Comparative study on higher order compact RBF-FD formulas with Gaussian and Multiquadric radial functions

TL;DR

This paper develops higher order compact RBF-FD formulas using Gaussian functions, deriving analytical weights and errors, and demonstrates improved accuracy over multiquadric RBF-FD and finite difference schemes.

Contribution

It introduces a method for deriving analytical weights for higher order Gaussian RBF-FD formulas and compares their accuracy with existing methods.

Findings

Formulas converge to polynomial-based schemes in the flat limit.

Demonstrated improved approximation accuracy over multiquadric RBF-FD.

Computed optimal shape parameters for each formula.

Abstract

We generate Gaussian radial function based higher order compact RBF-FD formulas for some differential operators. Analytical expressions for weights associated to first and second derivative formulas (up to order 10) and 2D-Laplacian formulas (up to order 6) are derived. Then these weights are used to obtain analytical expression for local truncation errors. The weights are obtained by symbolic computation of a linear system in Mathematica. Often such linear systems are not directly amenable to symbolic computation. We make use of symmetry of formula stencil along with Taylor series expansions for performing the computation. In the flat limit, the formulas converge to their respective order polynomial based compact FD formulas. We validate the formulas with standard test functions and demonstrate improvement in approximation accuracy with respect to corresponding order multiquadric based compact RBF-FD formulas and compact FD schemes. We also compute optimal value of shape parameter for each formula.