@Article{CiCP-18-1417,
author = {Qin Li, Pengxin Liu and Hanxin Zhang},
title = {Piecewise Polynomial Mapping Method and Corresponding WENO Scheme with Improved Resolution},
journal = {Communications in Computational Physics},
year = {2018},
volume = {18},
number = {5},
pages = {1417--1444},
abstract = {<p style="text-align: justify;">The method of mapping function was first proposed by Henrick et al. [J.
Comput. Phys. 207:542-547 (2005)] to adjust nonlinear weights in [0,1] for the fifth-order
WENO scheme, and through which the requirement of convergence order is
satisfied and the performance of the scheme is improved. Different from Henrick&#39;s
method, a concept of piecewise polynomial function is proposed in this study and
corresponding WENO schemes are obtained. The advantage of the new method is
that the function can have a gentle profile at the location of the linear weight (or the
mapped nonlinear weight can be close to its linear counterpart), and therefore is favorable
for the resolution enhancement. Besides, the function also has the flexibility
of quick convergence to identity mapping near two endpoints of [0,1], which is favorable
for improved numerical stability. The fourth-, fifth- and sixth-order polynomial
functions are constructed correspondingly with different emphasis on aforementioned
flatness and convergence. Among them, the fifth-order version has the flattest profile.
To check the performance of the methods, the 1-D Shu-Osher problem, the 2-D
Riemann problem and the double Mach reflection are tested with the comparison of
WENO-M, WENO-Z and WENO-NS. The proposed new methods show the best resolution
for describing shear-layer instability of the Riemann problem, and they also
indicate high resolution in computations of double Mach reflection, where only these
proposed schemes successfully resolved the vortex-pairing phenomenon. Other investigations
have shown that the single polynomial mapping function has no advantage
over the proposed piecewise one, and it is of no evident benefit to use the proposed
method for the symmetric fifth-order WENO. Overall, the fifth-order piecewise polynomial
and corresponding WENO scheme are suggested for resolution improvement.</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.150215.250515a},
url = {http://global-sci.org/intro/article_detail/cicp/11074.html}
}