.**13**(1976), no. 5, 775–813. MR**0440931****[6]**J. H. Freilich and E. L. Ortiz,*Numerical solution of systems of ordinary differential equations with the Tau method: an error analysis*, Math. Comp.**39**(1982), no. 160, 467–479. MR**669640**, 10.1090/S0025-5718-1982-0669640-6**[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. 1-14.**[8]**Karl G. Guderley,*A unified view of some methods for stiff two-point boundary value problems*, SIAM Rev.**17**(1975), 416–442. MR**0366046****[9]**Gershon Kedem,*A posteriori error bounds for two-point boundary value problems*, SIAM J. Numer. Anal.**18**(1981), no. 3, 431–448. MR**615524**, 10.1137/0718028**[10]**C. Lanczos, "Trigonometric interpolation of empirical and analytical functions,"*J. Math. Phys.*, v. 17, 1938, pp. 123-199.**[11]**Cornelius Lanczos,*Applied analysis*, Prentice Hall, Inc., Englewood Cliffs, N. J., 1956. MR**0084175****[12]**C. Lánczos,*Legendre versus Chebyshev polynomials*, Topics in numerical analysis (Proc. Roy. Irish Acad. Conf., University Coll., Dublin, 1972) Academic Press, London, 1973, pp. 191–201. MR**0341880****[13]**Yudell L. Luke,*The special functions and their approximations, Vol. I*, Mathematics in Science and Engineering, Vol. 53, Academic Press, New York-London, 1969. MR**0241700****[14]**P. Llorente and E. L. Ortiz,*Sur quelques aspects algébriques d’une méthode d’approximation de M. Lanczos*, Math. Notae**21**(1966/1967), 17–23. MR**0246519****[15]**Günter Meinardus,*Approximation of functions: Theory and numerical methods*, Expanded translation of the German edition. Translated by Larry L. Schumaker. Springer Tracts in Natural Philosophy, Vol. 13, Springer-Verlag New York, Inc., New York, 1967. MR**0217482****[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 05-09-81, 1981, pp. 1-5.**[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 04-08-82.**[18]**F. Aleixo Oliveira,*Collocation and residual correction*, Numer. Math.**36**(1980/81), no. 1, 27–31. MR**595804**, 10.1007/BF01395986**[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 and 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.**18**(1982), no. 5, 775–781. MR**664672**, 10.1002/nme.1620180512**[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. 282-286.**[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. 1-22.**[24]**Eduardo L. Ortiz,*The tau method*, SIAM J. Numer. Anal.**6**(1969), 480–492. MR**0258287****[25]**Eduardo L. Ortiz,*Canonical polynomials in the Lanczos tau method*, Studies in numerical analysis (papers in honour of Cornelius Lanczos on the occasion of his 80th birthday), Academic Press, London, 1974, pp. 73–93. MR**0474847****[26]**E. L. Ortiz,*Step by step Tau method. I. Piecewise polynomial approximations*, Computers and mathematics with applications, Pergamon, Oxford, 1976, pp. 381–392. MR**0464550****[27]**E. L. Ortiz,*Sur Quelques Nouvelles Applications de la Méthode Tau*, Séminaire Lions, Analyse et Contrôle de Systèmes, IRIA, Paris, 1975, pp. 247-257.**[28]**E. L. Ortiz,*On the numerical solution of nonlinear and functional differential equations with the tau method*, Numerical treatment of differential equations in applications (Proc. Meeting, Math. Res. Center, Oberwolfach, 1977) Lecture Notes in Math., vol. 679, Springer, Berlin, 1978, pp. 127–139. MR**515576****[29]**Eduardo L. Ortiz,*Polynomial and rational approximation of boundary layer problems with the tau method*, Boundary and interior layers—computational and asymptotic methods (Proc. Conf., Trinity College, Dublin, 1980) Boole, Dún Laoghaire, 1980, pp. 387–391. MR**589393****[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 and 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.**6**(1984), no. 4, 457–466. MR**771805**, 10.1002/mma.1670060128**[32]**E. L. Ortiz, F. J. Rodriguez Cañizares & W. F. C. Purser,*Automation of the Tau Method*, Imperial College Research Report NAS 01-72, 1972, pp. 1-54. (Presented to the Conference on Numerical Analysis organized by the Royal Irish Academy, Dublin, 1972.)**[33]**E. L. Ortiz and H. Samara,*An operational approach to the tau method for the numerical solution of nonlinear differential equations*, Computing**27**(1981), no. 1, 15–25 (English, with German summary). MR**623173**, 10.1007/BF02243435**[34]**R. D. Russell and L. F. Shampine,*A collocation method for boundary value problems*, Numer. Math.**19**(1972), 1–28. MR**0305607****[35]**Melvin R. Scott,*On the conversion of boundary-value problems into stable initial-value problems via several invariant imbedding algorithms*, Numerical solutions of boundary value problems for ordinary differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1974), Academic Press, New York, 1975, pp. 89–146. MR**0416044****[36]**Hans J. Stetter,*Economical global error estimation*, Stiff differential systems (Proc. Internat. Sympos., Wildbad, 1973), Plenum, New York, 1974, pp. 245–258. IBM Res. Sympos. Ser. MR**0405863****[37]**Pedro E. Zadunaisky,*On the estimation of errors propagated in the numerical integration of ordinary differential equations*, Numer. Math.**27**(1976/77), no. 1, 21–39. MR**0431696**

Retrieve articles in *Mathematics of Computation*
with MSC:
65L10

Retrieve articles in all journals with MSC: 65L10

Additional Information

DOI:
http://dx.doi.org/10.1090/S0025-5718-1984-0744930-9

Keywords:
Initial value problems,
boundary value problems,
systems of ordinary differential equations,
stiff problems,
singularly perturbed problems,
step-by-step approximation,
adaptive methods,
Tau method

Article copyright:
© Copyright 1984
American Mathematical Society