@Article{CiCP-30-1037,
author = {Nishikawa , Hiroaki},
title = {On False Accuracy Verification of UMUSCL Scheme},
journal = {Communications in Computational Physics},
year = {2021},
volume = {30},
number = {4},
pages = {1037--1060},
abstract = {<p style="text-align: justify;">In this paper, we reveal a mechanism behind a false accuracy verification
encountered with unstructured-grid schemes based on solution reconstruction such
as UMUSCL. Third- (or higher-) order of accuracy has been reported for the Euler
equations in the literature, but UMUSCL is actually second-order accurate at best for
nonlinear equations. False high-order convergence occurs generally for a scheme that
is high order for linear equations but second-order for nonlinear equations. It is caused
by unexpected linearization of a target nonlinear equation due to too small of a perturbation added to an exact solution used for accuracy verification. To clarify the mechanism, we begin with a proof that the UMUSCL scheme is third-order accurate only for
linear equations. Then, we derive a condition under which the third-order truncation
error dominates the second-order error and demonstrate it numerically for Burgers’
equation. Similar results are shown for the Euler equations, which disprove some
accuracy verification results in the literature. To be genuinely third-order, UMUSCL
must be implemented with flux reconstruction.</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.OA-2020-0198},
url = {http://global-sci.org/intro/article_detail/cicp/19393.html}
}