Towards the ultimate understanding of MUSCL: Pitfalls in achieving third-order accuracy

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 κ=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 κ=1/3 but κ=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.