Backward Differentiation Formulae

Synonyms

Extended backward differentiation formulae; Linear multistep methods

Definition

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

$$\displaystyle{ \frac{dy} {dx} = f(x,y),\quad y(a) = y_{a}. }$$

(1)

As we will see, the methods we will consider are also very efficient for the numerical solution of differential algebraic equations of the form

$$\displaystyle{ F(x,y,y\prime) = 0 }$$

(2)

for the important case where (2) has index 1. It is often the case that systems of the...