We present a proof by analysis and numerical results that Van Leer's MUSCL conservative scheme with the discretization parameter κ is third-order accurate for $\kappa =1/3$. We include both the original finite-volume MUSCL family, updating cell-averaged values of the solution, and the related finite-difference version, updating point values. The presentation is needed because in the CFD literature claims have been made that not $\kappa =1/3$ but $\kappa =1/2$ yields third-order accuracy, or even that no value of κ can yield third-order accuracy. These false claims are the consequence of mixing up finite-difference concepts with finite-volume concepts. In a series of Pitfalls, we show how incorrect conclusions can be drawn when pointwise values of the discrete solution are interchanged with cell-averaged values. All flawed schemes presented in the Pitfalls, and some correct ones for comparison, are tested numerically and shown to behave as predicted by the analysis. We conclude with firm recommendations on how to achieve third-order accuracy at all output times, or just in a steady state.

我们通过分析和数值结果证明了具有离散化参数κ 的Van Leer 的 MUSCL 保守方案对于$\kappa =1/3$. 我们包括原始有限体积 MUSCL 系列，更新解决方案的单元平均值，以及相关的有限差分版本，更新点值。需要进行介绍，因为在 CFD 文献中已经声明不$\kappa =1/3$ 但 $\kappa =1/2$产生三阶精度，甚至没有κ值可以产生三阶精度。这些错误的主张是将有限差分概念与有限体积概念混合在一起的结果。在一系列陷阱中，我们展示了当离散解的逐点值与像元平均值互换时如何得出错误的结论。陷阱中提出的所有有缺陷的方案，以及一些用于比较的正确方案，都经过数值测试，并显示其行为与分析预测的一样。我们最后给出了关于如何在所有输出时间或仅在稳定状态下实现三阶精度的坚定建议。