Backward differentiation formula

The backward differentiation formula (BDF) is a family of implicit methods for the numerical integration of ordinary differential equations. They are linear multistep methods that, for a given function and time, approximate the derivative of that function using information from already computed times, thereby increasing the accuracy of the approximation. These methods are especially used for the solution of stiff differential equations.

General formula

A BDF is used to solve the initial value problem

$y' = f(t,y), \quad y(t_0) = y_0.$

The general formula for a BDF can be written as ^[1]

$\sum_{k=0}^s a_k y_{n+k} = h \beta f(t_{n+s}, y_{n+s}),$

where $h$ denotes the step size and $t_n = t_0 + nh$. The coefficients $a_{k}$ and $\beta$ are chosen so that the method achieves order $s$, which is the maximum possible.

BDF methods are implicit and, as such, require the solution of nonlinear equations at each step. Typically, a modified Newton's method is used to solve these nonlinear equations.^[1]

Specific formulas

The s-step BDFs with s < 7 are:^[2]

• BDF1: $y_{n+1} - y_n = h f(t_{n+1}, y_{n+1})$; (this is the backward Euler method)
• BDF2: $y_{n+2} - \tfrac43 y_{n+1} + \tfrac13 y_n = \tfrac23 h f(t_{n+2}, y_{n+2});$
• BDF3: $y_{n+3} - \tfrac{18}{11} y_{n+2} + \tfrac9{11} y_{n+1} - \tfrac2{11} y_n = \tfrac6{11} h f(t_{n+3}, y_{n+3})$
• BDF4: $y_{n+4} - \tfrac{48}{25} y_{n+3} + \tfrac{36}{25} y_{n+2} - \tfrac{16}{25} y_{n+1} + \tfrac{3}{25} y_n = \tfrac{12}{25} h f(t_{n+4}, y_{n+4})$
• BDF5: $y_{n+5} - \tfrac{300}{137} y_{n+4} + \tfrac{300}{137} y_{n+3} - \tfrac{200}{137} y_{n+2} + \tfrac{75}{137} y_{n+1} - \tfrac{12}{137} y_n = \tfrac{60}{137} h f(t_{n+5}, y_{n+5})$
• BDF6: $y_{n+6} - \tfrac{360}{147} y_{n+5} + \tfrac{450}{147} y_{n+4} - \tfrac{400}{147} y_{n+3} + \tfrac{225}{147} y_{n+2} - \tfrac{72}{147} y_{n+1} + \tfrac{10}{147} y_n = \tfrac{60}{147} h f(t_{n+6}, y_{n+6}).$

Methods with s > 6 are not zero-stable so they cannot be used.^[3]

Stability

The stability of numerical methods for solving stiff equations is indicated by their region of absolute stability. For the BDF methods, these regions are shown in the plots below.

Ideally, the region contains the left half of the complex plane, in which case the method is said to be A-stable. However, linear multistep methods with an order greater than 2 cannot be A-stable. The stability region of the higher-order BDF methods contain a large part of the left half-plane and in particular the whole of the negative real axis. The BDF methods are the most efficient linear multistep methods of this kind.^[3]

The pink region shows the stability region of the BDF methods
BDF1  BDF2  BDF3  BDF4  BDF5  BDF6

References

Citations

1. Ascher 1998, §5.1.2, p. 129
2. Iserles 1996, p. 27 (for s = 1, 2, 3); Süli & Mayers 2003, p. 349 (for all s)
3. Süli & Mayers 2003, p. 349

Referred works

• Ascher, U. M.; Petzold, L. R. (1998), Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, SIAM, Philadelphia, ISBN 0-89871-412-5.
• Iserles, Arieh (1996), A First Course in the Numerical Analysis of Differential Equations, Cambridge University Press, ISBN 978-0-521-55655-2.
• Süli, Endre; Mayers, David (2003), An Introduction to Numerical Analysis, Cambridge University Press, ISBN 0-521-00794-1.

Further reading

• BDF Methods at the SUNDIALS wiki (SUNDIALS is a library implementing BDF methods and similar algorithms).