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.

A BDF is used to solve the initial value problem

The general formula for a BDF can be written as

where ${\displaystyle h}$ denotes the step size and ${\displaystyle t_{n}=t_{0}+nh}$. Since ${\displaystyle f}$ is evaluated for the unknown ${\displaystyle y_{n+s}}$, BDF methods are implicit and possibly require the solution of nonlinear equations at each step. The coefficients ${\displaystyle a_{k}}$ and ${\displaystyle \beta }$ are chosen so that the method achieves order ${\displaystyle s}$, which is the maximum possible.

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