Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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

 
 

 

The numerical solution of boundary value problems for stiff differential equations


Authors: Joseph E. Flaherty and R. E. O'Malley
Journal: Math. Comp. 31 (1977), 66-93
MSC: Primary 65L10
DOI: https://doi.org/10.1090/S0025-5718-1977-0657396-0
MathSciNet review: 0657396
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The numerical solution of boundary value problems for certain stiff ordinary differential equations is studied. The methods developed use singular perturbation theory to construct approximate numerical solutions which are valid asymptotically; hence, they have the desirable feature of becoming more accurate as the equations become stiffer. Several numerical examples are presented which demonstrate the effectiveness of these methods.


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

  • [1] L. R. ABRAHAMSSON, H. B. KELLER & H. O. KREISS, "Difference approximations for singular perturbations of systems of ordinary differential equations," Numer. Math., v. 22, 1974, pp. 367-391. MR 0388784 (52:9618)
  • [2] R. AIKEN & L. LAPIDUS, "An effective numerical integration method for typical stiff systems," A.I.Ch.E.J., v. 20, 1974, pp. 368-375. MR 0395228 (52:16026)
  • [3] P. T. BOGGS, "A minimization algorithm based on singular perturbation theory," SIAM J. Numer. Anal. (To appear.) MR 0519600 (58:24933)
  • [4] R. BULIRSCH & J. STOER, "Numerical treatment of ordinary differential equations by extrapolation methods," Numer. Math., v. 8, 1966, pp. 1-13. MR 32 #8504. MR 0191095 (32:8504)
  • [5] J. D. COLE, Perturbation Methods in Applied Mathematics, Blaisdell, Waltham, Mass., 1968. MR 39 #7841. MR 0246537 (39:7841)
  • [6] S. D. CONTE, "The numerical solution of linear boundary value problems," SIAM Rev., v. 8, 1966, pp. 309-321. MR 34 #3792. MR 0203945 (34:3792)
  • [7] S. D. CONTE & C. de BOOR, Elementary Numerical Analysis, 2nd ed., McGraw-Hill, New York, 1972, Chapter 5.
  • [8] F. W. DORR, "The numerical solution of singular perturbations of boundary value problems," SIAM J. Numer. Anal., v. 7, 1970, pp. 281-313. MR 42 #2683. MR 0267781 (42:2683)
  • [9] F. W. DORR, "An example of ill-conditioning in the numerical solution of singular perturbation problems," Math. Comp., v. 25, 1971, pp. 271-283. MR 45 #6200. MR 0297142 (45:6200)
  • [10] W. E. FERGUSON, JR., "A singularly perturbed linear two-point boundary-value problem," Ph.D. Dissertation, California Inst. Tech., 1975.
  • [11] P. C. FIFE, "Semilinear elliptic boundary value problems with small parameters," Arch. Rational Mech. Anal., v. 52, 1973, pp. 205-232. MR 51 #10863. MR 0374665 (51:10863)
  • [12] P. C. FIFE, "Transition layers in singular perturbation problems," J. Differential Equations, v. 15, 1974, pp. 77-105. MR 48 #9002. MR 0330665 (48:9002)
  • [13] N. FRÖMAN & P. O. FRÖMAN, JWKB Approximation. Contributions to the Theory, North-Holland, Amsterdam, 1965. MR 30 #3694. MR 0173481 (30:3694)
  • [14] C. W. GEAR, Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-Hall, Englewood Cliffs, N. J., 1971, Chapters 9, 11. MR 47 #4447. MR 0315898 (47:4447)
  • [15] P. W. HEMKER, A Method of Weighted One-Sided Differences for Stiff Boundary Value Problems with Turning Points, Report NW 9/74, Mathematisch Centrum, Amsterdam, 1974. MR 50 #3587. MR 0351096 (50:3587)
  • [16] F. A. HOWES, "The asymptotic solution of a class of singularly perturbed nonlinear second order boundary value problems via differential inequalities," SIAM J. Math. Anal. (To appear.) MR 0477345 (57:16877a)
  • [17] H. B. KELLER, "Accurate difference methods for linear ordinary differential systems subject to linear constraints," SIAM J. Numer. Anal., v. 6, 1969, pp. 8-30. MR 40 #6776. MR 0253562 (40:6776)
  • [18] H. B. KELLER, "Accurate difference methods for nonlinear two-point boundary value problems," SIAM J. Numer. Anal., v. 11, 1974, pp. 305-320. MR 50 #3589. MR 0351098 (50:3589)
  • [19] H. B. KELLER & T. CEBECI, "Accurate numerical methods for boundary-layer flows. II. Two-dimensional turbulent flows," AIAA J., v. 10, 1972, pp. 1193-1199. MR 46 #10300. MR 0311207 (46:10300)
  • [20] H. B. KELLER & A. B. WHITE, JR., "Difference methods for boundary value problems in ordinary differential equations," SIAM J. Numer. Anal., v. 12, 1975, pp. 791-802. MR 0413513 (54:1627)
  • [21] M. LENTINI & V. PEREYRA, "Boundary problem solvers for first order systems based on deferred corrections," in Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, (A. K. Aziz, Editor), Academic Press, New York, 1975. MR 0488787 (58:8297)
  • [22] B. LINDBERG, "On a dangerous property of methods for stiff differential equations," BIT, v. 14, 1974, pp. 430-436. MR 50 #15347. MR 0362909 (50:15347)
  • [23] D. B. MACMILLAN, "Asymptotic methods for systems of differential equations in which some variables have very short response times," SIAM J. Appl. Math., v. 16, 1968, pp. 704-722. MR 38 #2403. MR 0234084 (38:2403)
  • [24] J. A. M. McHUGH, "An historical survey of ordinary differential equations with a large parameter and turning points," Arch. History Exact Sci., v. 7, 1971, pp. 277-324. MR 1554147
  • [25] W. L. MIRANKER, "Numerical methods of boundary layer type for stiff systems of differential equations," Computing, v. 11, 1973, pp. 221-234. MR 0386276 (52:7134)
  • [26] W. L. MIRANKER & J. P. MORREEUW, "Semianalytic studies of turning points arising in stiff boundary value problems," Math. Comp., v. 28, 1974, pp. 1017-1034. MR 0381329 (52:2226)
  • [27] D. E. MULLER, "A method for solving algebraic equations using an automatic computer," MTAC, v. 10, 1956, pp. 208-215. MR 18, 766. MR 0083822 (18:766e)
  • [28] W. D. MURPHY, "Numerical analysis of boundary-layer problems in ordinary differential equations," Math. Comp., v. 21, 1967, pp. 583-596. MR 37 #1089. MR 0225496 (37:1089)
  • [29] R. E. O'MALLEY, JR.,Introduction to Singular Perturbations, Academic Press, New York, 1974. MR 0402217 (53:6038)
  • [30] R. E. O'MALLEY, JR., "On multiple solutions of a singular perturbation problem," Arch. Rational Mech. Anal., v. 49, 1972/73, pp. 89-98. MR 49 #761. MR 0335985 (49:761)
  • [31] R. E. O'MALLEY, JR., "Phase plane solutions to some singular perturbation problems," J. Math. Anal. Appl., v. 54, 1976, pp. 449-466. MR 0450722 (56:9015)
  • [32] R. E. O'MALLEY, JR., "Boundary layer methods for ordinary differential equations with small coefficients multiplying the highest derivatives," (Proc. Sympos. on Constructive and Computational Methods for Differential and Integral Equations), Lecture Notes in Math., vol. 430, Springer-Verlag, Berlin and New York, 1974, pp. 363-389. MR 0486872 (58:6566)
  • [33] R. E. O'MALLEY, JR. & J. B. KELLER, "Loss of boundary conditions in the asymptotic solution of linear ordinary differential equations. II. Boundary value problems," Comm. Pure Appl. Math., v. 21, 1968, pp. 263-270. MR 37 #528. MR 0224929 (37:528)
  • [34] F. W. J. OLVER, Asymptotics and Special Functions, Academic Press, New York, 1974. MR 0435697 (55:8655)
  • [35] S. V. PARTER, "Singular perturbations of second order differential equations." (Unpublished.)
  • [36] C. E. PEARSON, "On a differential equation of the boundary layer type," J. Math. and Phys., v. 47, 1968, pp. 134-154. MR 37 #3773. MR 0228189 (37:3773)
  • [37] C. E. PEARSON, "On non-linear ordinary differential equations of boundary layer type," J. Math. and Phys., v. 47, 1968, pp. 351-358. MR 38 #5400. MR 0237107 (38:5400)
  • [38] M. R. SCOTT & H. A. WATTS, Support-A Computer Code for Two-Point Boundary-Value Problems via Orthonormalization, Sandia Laboratories Report SAND 75-0198, June 1975.
  • [39] S. TIMOSHENKO, Strength of Materials, Part II, Advanced Theory and Problems, 3rd ed., Van Nostrand, Princeton, N. J., 1956.
  • [40] M. I. VIŠIK & L. A. LJUSTERNIK, "Initial jump for non-linear differential equations containing a small parameter," Dokl. Akad. Nauk SSSR, Tom 132, 1960, pp. 1242-1245 = Soviet Math. Dokl., v. 1, 1960, pp. 749-752. MR 22 #11181. MR 0120427 (22:11181)
  • [41] W. R. WASOW, "On the asymptotic solution of boundary value problems for ordinary differential equations containing a parameter," J. Math. and Phys., v. 23, 1944, pp. 173-183. MR 6, 86. MR 0010907 (6:86c)
  • [42] W. R. WASOW, "Singular perturbations of boundary value problems for nonlinear differential equations of the second order," Comm. Pure. Appl. Math., v. 9, 1956, pp. 93-113. MR 18, 39. MR 0079161 (18:39d)
  • [43] W. R. WASOW, "Connection problems for asymptotic series," Bull. Amer. Math. Soc., v. 74, 1968, pp. 831-853. MR 37 #4336. MR 0228757 (37:4336)
  • [44] R. WILLOUGHBY (Editor), Stiff Differential Systems, Plenum Press, New York, 1974. MR 0343619 (49:8359)
  • [45] J. YARMISH, Aspects of the Numerical and Theoretical Treatment of Singular Perturbation, Doctoral Dissertation, New York Univ., 1972.
  • [46] J. YARMISH, "Newton's method techniques for singular perturbations," SIAM J. Math. Anal., v. 6, 1975, pp. 661-680. MR 0426448 (54:14391)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65L10

Retrieve articles in all journals with MSC: 65L10


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1977-0657396-0
Article copyright: © Copyright 1977 American Mathematical Society

American Mathematical Society