Backward differentiation formula

The BDF methods (English Backward Differentiation Formulas ) are multi-step methods for the numerical solution of initial value problems:

It is calculated for an approximate solution for the intermediate locations:

The procedures were introduced in 1952 by Charles Francis Curtiss and Joseph Oakland Hirschfelder and since the publication of the works of C. William Gear in 1971 as a solver for stiff ordinary differential equations widespread.

Description

In contrast to the Adams - Moulton methods is not the right hand side is approximated by an interpolation method for BDF, but is defined by the last K approximations to the solution, and the unknown value, an interpolation polynomial. The unknown value is then obtained by requiring that the derivative of the polynomial satisfies the differential equation in point:

Here, the step size. After suitable initial values ​​are, for example, that generated by one-step method, we obtain the remaining approximations via the formula:

The coefficients resulting from the derivation of the interpolation. Here the coefficients of the first three methods for constant step size h are

Properties

The BDF methods are all implicit as the unknown value enters into the equation. BDF (k ) has exactly the consistency order k The method BDF (1 ) is the implicit Euler method. This and BDF ( 2) are A- stable, higher order methods A () - stable, with the opening angle decreases with higher order. For k > 6, the method is unstable. In particular BDF ( 2) is very popular due to its optimal properties with respect to the second Dahlquist barrier in the calculation stiff differential equations.

• Regions of stability of the BDF method

BDF2

BDF3

BDF4

BDF5

BDF6