On False Accuracy Verification of UMUSCL Scheme (Communications in Computational Physics, 2021)
In this paper, we reveal a mechanism behind a false accuracy verificationencountered with unstructured-grid schemes based on solution reconstruction suchas UMUSCL. Third- (or higher-) order of accuracy has been reported for the Eulerequations in the literature, but UMUSCL is actually second-order accurate at best fornonlinear equations. False high-order convergence occurs generally for a scheme thatis high order for linear equations but second-order for nonlinear equations. It is causedby 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 forlinear equations. Then, we derive a condition under which the third-order truncationerror dominates the second-order error and demonstrate it numerically for Burgers’equation. Similar results are shown for the Euler equations, which disprove someaccuracy verification results in the literature. To be genuinely third-order, UMUSCLmust be implemented with flux reconstruction.
Keywords: UMUSCL; high-order; finite-volume; finite-difference; unstructured grids; reconstruction schemes.