Exact numerical schemes and the three means

Abstract

Grids based on geometric and harmonic sequences are used here in defining approximations to derivatives which have second order accuracy. Such schemes are superior to the usual arithmetic difference grids for important classes of functions exhibiting significant variation—such as a singularity in a region. Furthermore, the approximations to derivatives are exact in certain cases. First and second difference operators based on the geometric mean give exact results for functions of the form ax + b + cx ^-1, and operators based on the harmonic mean give exact results for functions of the form a+bx ^-1+cx ^-2 This unifies the well-known result for arithmetic differences, which give exact derivatives for second degree polynomials, and demonstrates yet another surprising connection between arithmetic, geometric and harmonic means.

Keywords:

• Finite differences
• derivative approximations
• non-uniform grid
• geometric mean
• harmonic mean