The carbuncle phenomenon has been regarded as a spurious solution produced by most of contact-preserving methods.The hybrid method of combining high resolution flux with more dissipative solver is an attractive attempt to cure this kind of non-physical phenomenon.In this paper,a matrix-based stability analysis for 2-D Euler equations is performed to explore the cause of instability of numerical schemes.By combining the Roe with HLL flux in different directions and different flux components,we give an interesting explanation to the linear numerical instability.Based on such analysis,some hybrid schemes are compared to illustrate different mechanisms in controlling shock instability.Numerical experiments are presented to verify our analysis results.The conclusion is that the scheme of restricting directly instability source is more stable than other hybrid schemes.
In this paper, the extremum of second-order directional derivatives, i.e. the gradient of first-order derivatives is discussed. Given second-order directional derivatives in three nonparallel directions, or given second-order directional derivatives and mixed directional derivatives in two nonparallel directions, the formulae for the extremum of second-order directional derivatives are derived, and the directions corresponding to maximum and minimum are perpendicular to each other.
