On developing piecewise rational mapping with fine regulation capability for WENO schemes

TL;DR

This paper develops a new piecewise rational mapping for WENO schemes that enhances resolution, stability, and robustness, with explicit control over mapping properties to improve numerical performance.

Contribution

A novel piecewise rational mapping with fine regulation capabilities is proposed for WENO schemes, balancing resolution and stability while preserving optimal order at critical points.

Findings

New mappings improve numerical stability in long-term computations

Enhanced resolution and robustness demonstrated through numerical tests

Proposed schemes outperform recent mappings and WENO-Z type schemes

Abstract

On the idea of mapped WENO-JS scheme, properties of mapping methods are analyzed, uncertainties in mapping development are investigated, and new rational mappings are proposed. Based on our former understandings, i.e. mapping at endpoints {0, 1} tending to identity mapping, an integrated Cm,n condition is summarized for function development. Uncertainties, i.e., whether the mapping at endpoints would make mapped scheme behave like WENO or ENO, whether piecewise implementation would entail numerical instability, and whether WENO3-JS could preserve the third-order at first-order critical points by mapping, are analyzed and clarified. A new piecewise rational mapping with sufficient regulation capability is developed afterwards, where the flatness of mapping around the linear weights and its endpoint convergence toward identity mapping can be coordinated explicitly and simultaneously. Hence, the increase of resolution and preservation of stability can be balanced. Especially, concrete mappings are determined for WENO3,5,7-JS. Numerical cases are tested for the new mapped WENO-JS, which regards numerical stability including that in long time computation, resolution and robustness. In purpose of comparison, some recent mappings such as IM by [App. Math. Comput. 232, 2014:453-468], RM by [J. Sci. Comput. 67, 2016:540-580] and AIM by [J. Comput. Phys. 381, 2019:162-188] are chosen; in addition, some recent WENO-Z type scheme are selected also. Proposed new schemes can preserve optimal orders at corresponding critical points, achieve numerical stability and indicate overall comparative advantages regarding accuracy, resolution and robustness.