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Stiff ODE solvers: A review of current and coming attractions. (English) Zbl 0614.65078
This article (62 pages) gives a review of methods and software solvers that are currently in use for stiff ordinary differential equations. Further, it presents a set of 10 stiff problems which show the challenges to the software and the capabilities of the solvers.
After an introduction into the notion of stiffness and into related problems (stability, method of lines, differential-algebraic systems), Section 2 presents a survey of methods for stiff problems including some comments on their relative merits. Section 3 is a description of available software. It begins with the history of ODE-solvers based on backward differentiation formulas and describes mainly the codes LSODE, EPISODE and related ones. Other codes (Runge-Kutta, blended and composite multistep, extrapolation, second-derivative) are mentioned briefly. An interesting and important part of this review is Section 4, where a set of 10 stiff problems is given. These examples are fairly representative with respect to many features and include problems arising in partial differential equations. It is shown how they can be solved by LSODE and its variants.
Reviewer: E.Hairer

MSC:
65L05 Numerical methods for initial value problems
65N40 Method of lines for boundary value problems involving PDEs
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
34A34 Nonlinear ordinary differential equations and systems, general theory
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References:
[1] Alexander, R., SIAM J. numer. anal., 14, 1006, (1977)
[2] Alfeld, P.; Lambert, J.D., Math. comput., 31, 922, (1977)
[3] Andria, G.D.; Byrne, G.D.; Hill, D.R., Bit, 13, 131, (1973)
[4] Axelrod, T.S.; Dubois, P.F.; Hickman, R.B.; Hindmarsh, A.C.; Painter, J.F., Lawrence livermore laboratory report no. UCID-30190, (January 1983), (unpublished)
[5] Bader, G.; Deuflhard, P., Numer. math., 41, 373, (1983)
[6] Benton, E.R.; Platzman, G.W., Quart. appl. math., 30, 195, (1972)
[7] Berzins, M.; Dew, P.M.; Furzeland, R.M., Software for time-dependent problems, (), 309 · Zbl 0679.65071
[8] Bickart, T.A.; Picel, Z., Bit, 13, 272, (1973)
[9] Bird, R.B.; Stewart, W.E.; Lightfoot, E.N., Transport phenomena, (1960), Wiley New York
[10] \scA. Björck, in “Some Methods for Separating Stiff Components in Initial Value Problems,” Proceedings of Dundee Numerical Analysis Conference 1983 (Springer-Verlag, Berlin, to appear).
[11] Brayton, R.K.; Gustavson, F.G.; Hachtel, H.D., (), 98
[12] Brown, P.N.; Hindmarsh, A.C., SIAM J. numer. anal., 23, 610, (1986)
[13] Burrage, K., Bit, 18, 22, (1978)
[14] Burrage, K.; Butcher, J.C.; Chipman, F.H., Bit, 20, 326, (1980)
[15] Butcher, J.C., Math. comput., 27, 793, (1973)
[16] Butcher, J.C., General linear methods: prospects as stiff solvers, () · Zbl 0983.65085
[17] Butcher, J.C., Appl. numer. math., 1, 273, (1985)
[18] Byrne, G.D.; Hindmarsh, A.C.; Jackson, K.R.; Brown, H.G., Argonne national laboratory report no. ANL 77-19, (March 1977), (unpublished)
[19] Byrne, G.D., The ODE solver EPISODE, its variants and their use, (), 8-17
[20] Byrne, G.D.; Hindmarsh, A.C., ACM trans. math. software, 1, 71, (1975)
[21] Byrne, G.D.; Hindmarsh, A.C., Lawrence livermore laboratory report no. UCID-30132, (April 30, 1976), (unpublished)
[22] Byrne, G.D.; Hindmarsh, A.C., Numerical solution of stiff ordinary differential equations, aiche today series, (1977), American Institute of Chemical Engineers New York, (unpublished)
[23] Byrne, G.D.; Hindmarsh, A.C., Appl. numer. math., 1, 29, (1985)
[24] Byrne, G.D.; Ho, W.S., Exxon research and engineering company report no. CTSD.2DI.83, (May 1983), (unpublished)
[25] Byrne, G.D.; Lambert, R.J., J. assoc. comput. Mach., 13, 114, (1966)
[26] Carver, M.B., Comput. phys. comm., 17, 239, (1979)
[27] Cash, J.R., Numer. math., 36, 253, (1981)
[28] Cash, J.R., Appl. numer. math., 1, 195, (1985)
[29] Cash, J.R.; Singhal, A., IMA J. numer. anal., 2, 211, (1982)
[30] Chan, T.F.; Jackson, K.R., SIAM J. sci. stat. comput., 7, 378, (1986)
[31] Chang, J.S.; Hindmarsh, A.C.; Madsen, N.K., Simulation of chemical kinetics transport in the stratosphere, (), 51
[32] Chin, R.C.Y.; Hestrom, G.W.; Karlsson, K.E., Math. comput., 33, 647, (1979)
[33] Curtis, A.R., Solution of large, stiff initial value problems-the state of the art, (), 257
[34] Curtiss, C.F.; Hirschfelder, J.O., (), 235
[35] Stetter, H., Global error estimation in ODE-solvers, (), 179 · Zbl 0404.65041
[36] Dahlquist, G., On the control of global error in stiff initial value problems, (), 38
[37] Dahlquist, G., On the local and global errors of one-leg methods, (), (unpublished) · Zbl 0475.65044
[38] Dahlquist, G., SIAM J. numer. anal., 20, 1130, (1983)
[39] Dfuflhard, P.; Bader, G.; Nowak, U., LARKIN-A software package for the numerical simulation of large systems arising in chemical reaction kinetics, (), 38
[40] Deuflhard, P., SIAM rev., 27, 505, (1985)
[41] Deuflhard, P.; Hairer, E.; Zugck, J., Institüt für angewandte Mathematik, universität Heidelberg preprint no. 318, (June 1985), (unpublished)
[42] Dew, P.M.; West, M.R., Bit, 19, 135, (1979)
[43] Dickinson, R.P.; Gelinas, R.J., J. comput. phys., 21, 123, (1976)
[44] Douglas, J., The numerical solution of a composition model in petroleum reservoir engineering, (), 54
[45] Eisfnstat, S.C.; Gursky, M.C.; Schultz, M.H.; Sherman, A.H., Yale university computer sciences dept. report no. 114, (1977), (unpublished)
[46] Eisenstat, S.C.; Gursky, M.C.; Schultz, M.H.; Sherman, A.H., Int. J. numer. methods eng., 18, 1145, (1982)
[47] Eitelberg, E., Modular simulation of large stiff systems, (), 281
[48] Enright, W.H., SIAM J. numer. anal., 11, 321, (1974)
[49] Enright, W.H.; Hull, T.E.; Lindberg, B., Bit, 15, 10, (1975)
[50] Enright, W.H.; Hull, T.E., Comparing numerical methods for the solution of stiff systems of ODEs arising in chemistry, (), 45
[51] Enright, W.H.; Kamel, M.S., ACM trans. math. software, 5, 374, (1979)
[52] Field, R.J.; Noyes, R.M., J. chem. phys., 60, 1877, (1974)
[53] Gaffney, P.W., ACM trans. math. software, 10, 58, (1984)
[54] Gear, C.W., The automatic integration of stiff ordinary differential equations, (), 187 · Zbl 0195.45702
[55] Gear, C.W., Numerical initial value problems in ordinary differential equations, (1971), Prentice-Hall Englewood Cliffs, N.J · Zbl 0217.21701
[56] Gear, C.W., IEEE trans. circuit theory, 18, 89, (1971)
[57] Gear, C.W., ACM trans. math. software, 6, 263, (1980)
[58] Gear, C.W.; Petzold, L.R., SIAM J. numer. anal., 21, 716, (1984)
[59] Gelinas, R.J., Lawrence livermore laboratory report no. UCRL-75373, (January 1974), (unpublished)
[60] Gladwell, I., ACM trans. math. software, 5, 386, (1979)
[61] Gottfried, B.S., Soc. pet. eng. J., 5, 196, (1965)
[62] Gottwald, B.A.; Wanner, G., Computing, 26, 355, (1981)
[63] Gupta, G.K.; Sacks-Davis, R.; Tischer, P.E., ACM computing surveys, 17, 5, (1985)
[64] Hachtel, G.D.; Brayton, R.K.; Gustavson, F.G., IEEE trans. circuit theory, 18, 101, (1971)
[65] Hairier, E.; Wanner, G., Computing, 11, 287, (1973)
[66] Henrici, P., Discrete variable methods in ordinary differential equations, (1962), Wiley New York · Zbl 0112.34901
[67] Hindmarsh, A.C., Lawrence livermore laboratory report no. UCID-30001, rev. 3, (December 1974), (unpublished)
[68] Hindmarsh, A.C., Lawrence livermore laboratory report no. UCID-30130, (February 1976), (unpublished)
[69] Hindmarsh, A.C., Lawrence livermore laboratory report no. UCRL-78129, (April 1976), (unpublished)
[70] Hindmarsh, A.C., Lawrence livermore laboratory report no. UCID-30149, (December 1976), (unpublished)
[71] Hindmarsh, A.C., Lawrence livermore laboratory report no. UCID-30059, rev. 2, (June 1977), (unpublished)
[72] Hindmarsh, A.C., Lawrence livermore laboratory report no. UCID-17954, (October 1978), (unpublished)
[73] Hindmarsh, A.C., A collection of software for ordinary differential equations, (), 1-8
[74] Hindmarsh, A.C., ACM SIGNUM newsletter, 15, No. 4, 10, (1980)
[75] Hindmarsh, A.C., ODE solvers for use with the method of lines, (), 312
[76] Hindmarsh, A.C., ODEPACK, a systematized collection of ODE solvers, (), 55
[77] Hindmarsh, A.C.; Byrne, G.D., Applications of EPISODE: an experimental package for the integration of systems of ordinary differential equations, (), 147
[78] Hindmarsh, A.C.; Byrne, G.D., Lawrence livermore laboratory report no. UCID-30134, (May 1976), (unpublished)
[79] Hindmarsh, A.C.; Byrne, G.D., Lawrence livermore laboratory report no. UCID-30112, rev. 1, (April 1977), (unpublished)
[80] Hofer, E., SIAM J. numer. anal, 13, 645, (1976)
[81] Hull, T.E.; Enright, W.H.; Jackson, K.R., Department of computer science, univ. of Toronto, report no. TR no. 100, (October 1976), (unpublished)
[82] Hulme, B.L., Math. comput., 26, 881, (1972)
[83] Hulme, B.L.; Daniel, S.L., Sandia laboratories report no. SAND74-0380, (December 1974), (unpublished)
[84] Hulme, B.L.; Daniel, S.L., Sandia laboratories report no. SAND74-0381, (December 1974), (unpublished)
[85] ()
[86] Jackson, K.R.; Sacks-Davis, R., ACM trans. math. software, 6, 295, (1980)
[87] Kaps, P.; Rentrop, P., Numer. math., 33, 55, (1979)
[88] Kaps, P.; Wanner, G., Numer. math., 38, 279, (1981)
[89] Kaps, P.; Poon, S.W.H.; But, T.D., Computing, 34, 17, (1985)
[90] Klopfenstein, R.W., RCA rev., 32, 447, (1971)
[91] \scR. W. Klopfenstein, RCA David Sarnoff Laboratory, private communication on IMPEQ (1974).
[92] Klopfenstein, R.W.; Dams, C.B., PECE algorithms for the solution of stiff systems of ordinary differential equations, Math. comput., 25, 457, (1971) · Zbl 0228.65059
[93] Krogh, F.T., TRW report no. 99900-6573-R000, (1968), (unpublished)
[94] Krogh, F.T.; Stewart, K., ACM trans. math. software, 10, 45, (1984)
[95] Lambert, J.D.; Sigurdsson, S.T., SIAM J numer. anal., 9, 715, (1972)
[96] Leaf, G.K.; Minkoff, M., DISPL2: A software package for solving one and T’wo spatially dimonsioned convection-diffusion kinetics nonlinear pdes, (), 429
[97] Leaf, G.K.; Minkoff, M., Argonne national laboratory report no. ANL-84-56, (September 1984), (unpublished)
[98] Lindberg, B., A stiff system package based on the implicit midpoint method, (), 201
[99] Liniger, W., High-order A-stable averaging algorithms for stiff differential systems, (), 1
[100] Liniger, W.; Willoughby, R.A., SIAM J. numer. anal., 7, 47, (1970)
[101] Madsen, N.K.; Sincovec, R.F., The numerical method of lines for the solution of nonlinear partial differential equations, (), 371 · Zbl 0314.65048
[102] Madsen, N.K.; Sincovec, R.F., ACM trans. math. software, 5, 326, (1979) · Zbl 0426.35005
[103] Morris, D.B.; Hindmarsh, A.C.; Dubois, P.F., Lawrence livermore laboratory report no. UCID-30119, rev. 1, (December 1977), (unpublished)
[104] NAG Library, Mark 9, (1982), Oxford
[105] Norsett, S.P.; Wolfbrandt, A., Numer. math., 32, 1, (1979)
[106] Peaceman, D.W., Fundamentals of reservoir simulation, (1977), Elsevier New York · Zbl 0204.28001
[107] Pelios, A.; Klopfenstein, R.W., Math. comput., 26, 467, (1972)
[108] Petzold, L.R., SIAM J. sci. stat. comput., 3, 367, (1982)
[109] Petzold, L.R., A description of DASSL: A differential-algebraic system solver, (), 65
[110] Petzold, L.R., SIAM J. sci. stat. comput., 4, 136, (1983)
[111] Price, H.S.; Varga, R.S.; Warren, J.E., J. math. phys., 45, 301, (1966)
[112] Prothero, A., Estimating the accuracy of numerical solutions to ordinary differential equations, (), 103 · Zbl 0488.65025
[113] Robertson, H.H., The solution of a set of reaction rate equations, (), 178
[114] Robinson, A.; Prothero, A., Global error estimates for solutions to stiff systems of ordinary differential equations, () · Zbl 0309.65034
[115] Rosser, J.B., SIAM rev., 9, 417, (1967)
[116] Rübner-Peterson, T., An efficient algorithm using backward time-scaled differences for solving stiff differential-algebraic systems, (), (unpublished)
[117] Sacks-Davis, R., ACM trans. math. software, 6, 540, (1980)
[118] Schryer, N.L., Bell laboratories computer science technical report no. 53, (1977), (unpublished)
[119] Scott, M.R.; Watts, H.A., A systematized collection of codes for solving two-point boundary-value problems, (), 197
[120] Seward, W.L.; Fairweather, G.; Johnston, R.L., IMA J. numer. anal., 4, 375, (1984)
[121] Shampine, L.F., Math. comput., 36, 499, (1981)
[122] Shampine, L.F., Sandia national laboratory report no. SAND 82-2517, (December 1982), (unpublished)
[123] Shampine, L.F., ACM trans. math. software, 8, 93, (1982)
[124] Shampine, L.F., Math. comput., 39, 109, (1982)
[125] Shampine, L.F., Appl. numer. math., 1, 107, (1985)
[126] Shampine, L.F.; Gordon, M.K., SIAM rev., 21, 1, (1979)
[127] Shampine, L.F.; Gordon, M.K., Computer solution of ordinary differential equations. the initial value problem, (1975), Freeman San Francisco · Zbl 0347.65001
[128] Shampine, L.F.; Watts, H.A., Sandia national laboratories report no. SAND79-2374, (September 1980), (unpublished)
[129] Sherman, A.H.; Hindmarsh, A.C., GEARS: A package for the solution of sparse, stiff ordinary differential equations, (), 190
[130] Sincovec, R.F.; Madsen, N.K., ACM trans. math. software, 1, 232, (1975)
[131] Sincovec, R.F.; Dembart, B.; Epton, M.A.; Erisman, A.M.; Manke, S.W.; Yip, E.L., IEEE trans. on auto control, 26, 139, (1981)
[132] Skeel, R.D.; Kong, A.K., ACM trans. math. software, 3, 326, (1977)
[133] Stetter, H., Global error estimation in ODE-solvers, (), 179 · Zbl 0404.65041
[134] Strang, G.; Fix, G.J., An analysis of the finite element method, (1973), Prentice-Hall Englewood Cliffs, N.J · Zbl 0278.65116
[135] Swartz, B.; Wendroff, B., Math. comput., 23, 37, (1969)
[136] Tendler, J.M.; Bickart, T.A.; Picel, Z., ACM trans. math. soft., 4, 339, (1978)
[137] Tendler, J.M.; Bickart, T.A.; Picel, Z., ACM trans. math. software, 4, 399, (1978)
[138] Thompson, S.; Tuttle, P.G., Automatic ODE software for use in an industrial environment, (), 440
[139] Tischer, P.E.; Gupta, G.K., Cosiderations in designing a cyclic method stiff ODE solver, Monash univ. computer science dept. report, clayton, Victoria, Australia, (1984), (unpublished)
[140] Tischer, R.; Sacks-Davis, R., SIAM J. sci. stat. comput., 4, 733, (1983)
[141] Varga, R.S., Matrix iterative analysis, (1962), Prentice-Hall Englewood Cliffs, N.J, Chap. 6 · Zbl 0133.08602
[142] Watanabe, D.S.; Sheikh, Q.M., SIAM J. sci. stat. comput., 5, 489, (1984)
[143] Watkins, D.S.; Hansonsmith, R.W., ACM trans. math. software, 9, 293, (1983)
[144] Watts, H.A.; Shampine, L.F., Bit, 12, 252, (1972)
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