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), 6693
MSC:
Primary 65L10
MathSciNet review:
0657396
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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.
 [1]
L.
R. Abrahamsson, H.
B. Keller, and H.
O. Kreiss, Difference approximations for singular perturbations of
systems of ordinary differential equations, Numer. Math.
22 (1974), 367–391. MR 0388784
(52 #9618)
 [2]
Richard
C. Aiken and Leon
Lapidus, An effective numerical integration method for typical
stiff systems, AIChE J. 20 (1974), no. 2,
368–375. MR 0395228
(52 #16026)
 [3]
Paul
T. Boggs, An algorithm, based on singular perturbation theory, for
illconditioned minimization problems, SIAM J. Numer. Anal.
14 (1977), no. 5, 830–843. MR 0519600
(58 #24933)
 [4]
Roland
Bulirsch and Josef
Stoer, Numerical treatment of ordinary differential equations by
extrapolation methods, Numer. Math. 8 (1966),
1–13. MR
0191095 (32 #8504)
 [5]
Julian
D. Cole, Perturbation methods in applied mathematics,
Blaisdell Publishing Co. Ginn and Co., Waltham, Mass.Toronto, Ont.London,
1968. MR
0246537 (39 #7841)
 [6]
S.
D. Conte, The numerical solution of linear boundary value
problems, SIAM Rev. 8 (1966), 309–321. MR 0203945
(34 #3792)
 [7]
S. D. CONTE & C. de BOOR, Elementary Numerical Analysis, 2nd ed., McGrawHill, New York, 1972, Chapter 5.
 [8]
Fred
Dorr, The numerical solution of singular perturbations of boundary
value problems, SIAM J. Numer. Anal. 7 (1970),
281–313. MR 0267781
(42 #2683)
 [9]
Fred
W. Dorr, An example of illconditioning in the
numerical solution of singular perturbation problems, Math. Comp. 25 (1971), 271–283. MR 0297142
(45 #6200), http://dx.doi.org/10.1090/S00255718197102971420
 [10]
W. E. FERGUSON, JR., "A singularly perturbed linear twopoint boundaryvalue problem," Ph.D. Dissertation, California Inst. Tech., 1975.
 [11]
Paul
C. Fife, Semilinear elliptic boundary value problems with small
parameters, Arch. Rational Mech. Anal. 52 (1973),
205–232. MR 0374665
(51 #10863)
 [12]
Paul
C. Fife, Transition layers in singular perturbation problems,
J. Differential Equations 15 (1974), 77–105. MR 0330665
(48 #9002)
 [13]
Nanny
Fröman and Per
Olof Fröman, JWKB approximation. Contributions to the
theory, NorthHolland Publishing Co., Amsterdam, 1965. MR 0173481
(30 #3694)
 [14]
C.
William Gear, Numerical initial value problems in ordinary
differential equations, PrenticeHall Inc., Englewood Cliffs, N.J.,
1971. MR
0315898 (47 #4447)
 [15]
P.
W. Hemker, A method of weighted onesided differences for stiff
boundary value problems with turning points, Mathematisch Centrum,
Amsterdam, 1974. Mathematisch Centrum, Afdeling Numerieke Wiskunde NW 9/74.
MR
0351096 (50 #3587)
 [16]
F.
A. Howes, The asymptotic solution of a class of singularly
perturbed nonlinear boundary value problems via differential
inequalities, SIAM J. Math. Anal. 9 (1978),
no. 2, 215–249. MR 0477345
(57 #16877a)
 [17]
Herbert
B. Keller, Accurate difference methods for linear ordinary
differential systems subject to linear constraints, SIAM J. Numer.
Anal. 6 (1969), 8–30. MR 0253562
(40 #6776)
 [18]
Herbert
B. Keller, Accurate difference methods for nonlinear twopoint
boundary value problems, SIAM J. Numer. Anal. 11
(1974), 305–320. MR 0351098
(50 #3589)
 [19]
Herbert
B. Keller and Tuncer
Cebeci, Accurate numerical methods for boundarylayer flows. II.
Twodimensional turbulent flows, AIAA J. 10 (1972),
1193–1199. MR 0311207
(46 #10300)
 [20]
H.
B. Keller and A.
B. White Jr., Difference methods for boundary value problems in
ordinary differential equations, SIAM J. Numer. Anal.
12 (1975), no. 5, 791–802. MR 0413513
(54 #1627)
 [21]
M.
Lentini and V.
Pereyra, Boundary problem solvers for first order systems based on
deferred corrections, Numerical solutions of boundary value problems
for ordinary differential equations (Proc. Sympos., Univ. Maryland,
Baltimore, Md., 1974), Academic Press, New York, 1975,
pp. 293–315. MR 0488787
(58 #8297)
 [22]
Bengt
Lindberg, On a dangerous property of methods for stiff differential
equations, Nordisk Tidskr. Informationsbehandling (BIT)
14 (1974), 430–436. 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. 16 (1968), 704–722. MR 0234084
(38 #2403)
 [24]
James
A. M. McHugh, An historical survey of ordinary linear differential
equations with a large parameter and turning points, Arch. History
Exact Sci. 7 (1971), no. 4, 277–324. MR
1554147, http://dx.doi.org/10.1007/BF00328046
 [25]
W.
L. Miranker, Numerical methods of boundary layer type for stiff
systems of differential equations, Computing (Arch. Elektron. Rechnen)
11 (1973), no. 3, 221–234 (English, with German
summary). MR
0386276 (52 #7134)
 [26]
W.
L. Miranker and J.
P. Morreeuw, Semianalytic numerical studies of
turning points arising in stiff boundary value problems, Math. Comput. 28 (1974), 1017–1034. MR 0381329
(52 #2226), http://dx.doi.org/10.1090/S00255718197403813295
 [27]
David
E. Muller, A method for solving algebraic
equations using an automatic computer, Math.
Tables Aids Comput. 10 (1956), 208–215. MR 0083822
(18,766e), http://dx.doi.org/10.1090/S00255718195600838220
 [28]
W.
D. Murphy, Numerical analysis of boundarylayer
problems in ordinary differential equations, Math. Comp. 21 (1967), 583–596. MR 0225496
(37 #1089), http://dx.doi.org/10.1090/S00255718196702254969
 [29]
Robert
E. O’Malley Jr., Introduction to singular perturbations,
Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New
YorkLondon, 1974. Applied Mathematics and Mechanics, Vol. 14. MR 0402217
(53 #6038)
 [30]
R.
E. O’Malley Jr., On multiple solutions of a singular
perturbation problem, Arch. Rational Mech. Anal. 49
(1972/73), 89–98. MR 0335985
(49 #761)
 [31]
R.
E. O’Malley Jr., Phaseplane solutions to some singular
perturbation problems, J. Math. Anal. Appl. 54
(1976), no. 2, 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, equations (Sympos., Indiana Univ., Bloomington, Ind.,
1974) Springer, Berlin, 1974, pp. 363–389. Lecture Notes in
Math., Vol. 430. MR 0486872
(58 #6566)
 [33]
Robert
E. O’Malley Jr. and Joseph
B. Keller, Loss of boundary conditions in the asymptotic solution
of linear ordinary differential equations. II. Boundary value
problems, Comm. Pure Appl. Math. 21 (1968),
263–270. MR 0224929
(37 #528)
 [34]
F.
W. J. Olver, Asymptotics and special functions, Academic Press
[A subsidiary of Harcourt Brace Jovanovich, Publishers], New YorkLondon,
1974. Computer Science and Applied Mathematics. MR 0435697
(55 #8655)
 [35]
S. V. PARTER, "Singular perturbations of second order differential equations." (Unpublished.)
 [36]
Carl
E. Pearson, On a differential equation of boundary layer type,
J. Math. and Phys. 47 (1968), 134–154. MR 0228189
(37 #3773)
 [37]
Carl
E. Pearson, On nonlinear ordinary differential equations of
boundary layer type., J. Math. and Phys. 47 (1968),
351–358. MR 0237107
(38 #5400)
 [38]
M. R. SCOTT & H. A. WATTS, SupportA Computer Code for TwoPoint BoundaryValue Problems via Orthonormalization, Sandia Laboratories Report SAND 750198, 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 and L.
A. Lyusternik, Initial jump for nonlinear differential equations
containing a small parameter, Soviet Math. Dokl. 1
(1960), 749–752. MR 0120427
(22 #11181)
 [41]
Wolfgang
Wasow, On the asymptotic solution of boundary value problems for
ordinary differential equations containing a parameter, J. Math. Phys.
Mass. Inst. Tech. 23 (1944), 173–183. MR 0010907
(6,86c)
 [42]
Wolfgang
Wasow, Singular perturbations of boundary value problems for
nonlinear differential equations of the second order, Comm. Pure Appl.
Math. 9 (1956), 93–113. MR 0079161
(18,39d)
 [43]
Wolfgang
Wasow, Connection problems for asymptotic
series, Bull. Amer. Math. Soc. 74 (1968), 831–853. MR 0228757
(37 #4336), http://dx.doi.org/10.1090/S000299041968120555
 [44]
Ralph
A. Willoughby (ed.), Stiff differential systems, Plenum Press,
New York, 1974. The IBM Research Symposia Series. MR 0343619
(49 #8359)
 [45]
J. YARMISH, Aspects of the Numerical and Theoretical Treatment of Singular Perturbation, Doctoral Dissertation, New York Univ., 1972.
 [46]
Joshua
Yarmish, Newton’s method techniques for singular
perturbations, SIAM J. Math. Anal. 6 (1975),
661–680. MR 0426448
(54 #14391)
 [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. 367391. 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. 368375. 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. 113. 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. 309321. MR 34 #3792. MR 0203945 (34:3792)
 [7]
 S. D. CONTE & C. de BOOR, Elementary Numerical Analysis, 2nd ed., McGrawHill, 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. 281313. MR 42 #2683. MR 0267781 (42:2683)
 [9]
 F. W. DORR, "An example of illconditioning in the numerical solution of singular perturbation problems," Math. Comp., v. 25, 1971, pp. 271283. MR 45 #6200. MR 0297142 (45:6200)
 [10]
 W. E. FERGUSON, JR., "A singularly perturbed linear twopoint boundaryvalue 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. 205232. MR 51 #10863. MR 0374665 (51:10863)
 [12]
 P. C. FIFE, "Transition layers in singular perturbation problems," J. Differential Equations, v. 15, 1974, pp. 77105. MR 48 #9002. MR 0330665 (48:9002)
 [13]
 N. FRÖMAN & P. O. FRÖMAN, JWKB Approximation. Contributions to the Theory, NorthHolland, Amsterdam, 1965. MR 30 #3694. MR 0173481 (30:3694)
 [14]
 C. W. GEAR, Numerical Initial Value Problems in Ordinary Differential Equations, PrenticeHall, Englewood Cliffs, N. J., 1971, Chapters 9, 11. MR 47 #4447. MR 0315898 (47:4447)
 [15]
 P. W. HEMKER, A Method of Weighted OneSided 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. 830. MR 40 #6776. MR 0253562 (40:6776)
 [18]
 H. B. KELLER, "Accurate difference methods for nonlinear twopoint boundary value problems," SIAM J. Numer. Anal., v. 11, 1974, pp. 305320. MR 50 #3589. MR 0351098 (50:3589)
 [19]
 H. B. KELLER & T. CEBECI, "Accurate numerical methods for boundarylayer flows. II. Twodimensional turbulent flows," AIAA J., v. 10, 1972, pp. 11931199. 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. 791802. 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. 430436. 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. 704722. 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. 277324. MR 1554147
 [25]
 W. L. MIRANKER, "Numerical methods of boundary layer type for stiff systems of differential equations," Computing, v. 11, 1973, pp. 221234. 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. 10171034. MR 0381329 (52:2226)
 [27]
 D. E. MULLER, "A method for solving algebraic equations using an automatic computer," MTAC, v. 10, 1956, pp. 208215. MR 18, 766. MR 0083822 (18:766e)
 [28]
 W. D. MURPHY, "Numerical analysis of boundarylayer problems in ordinary differential equations," Math. Comp., v. 21, 1967, pp. 583596. 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. 8998. 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. 449466. 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, SpringerVerlag, Berlin and New York, 1974, pp. 363389. 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. 263270. 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. 134154. MR 37 #3773. MR 0228189 (37:3773)
 [37]
 C. E. PEARSON, "On nonlinear ordinary differential equations of boundary layer type," J. Math. and Phys., v. 47, 1968, pp. 351358. MR 38 #5400. MR 0237107 (38:5400)
 [38]
 M. R. SCOTT & H. A. WATTS, SupportA Computer Code for TwoPoint BoundaryValue Problems via Orthonormalization, Sandia Laboratories Report SAND 750198, 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 nonlinear differential equations containing a small parameter," Dokl. Akad. Nauk SSSR, Tom 132, 1960, pp. 12421245 = Soviet Math. Dokl., v. 1, 1960, pp. 749752. 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. 173183. 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. 93113. MR 18, 39. MR 0079161 (18:39d)
 [43]
 W. R. WASOW, "Connection problems for asymptotic series," Bull. Amer. Math. Soc., v. 74, 1968, pp. 831853. 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. 661680. MR 0426448 (54:14391)
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DOI:
http://dx.doi.org/10.1090/S00255718197706573960
PII:
S 00255718(1977)06573960
Article copyright:
© Copyright 1977 American Mathematical Society
