Optimal Collocation Nodes for Fractional Derivative Operators

TL;DR

This paper investigates optimal collocation node distributions for spectral discretizations of fractional derivative operators, improving accuracy and efficiency in solving fractional PDEs through strategic node placement.

Contribution

It introduces a novel approach to selecting collocation nodes based on the operator, enhancing spectral method performance for fractional derivatives.

Findings

Optimal node distributions improve approximation accuracy.

Strategic node choices can accelerate computations.

Enhanced methods outperform traditional collocation approaches.

Abstract

Spectral discretizations of fractional derivative operators are examined, where the approximation basis is related to the set of Jacobi polynomials. The pseudo-spectral method is implemented by assuming that the grid, used to represent the function to be differentiated, may not be coincident with the collocation grid. The new option opens the way to the analysis of alternative techniques and the search of optimal distributions of collocation nodes, based on the operator to be approximated. Once the initial representation grid has been chosen, indications on how to recover the collocation grid are provided, with the aim of enlarging the dimension of the approximation space. As a results of this process, performances are improved. Applications to fractional type advection-diffusion equations, and comparisons in terms of accuracy and efficiency are made. As shown in the analysis, special choices of the nodes can also suggest tricks to speed up computations.