Finite Difference Methods

  • The key idea for approximating the derivative using Finite Difference Methods is as follows:
    • Split the domain into sub-intervals.
    • Define the distance between two points in the partition as .
    • Approximate the derivative at in the domain as a linear combination of , , and other points in the partition.

First Derivatives

  • A simple approximation to the derivative of a function can be made as follows

    For sufficiently small .

    • The above is a linear approximation of the derivative .
    • Note that this approximation fails when the error terms (as seen in the Taylor Series) are non-negligible.
    • (Sullivan 3.1) The approximation above is a first order approximation. That is
  • We can get a second order approximation as follows using the Taylor Series.

    Note that

    Observe that

    And we can verify that

    So

Second Derivatives

  • We can obtain approximations for the second derivative using the first degree approximations. In particular, it can be shown that the following three hold (respectively called, central, forward and backward difference approximations)

Higher Order Derivatives

  • We can, in general, obtain approximations for higher order derivatives as follows.
  • The forward approximation for the -th order derivative is given by
  • The backward approximation is given by
  • The central approximation is given by

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