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

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

BDFs are a kind of linear multistep method that are particularly useful for stiff equations. The general formula for a linear mulitistep method can be written as

$\sum_{i=0}^k a_i y_{n-i} = h \sum_{i=0}^k b_i f_{n-i}$

where h denotes the step size, $f_l$ denotes $f(t_l, y_l)$, and the coefficients, $a_i$ and $b_i$, determine the particular linear multistep method. The family of BDFs consist of the methods arising from the case $b_i = 0$ for $i > 0$. The general formula for a BDF can be written as

$\sum_{i=0}^k a_i y_{n-i} = h b_0 f_n$

BDF methods are implicit and, as such, require the solution of non-linear equations at each step. Typically, a modified Newton's method is used to solve these non-linear equations.

References

Further Help: http://sundials.wikidot.com/bdf-method

Referred work

• Ascher, U. M.; Petzold, L. R. (1998), Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, SIAM, Philadelphia, ISBN 0-89871-412-5 .