Affordable, entropy-consistent Euler flux functions. II: Entropy production at shocks.

*(English)*Zbl 1280.76015Summary: In this paper, an entropy-consistent flux is developed, continuing from the work of the previous paper [the second author, Affordable, entropy-consistent, Euler flux functions. I: Analytical results. J. Comput. Phys. (submitted)]. To achieve entropy consistency, a second and third-order differential terms are added to the entropy-conservative flux. This new flux function is tested on several one dimensional problems and compared with the original Roe flux. The new flux function exactly preserves the stationary contact discontinuity and does not capture the unphysical rarefaction shock. For steady shock problems, the new flux predicts a slightly more diffused profile whereas for unsteady cases, the captured shock is very similar to those produced by the Roe flux. The shock stability is also studied in one dimension. Unlike the original Roe flux, the new flux is completely stable which will provide as a candidate to combat multidimensional shock instability, particularly the carbuncle phenomenon.

##### MSC:

76L05 | Shock waves and blast waves in fluid mechanics |

76M25 | Other numerical methods (fluid mechanics) (MSC2010) |

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

##### Keywords:

entropy conservation; entropy production; entropy-stability; entropy consistency; shock stability
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\textit{F. Ismail} and \textit{P. L. Roe}, J. Comput. Phys. 228, No. 15, 5410--5436 (2009; Zbl 1280.76015)

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