An extension of the order-preserving mapping to the WENO-Z-type schemes

TL;DR

This paper extends the order-preserving criterion to WENO-Z-type schemes, enhancing their ability to achieve high resolution and suppress oscillations in long-term hyperbolic system simulations.

Contribution

The study introduces a generalized mapped WENO framework and applies the OP criterion to improve WENO-Z schemes, reducing oscillations while maintaining convergence.

Findings

Enhanced schemes achieve high resolution and suppress oscillations.

The new schemes reduce post-shock oscillations in 2D Euler problems.

Convergence properties are maintained compared to original WENO-X schemes.

Abstract

In our latest studies, by introducing the novel order-preserving (OP) criterion, we have successfully addressed the widely concerned issue of the previously published mapped weighted essentially non-oscillatory (WENO) schemes that it is rather difficult to achieve high resolutions on the premise of removing spurious oscillations for long-run simulations of the hyperbolic systems. In the present study, we extend the OP criterion to the WENO-Z-type schemes as the forementioned issue has also been extensively observed numerically for these schemes. Firstly, we innovatively present the concept of the generalized mapped WENO schemes by rewriting the Z-type weights in a uniform formula from the perspective of the mapping relation. Then, we naturally introduce the OP criterion to improve the WENO-Z-type schemes, and the resultant schemes are denoted as MOP-GMWENO-X. Finally, extensive numerical experiments have been conducted to demonstrate the benefits of these new schemes. We draw the conclusion that, the convergence propoties of the proposed schemes are equivalent to the corresponding WENO-X schemes. The major benefit of the new schemes is that they have the capacity to achieve high resolutions and simultaneously remove spurious oscillations for long simulations. The new schemes have the additional benefit that they can greatly decrease the post-shock oscillations on solving 2D Euler problems with strong shock waves.