Towards building the OP-Mapped WENO schemes: A general methodology

TL;DR

This paper introduces a general methodology for designing OP-Mapped WENO schemes that preserve high resolution and robustness over long simulations by extending the OP criterion and devising improved mappings.

Contribution

It defines the OP-Mapped WENO family, develops a general formula for mappings, and proposes a unified implementation for enhanced long-term performance.

Findings

MOP-WENO-X schemes outperform existing mapped WENO schemes in long-term simulations.

The generalized mapping formula improves the balance between resolution and oscillation control.

Extensive experiments validate the superior robustness and accuracy of the proposed schemes.

Abstract

A serious and ubiquitous issue in existing mapped WENO schemes is that most of them can hardly preserve high resolutions and in the meantime prevent spurious oscillations on solving hyperbolic conservation laws with long output times. Our goal in this article is to address this widely concerned problem [3, 4, 15, 29, 16, 18]. In our previous work [18], the order-preserving (OP) criterion was originally introduced and carefully used to devise a new mapped WENO scheme that performs satisfactorily in long-run simulations, and hence it was indicated that the OP criterion plays a critical role in the maintenance of low-dissipation and robustness for the mapped WENO schemes. Thus, in our present work, we firstly define the family of the mapped WENO schemes, whose mappings meet the OP criterion, as OP-Mapped WENO. Next, we attentively take a closer look at the mappings of various existing mapped WENO schemes and devise a general formula for them. It helps us to extend the OP criterion into the design of the improved mappings. Then, we propose the generalized implementation of obtaining a group of OP-Mapped WENO schemes, named MOP-WENO-X as they are developed from the existing mapped WENO-X schemes, where the notation "X" is used to identify the version of the existing mapped WENO scheme. Finally, extensive numerical experiments and comparisons with competing schemes are conducted to demonstrate the enhanced performances of the MOP-WENO-X schemes.