A modified adaptive improved mapped WENO method

TL;DR

This paper introduces a new family of adaptive mapped WENO schemes that enhance solution accuracy near discontinuities and critical points by using adaptive control functions and smoothing techniques.

Contribution

The paper presents a novel adaptive mapped WENO framework with flexible parameters, improving convergence and solution quality for problems with discontinuities.

Findings

Achieves optimal convergence rates near critical points.

Produces high-resolution solutions without spurious oscillations.

Excels in long-time simulations with discontinuities.

Abstract

We propose a new family of mapped WENO schemes by using several adaptive control functions and a smoothing approximation of the signum function. The proposed schemes introduce the adaptivity and admit an extensive permitted range of the parameters in the mapping functions. Consequently, they have the capacity to achieve optimal convergence rates, even near critical points. Particularly, one of these new schemes with fine-tuned parameters illustrates a significant advantage when solving problems with discontinuities. It produces numerical solutions with high resolution without generating spurious oscillations, especially for long output times.