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An MEBDF code for stiff initial value problems

Editor: John R. Rice Authors:

J. R. Cash,

S. ConsidineAuthors Info & Claims

ACM Transactions on Mathematical Software (TOMS), Volume 18, Issue 2

Pages 142 - 155

https://doi.org/10.1145/146847.146922

Published: 01 June 1992 Publication History

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Abstract

In two recent papers one of the present authors has proposed a class of modified extended backward differentiation formulae for the numerical integration of stiff initial value problems. In this paper we describe a code based on this class of formulae and discuss its performance on a large set of stiff test problems.

References

[1]

BuI, T. D. Some A~stable and L-stable methods for the numerical integration of stiff ordinary differential equations. JACM 26, 3 (July 1979), 483 493.]]

Crossref

Google Scholar

[2]

CASH, J. R On the integration of stiff systems of ODEs using extended backward differentiation formulae. Namer. Math. 34, 2 (1980), 235-246.]]

Google Scholar

[3]

CASH, J R. The integration of stiff initial value problems in ODEs using modified extended backward differentiation formulae. Compat. Math. Appl. 9, 5 (1983), 645-657.]]

Google Scholar

[4]

CHAMBERS, T. The use of numerical software in the digital simulation language PMSP. In Numerical Software--Needs and Availability, D. Jacobs, Ed., Academic Press, New York, 1978, 237-253.]]

Google Scholar

[5]

CONSImNE, S. Modified linear multistep methods for the numerical integration of stiff initial value problems. PhD Thesis, Univ. of London, 1988.]]

Google Scholar

[6]

CURTIS, A. R. Solution of large, stiff initial value problems--the state of the art. In Numerical Software--Needs and Availability, D. Jacobs, Ed., Academic Press, New York, 1978, 253-267.]]

Google Scholar

[7]

DAHLQUIST, G, A special stability problem for linear multistep methods. BIT 3, 1 (1963), 27 -43.]]

Google Scholar

[8]

ENmGHT, W.H. Second derivative methods for stiff ordinary differential equations. SIAM J. Numer. Anal. 11, 2 (Apr. 1974), 321-331.]]

Google Scholar

[9]

ENRIGET, W. H. Using a testing package for the automatic assessment of numermal methods for ODEs. In Performance Evaluation of Numerical Software--Proceedings IFIP TC2.5 Working Conference (Baden, Austria 1978), L, Fosdick, Ed., North Holland, Amsterdam, 1979, 199-213.]]

Google Scholar

[10]

ENRIGHT, W. H., AND HULL, T. E. Test results on initial value methods for nomstiff ordinary differential equations. SIAM J. Numer. Anal. 13, 6 (Dec. 1976), 944-961.]]

Google Scholar

[11]

ENRIGHT, W. H., AND HULL, T.E. Comparing numerical methods for the solution of stiff systems of ODEs arising in chemistry. In Numerical Methods for Differential Systems, L. Lapidus and W. E. Schiesser, Eds., Academic Press, New York, 1976, 45-66.]]

Google Scholar

[12]

ENRIGHT, W. H., HULL, T. E., AND LINDBE~G, B. Comparing numerical methods for stiff systems of ODEs. BIT 15, i (1975), 10-48.]]

Google Scholar

[13]

ENRI~HT, W. H., AND PRYCE, J. D. Two Fortran packages for assessing initial value methods. ACM Trans. Math. Softw. 13, I (Mar 1987), 1-27.]]

Crossref

Google Scholar

[14]

GE^~, C.W. Algorithm 407--DIFSUB for the solution of ordinary differential equations, Commun. ACM 14, 3 (Mar. 1971), 185-190.]]

Crossref

Google Scholar

[15]

GOTTWALD, B. A., AND WANNE~, G. A reliable Rosenbrock integrator for stiff differential equations. Computing 26, 2 (1981), 355-360.]]

Google Scholar

[16]

H~NDMARSH, A. C. LSODE and LSODI: Two new initial value ordinary differential equation solvers. SIGNUM News 15, 10 (Dec. 1980).]]

Crossref

Google Scholar

[17]

HULL, T. E., ENR~CRT, W. H., FELLEN, B. M., AND SEDGWICK, A.E. Comparing numerical methods for ordinary differential equations. SIAM J. Numer. Anal. 9, 4 (Dec. 1972), 603-637.]]

Google Scholar

[18]

KAPS, P., POON, S. W. H., AND BuI, T.D. Rosenbrock methods for stiff ODEs: A comparison of Richardson extrapolation and embedding techniques. Computing 34, i (1985), 17-40.]]

Google Scholar

[19]

KROGH, F.T. On testing a subroutine for the numerical integration of ordinary differential equations. JACM 20, 4 (Oct. 1973), 545-562.]]

Crossref

Google Scholar

[20]

KROGH, F. T. What I would put into an ODE solver. In Proceedings ODE Conference (Albuquerque, New Mex., July, 1986).]]

Google Scholar

[21]

ROBERTSON, H.H. The solution of a set of reaction rate equations. In Numerical Analysts, J. Walsh, Ed., Academic Press, London, 1966, 178-182.]]

Google Scholar

[22]

SCOTT, M. R., AND WATTS, H.A. In Numerical Methods for Differential Systems, L. Lapidus and W. E. Schiesser, Eds., Academic Press, New York, 1976, 197-227.]]

Google Scholar

[23]

SHAMPIXE, L. F. Local error control in codes for ordinary differential equations. Appl. Math. Camput. 3, 2 (1977), 189-210.]]

Google Scholar

[24]

SHAMP~NE, L.F. Evaluation of implicit formulas for the solution of ODEs. BIT 19, 4 (1979), 495-502.]]

Google Scholar

[25]

SHAMPINE, L.F. ImplemenLa~lon of implicit formulas for the solution of ODEs. SIAM J. Sci. Stat. Comput. 1, i (Mar. 1980), 103-118.]]

Google Scholar

[26]

SHAMPINE, L.F. Evaluation of a test set for stiff ODE solvers. ACM Trans. Math. Saftw. 7, 4 (Dec. 1981), 409-420.]]

Crossref

Google Scholar

[27]

SHAMPINE, L. F. ^ND GORDON, M.K. Computer Solution of Ordinary Differential Equations: The Initial Vcdue Problem. Freeman, San Francisco, 1975.]]

Google Scholar

[28]

WmLOUG~Y, R.A. Stiff Differential Systems. Plenum Press, New York, 1974.]]

Google Scholar

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Index Terms

An MEBDF code for stiff initial value problems
1. Mathematics of computing
  1. Mathematical analysis
    1. Differential equations
      1. Ordinary differential equations
    2. Numerical analysis
      1. Interpolation
  2. Mathematical software

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Reviews

Reviewer: Lawrence Shampine

The most popular methods for the solution of stiff initial value problems for ordinary differential equations are the backward differentiation formulas (BDFs). Because the stability of these formulas deteriorates rapidly as the order increases within the family, a great deal of effort has been devoted to finding formulas of moderate to high order with better stability. Stability is not the only issue, however: the formulas also have to be computationally efficient. In two earlier papers, Cash proposed and developed some formulas known as modified extended backward differentiation formulas (MEBDFs). They have good properties, but developing a production-grade code based on the formulas that could compete fairly with the highly polished BDF codes in use is a task of considerable magnitude. This paper presents such a code. The authors describe some of the algorithmic developments that are so important to quality software. They present substantial experiments comparing the code to the popular code LSODE, based on the BDFs, and to SECDER, based on second derivative methods. No way of solving stiff initial value problems is best in general. The evidence presented in this paper makes it clear that the MEBDFs as implemented in this code compete well with the BDFs and are superior for certain kinds of problems.

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Published In

cover image ACM Transactions on Mathematical Software

ACM Transactions on Mathematical Software Volume 18, Issue 2

June 1992

124 pages

ISSN:0098-3500

EISSN:1557-7295

DOI:10.1145/146847

Editor:
John R. Rice
Purdue Univ., West Lafayette, IN

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 June 1992

Published in TOMS Volume 18, Issue 2

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https://dl.acm.org/doi/10.1007/s11227-024-06137-2
Tang JLu J(2023)Modified Extended Lie-Group Method for Hessenberg Differential Algebraic Equations with Index-3Mathematics10.3390/math1110236011:10(2360)Online publication date: 18-May-2023
https://doi.org/10.3390/math11102360
Zhou YShi XGuo DDazer MBertsche BMei Z(2023)Nonlinear dynamic behaviour and severity of lightly loaded gear rattle under different vibro-impact models and internal excitationsNonlinear Dynamics10.1007/s11071-023-09113-2112:2(961-993)Online publication date: 13-Dec-2023
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Abstract

References

Cited By

Index Terms

Recommendations

Algorithm 703: MEBDF: a FORTRAN subroutine for solving first-order systems of stiff initial value problems for ordinary differential equations

Two classes of implicit-explicit multistep methods for nonlinear stiff initial-value problems

An automatic multistep method for solving stiff initial value problems

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