Stability with large step sizes for multistep discretizations of stiff ordinary differential equations

Author:
George Majda

Journal:
Math. Comp. **47** (1986), 473-502, S41

MSC:
Primary 65L20

MathSciNet review:
856698

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we consider a large set of variable coefficient linear systems of ordinary differential equations which possess two different time scales, a slow one and a fast one. A small parameter characterizes the stiffness of these systems. We approximate a system of ODE's in this set by a general class of multistep discretizations which includes both one-leg and linear multistep methods. We determine sufficient conditions under which each solution of a multistep method is uniformly bounded, with a bound which is independent of the stiffness of the system of ODE's, when the step size resolves the slow time scale but not the fast one. We call this property stability with large step sizes.

The theory presented in this paper lets us compare properties of one-leg methods and linear multistep methods when they approximate variable coefficient systems of stiff ODE's. In particular, we show that one-leg methods have better stability properties with large step sizes than their linear multistep counterparts. This observation is consistent with results obtained by Dahlquist and Lindberg [11], Nevanlinna and Liniger [33] and van Veldhuizen [42]. Our theory also allows us to relate the concept of *D*-stability (van Veldhuizen [42]) to the usual notions of stability and stability domains and to the propagation of errors for multistep methods which use large step sizes.

**[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****[2]**G. Bjurel, G. Dahlquist, B. Lindberg, S. Linde & L. Oden,*Survey of Stiff Ordinary Differential Equations*, Report NA 70.11, Dept. of Information Processing, Royal Institute of Technology, S-100 44 Stockholm 70, Sweden, 1970.**[3]**Kevin Burrage,*High order algebraically stable Runge-Kutta methods*, BIT**18**(1978), no. 4, 373–383. MR**520749**, 10.1007/BF01932017**[4]**Kevin Burrage and J. C. Butcher,*Stability criteria for implicit Runge-Kutta methods*, SIAM J. Numer. Anal.**16**(1979), no. 1, 46–57. MR**518683**, 10.1137/0716004**[5]**J. C. Butcher, "A stability property of implicit Runge-Kutta methods,"*BIT*, v. 15, 1975, pp. 358-361.**[6]**Michel Crouzeix,*Sur la 𝐵-stabilité des méthodes de Runge-Kutta*, Numer. Math.**32**(1979), no. 1, 75–82 (French, with English summary). MR**525638**, 10.1007/BF01397651**[7]**Germund G. Dahlquist,*A special stability problem for linear multistep methods*, Nordisk Tidskr. Informations-Behandling**3**(1963), 27–43. MR**0170477****[8]**G. G. Dahlquist,*On Stability and Error Analysis for Stiff Non-linear Problems*, Part 1, Report TRITA-NA-7508, Dept. of Information Processing, Royal Institute of Technology, Stockholm, Sweden, 1975.**[9]**Germund Dahlquist,*The sets of smooth solutions of differential and difference equations*, Stiff differential systems (Proc. Internat. Sympos., Wildbad, 1973), Plenum, New York, 1974, pp. 67–80. IBM Res. Sympos. Ser. MR**0405860****[10]**G. G. Dahlquist,*Some Properties of Linear Multistep and One-leg Methods for Ordinary Differential Equations*, Report TRITA-NA-7904, Dept. of Numerical Analysis and Computing Science, Royal Institute of Technology, Stockholm, Sweden, 1979.**[11]**G. G. Dahlquist & B. Lindberg,*On Some Implicit One-step Methods for Stiff Differential Equations*, Report TRITA-NA-7302, Dept. of Information Processing, Royal Institute of Technology, Stockholm, Sweden, 1973.**[12]**G. G. Dahlquist & G. Söderland,*Error Propagation in Stiff Differential Systems of Singular Perturbation Type*, Report TRITA-NA-8108, Dept. of Information Processing, Royal Institute of Technology, Stockholm, Sweden, 1981.**[13]**B. L. Ehle,*On Padé Approximations to the Exponential Function and A-stable Methods for the Numerical Solution of Initial Value Problems*, Dept. of Applied Analysis and Computer Science, University of Waterloo, Research Report No. CSRR 2010, 1969.**[14]**W. H. Enright & T. E. Hull, "Comparing numerical methods for the solution of stiff systems of ODE's arising in chemistry," in*Numerical Methods for Differential Systems*(L. Lapidus and W. E. Schiesser, eds.), Academic Press, New York, 1976, pp. 45-66.**[15]**W. H. Enright, T. E. Hull & B. Lindberg, "Comparing numerical methods for stiff systems of ODE's,"*BIT*, v. 15, 1975, pp. 10-48.**[16]**S. F. Feshchenko, N. I. Shkil′, and L. D. Nīkolenko,*Asymptotic methods in the theory of linear differential equations*, Translated from the Russian by Scripta Technica, Inc. Translation editor, Herbert Eagle. Modern Analytic and Computational Methods in Science and Mathematics, No. 10, American Elsevier Publishing Co., Inc., New York, 1967. MR**0221029****[17]**Reinhard Frank, Josef Schneid, and Christoph W. Ueberhuber,*The concept of 𝐵-convergence*, SIAM J. Numer. Anal.**18**(1981), no. 5, 753–780. MR**629662**, 10.1137/0718051**[18]**C. William Gear,*Numerical initial value problems in ordinary differential equations*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. MR**0315898****[19]**A. R. Gourlay,*A note on trapezoidal methods for the solution of initial value problems*, Math. Comp.**24**(1970), 629–633. MR**0275680**, 10.1090/S0025-5718-1970-0275680-3**[20]**Frank Hoppensteadt,*Properties of solutions of ordinary differential equations with small parameters*, Comm. Pure Appl. Math.**24**(1971), 807–840. MR**0288378****[21]**F. Hoppensteadt & W. L. Miranker,*Numerical Solution of Differential Equations with Rapidly Changing Solutions*, IBM Report RC4792, IBM Research Center, Yorktown Heights, New York, 1974.**[22]**Heinz-Otto Kreiss,*Difference methods for stiff ordinary differential equations*, SIAM J. Numer. Anal.**15**(1978), no. 1, 21–58. MR**486570**, 10.1137/0715003**[23]**Heinz-Otto Kreiss,*Problems with different time scales for ordinary differential equations*, SIAM J. Numer. Anal.**16**(1979), no. 6, 980–998. MR**551320**, 10.1137/0716072**[24]**J. D. Lambert,*Computational methods in ordinary differential equations*, John Wiley & Sons, London-New York-Sydney, 1973. Introductory Mathematics for Scientists and Engineers. MR**0423815****[25]**L. Lapidus and W. E. Schiesser (eds.),*Numerical methods for differential systems*, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. Recent developments in algorithms, software, and applications; Papers presented at the 80th National Meeting of the American Institute of Chemical Engineers in Boston, Mass., September 7–10, 1975. MR**0438721****[26]**B. Lindberg,*IMPEX--A Program Package for the Solution of Systems of Stiff Differential Equations*, Report TRITA-NA-72.50, Royal Institute of Technology, Stockholm, Sweden, 1972.**[27]**B. Lindberg,*IMPEX 2--A Procedure for Solution of Systems of Stiff Differential Equations*, Report TRITA-NA-7303, Royal Institute of Technology, Stockholm, Sweden, 1973.**[28]**George Majda,*Filtering techniques for systems of stiff ordinary differential equations. I*, SIAM J. Numer. Anal.**21**(1984), no. 3, 535–566. MR**744172**, 10.1137/0721038**[29]**George Majda,*Filtering techniques for systems of stiff ordinary differential equations. II. Error estimates*, SIAM J. Numer. Anal.**22**(1985), no. 6, 1116–1134. MR**811187**, 10.1137/0722067**[30]**G. Majda,*A New Theory for Multistep Discretizations of Stiff Ordinary Differential Equations*I:*Stability with Large Step Sizes*, Report NA-3, Division of Applied Mathematics, Brown University, Providence, R.I., 1983.**[31]**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****[32]**Willard L. Miranker,*Numerical methods for stiff equations and singular perturbation problems*, Mathematics and its Applications, vol. 5, D. Reidel Publishing Co., Dordrecht-Boston, Mass., 1981. MR**603627****[33]**Olavi Nevanlinna and Werner Liniger,*Contractive methods for stiff differential equations. I*, BIT**18**(1978), no. 4, 457–474. MR**520755**, 10.1007/BF01932025**[34]**F. Odeh and W. Liniger,*A note on unconditional fixed-ℎ stability of linear multistep formulae*, Computing (Arch. Elektron. Rechnen)**7**(1971), 240–253 (English, with German summary). MR**0298957****[35]**F. Odeh and W. Liniger,*Nonlinear fixed-ℎ stability of linear multistep formulas*, J. Math. Anal. Appl.**61**(1977), no. 3, 691–712. MR**0468197****[36]**A. Prothero and A. Robinson,*On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations*, Math. Comp.**28**(1974), 145–162. MR**0331793**, 10.1090/S0025-5718-1974-0331793-2**[37]**Hans J. Stetter,*Towards a theory for discretizations of stiff differential systems*, Numerical analysis (Proc. 6th Biennial Dundee Conf., Univ. Dundee, Dundee, 1975) Springer, Berlin, 1976, pp. 190–201. Lecture Notes in Math., Vol. 506. MR**0455414****[38]**Hans J. Stetter,*Analysis of discretization methods for ordinary differential equations*, Springer-Verlag, New York-Heidelberg, 1973. Springer Tracts in Natural Philosophy, Vol. 23. MR**0426438****[39]**Gilbert Strang,*Accurate partial difference methods. II. Non-linear problems*, Numer. Math.**6**(1964), 37–46. MR**0166942****[40]**M. van Veldhuizen,*Convergence of One-step Discretizations for Stiff Differential Equations*, Ph.D. Thesis, Mathematical Institute, University of Utrecht, Netherlands, 1973.**[41]**M. van Veldhuizen,*Consistency and stability for one-step discretizations of stiff differential equations*, Stiff differential systems (Proc. Internat. Sympos., Wildbad, Germany, 1973), IBM Res. Sympos. Ser., Plenum, New York, 1974, pp. 259–270. MR**0431694****[42]**M. van Veldhuizen,*𝐷-stability*, SIAM J. Numer. Anal.**18**(1981), no. 1, 45–64. MR**603430**, 10.1137/0718005**[43]**Wolfgang Wasow,*Asymptotic expansions for ordinary differential equations*, Pure and Applied Mathematics, Vol. XIV, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1965. MR**0203188****[44]**Olof B. Widlund,*A note on unconditionally stable linear multistep methods*, Nordisk Tidskr. Informations-Behandling**7**(1967), 65–70. MR**0215533****[45]**Ralph A. Willoughby (ed.),*Stiff differential systems*, Plenum Press, New York-London, 1974. The IBM Research Symposia Series. MR**0343619**

Retrieve articles in *Mathematics of Computation*
with MSC:
65L20

Retrieve articles in all journals with MSC: 65L20

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1986-0856698-8

Article copyright:
© Copyright 1986
American Mathematical Society