Understanding Derivatives on Unevenly Spaced Points

In the realm of numerical analysis, the calculation of derivatives is a fundamental aspect, particularly when dealing with simulations that utilize unevenly spaced points. This method can reflect either a smooth transformation or a more arbitrary spacing along either x or t. Recent studies, particularly the work by Rudolph, have highlighted the advantages of applying direct discretization of derivatives on an exponentially expanding grid over the traditional equally spaced transformed grid. This shift in perspective marks an interesting development in computational techniques.

One notable insight from Rudolph's findings is the behavior of concentration profiles in electrochemical systems. Near the electrode, these profiles often exhibit nearly linear characteristics, which suggests that current approximations and first derivative calculations are more reliable when they account for the actual curvature of the function in the transformed space. This adjustment to a non-uniform grid can improve the accuracy of second derivative calculations as well, as evidenced by various numerical experiments.

Several academic contributions have emerged around the topic of derivative approximations on uneven grids. For instance, Bieniasz introduced a four-point formula for second derivatives, initially based on the work of Blom. Although the original formula proved inconsistent, Bieniasz later rectified it to provide a more dependable solution. Additionally, researchers like Britz and Strutwolf have presented derivations of derivative formulae, showcasing practical examples that highlight the effectiveness of this approach.

Mathematical expressions for calculating derivatives on non-uniform grids utilize linear combinations of function values at specific points, denoted as ( u' ) and ( u'' ) for first and second derivatives, respectively. These calculations not only aid in determining derivative values but also play a significant role in establishing boundary conditions and constructing discretization matrices for various problems.

To facilitate these calculations, a sequence of displacements from a reference point is defined. This framework allows for Taylor expansions to be applied, incorporating higher-order derivatives as necessary. The resulting system can be expressed in matrix form, providing a structured approach to organize the calculations and manage error terms effectively.

The exploration of derivatives on unevenly spaced points is not merely an academic exercise; it addresses practical challenges in fields like electrochemistry and numerical modeling. By embracing non-uniform grids, researchers can enhance the precision of their simulations, paving the way for more accurate and reliable results in their respective domains.