An Exploration of the Approximation of Derivative Functions via Finite Differences

Details An Exploration of the Approximation of Derivative Functions via Finite Differences An Exploration of the Approximation of Derivative Functions via Finite Differences

2010-06-08

arxiv.org/abs/1006.1620v1

Rose-Hulman Undergraduate Mathematics Journal,Vol.8-2-2007

Mathematics

Numerical Analysis

16 pages

Scientific paper

Finite differences have been widely used in mathematical theory as well as in scientific and engineering computations. These concepts are constantly mentioned in calculus. Most frequently-used difference formulas provide excellent approximations to various derivative functions, including those used in modeling important physical processes on uniform grids. However, our research reveals that difference approximations on uniform grids cannot be applied blindly on nonuniform grids, nor can difference formulas to form consistent approximations to second derivatives. At best, they may lose accuracy; at worst they are inconsistent. Detailed consistency and error analysis, together with simulated examples, will be given.

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