On False Accuracy Verification of UMUSCL Scheme

TL;DR

This paper uncovers why UMUSCL scheme falsely appears to have high-order accuracy for nonlinear equations, revealing it is actually only second-order accurate and emphasizing the need for flux reconstruction for true third-order accuracy.

Contribution

The paper demonstrates that UMUSCL is only third-order accurate for linear equations and clarifies the mechanism behind false high-order accuracy verification in nonlinear cases.

Findings

UMUSCL is third-order accurate only for linear equations.

False high-order convergence occurs due to linearization effects.

Flux reconstruction is necessary for genuine third-order accuracy.

Abstract

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.