A Note on the Direct Approximation of Derivatives in Rational Radial Basis Functions Partition of Unity Method

TL;DR

This paper introduces a new method for efficiently approximating derivatives of functions with steep gradients or discontinuities using rational radial basis functions, simplifying computations and improving speed.

Contribution

The paper presents a novel D-RRBF-PU approach that eliminates derivatives of weight functions, enhancing computational efficiency in derivative approximation.

Findings

Derivatives are computed more easily and quickly.

The method maintains accuracy with discussed error bounds.

Numerical results demonstrate the technique's potential.

Abstract

This paper proposes a Direct Rational Radial Basis Functions Partition of Unity (D-RRBF-PU) approach to compute derivatives of functions with steep gradients or discontinuities. The novelty of the method concerns how derivatives are approximated. More precisely, all derivatives of the partition of unity weight functions are eliminated while we compute the derivatives of the local rational approximants in each patch. As a result, approximate derivatives are obtained more easily and quickly than those obtained in the standard formulation. The corresponding error bounds are briefly discussed. Some numerical results are presented to show the technique's potential.