论文标题
关于UMUSCL方案的错误准确性验证
On False Accuracy Verification of UMUSCL Scheme
论文作者
论文摘要
在本文中，我们揭示了基于解决方案重建（例如UMUSCL）的非结构性网格方案遇到的虚假准确性验证背后的机制。已经报道了文献中的Euler方程的第三（或更高）准确性，但是对于非线性方程来说，UMUSCL实际上是二阶精度。错误的高阶收敛通常是针对线性方程的高阶方案而发生的，但对于非线性方程的二阶。它是由于目标非线性方程的意外线性化引起的，这是由于添加到用于准确验证的精确解决方案中的扰动太小而引起的。为了澄清机制，我们首先证明了UMUSCL方案仅适用于线性方程式。然后，我们得出了一个条件，在该条件下，三阶截断误差主导二阶误差并以数值为汉堡方程的情况。为Euler方程显示了类似的结果，这在文献中反驳了一些准确性验证。为了真正的三阶，必须通过通量重建实现UMUSCL。
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.
