  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.
