Radial basis function ENO and WENO finite difference methods based on the optimization of shape parameters

TL;DR

This paper introduces adaptive RBF-ENO/WENO finite difference methods that improve accuracy and sharpness near discontinuities by optimizing shape parameters, extending previous ENO/WENO approaches with smooth RBFs.

Contribution

The paper develops RBF-ENO/WENO finite difference methods that enhance accuracy and implementation simplicity over traditional methods by locally optimizing shape parameters.

Findings

RBF-ENO/WENO methods outperform regular ENO/WENO in accuracy.

Enhanced sharpness of solutions near discontinuities.

Methods are easy to implement in existing code.

Abstract

We present adaptive finite difference ENO/WENO methods by adopting infinitely smooth radial basis functions (RBFs). This is a direct extension of the non-polynomial finite volume ENO/WENO method proposed by authors in \cite{GuoJung} to the finite difference ENO/WENO method based on the original smoothness indicator scheme developed by Jiang and Shu \cite{WENO}. The RBF-ENO/WENO finite difference method slightly perturbs the reconstruction coefficients with RBFs as the reconstruction basis and enhances accuracy in the smooth region by locally optimizing the shape parameters. The RBF-ENO/WENO finite difference methods provide more accurate reconstruction than the regular ENO/WENO reconstruction and provide sharper solution profiles near the jump discontinuity. Furthermore the RBF-ENO/WENO methods are easy to implement in the existing regular ENO/WENO code. The numerical results in 1D and 2D presented in this work show that the proposed RBF-ENO/WENO finite difference method better performs than the regular ENO/WENO method.