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Backward differentiation formula

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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.

Contents

General formula

A BDF is used to solve the initial value problem

y ′ = f ( t , y ) , y ( t 0 ) = y 0 .

The general formula for a BDF can be written as

∑ k = 0 s a k y n + k = h β f ( t n + s , y n + s ) ,

where h denotes the step size and t n = t 0 + n h . The coefficients a k and β 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.

Specific formulas

The s-step BDFs with s < 7 are:

  • BDF1: y n + 1 − y n = h f ( t n + 1 , y n + 1 ) ; (this is the backward Euler method)
  • BDF2: y n + 2 − 4 3 y n + 1 + 1 3 y n = 2 3 h f ( t n + 2 , y n + 2 ) ;
  • BDF3: y n + 3 − 18 11 y n + 2 + 9 11 y n + 1 − 2 11 y n = 6 11 h f ( t n + 3 , y n + 3 )
  • BDF4: y n + 4 − 48 25 y n + 3 + 36 25 y n + 2 − 16 25 y n + 1 + 3 25 y n = 12 25 h f ( t n + 4 , y n + 4 )
  • BDF5: y n + 5 − 300 137 y n + 4 + 300 137 y n + 3 − 200 137 y n + 2 + 75 137 y n + 1 − 12 137 y n = 60 137 h f ( t n + 5 , y n + 5 )
  • BDF6: y n + 6 − 360 147 y n + 5 + 450 147 y n + 4 − 400 147 y n + 3 + 225 147 y n + 2 − 72 147 y n + 1 + 10 147 y n = 60 147 h f ( t n + 6 , y n + 6 ) .

    • Methods with s > 6 are not zero-stable so they cannot be used.

    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.

    References

    Backward differentiation formula Wikipedia
    (Text) CC BY-SA


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