TY - JOUR T1 - On False Accuracy Verification of UMUSCL Scheme AU - Nishikawa , Hiroaki JO - Communications in Computational Physics VL - 4 SP - 1037 EP - 1060 PY - 2021 DA - 2021/08 SN - 30 DO - http://doi.org/10.4208/cicp.OA-2020-0198 UR - https://global-sci.org/intro/article_detail/cicp/19393.html KW - UMUSCL, high-order, finite-volume, finite-difference, unstructured grids, reconstruction schemes. AB -

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.