Encyclopedia of Applied and Computational Mathematics

2015 Edition
| Editors: Björn Engquist

Backward Differentiation Formulae

  • Jeff R Cash
Reference work entry
DOI: https://doi.org/10.1007/978-3-540-70529-1_94


Extended backward differentiation formulae; Linear multistep methods


Backward differentiation formulae (BDF) are linear multistep methods suitable for solving stiff initial value problems and differential algebraic equations. The extended formulae (MEBDF) have considerably better stability properties than BDF.

Review of Stiffness

We derive BDF and MEBDF suitable for solving stiff initial value problems and differential algebraic equations. In this section, we will be concerned with a special class of multistep methods for the approximate numerical integration of first-order systems of ordinary differential equations of the form
$$\displaystyle{ \frac{dy} {dx} = f(x,y),\quad y(a) = y_{a}. }$$
This is a preview of subscription content, log in to check access.


  1. 1.
    Ascher, U.M., Mattheij, R.M.M., Russell, R.D.: Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, Classics in Applied Mathematics, vol. 13. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1995)Google Scholar
  2. 2.
    Brenan, K.E., Campbell, S.L., Petzold, L.R.: Numerical Solution of Initial Value Problems in Differential-Algebraic Equations. North-Holland, New York (1989)zbMATHGoogle Scholar
  3. 3.
    Cash, J.R.: Second derivative extended backward differentiation formulas for the numerical integration of stiff systems. SIAM J. Numer. Anal. 18(1), 21–36 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Cash, J.R.: The integration of stiff initial value problems in ODEs using modified extended backward differentiation formulae. Comput. Math. Appl. 9(5), 645–657 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Dahlquist, G.: A special stability problem for linear multistep methods. BIT 3, 27–43 (1963)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Gear, C.W.: Numerical Initial Value Problems in Ordinary Differential Equations. Prentice-Hall, Englewood Cliffs (1971)zbMATHGoogle Scholar
  7. 7.
    Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, 2nd Rev. edn. Springer, Heidelberg (1996)Google Scholar
  8. 8.
    Mazzia, F., Magherini, C.: Test Set for Initial Value Problem Solvers, release 2.4. Department of Mathematics, University of Bari and INdAM, Research Unit of Bari. Available at http://www.dm.uniba.it/ testset (2008)
  9. 9.
    Widlund, O.B.: A note on unconditionally stable linear multistep methods. BIT 7, 65–70 (1967)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Jeff R Cash
    • 1
  1. 1.Department of MathematicsImperial CollegeLondonEngland