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**On entropy generation and dissipation of kinetic energy in high-resolution shock-capturing schemes.**
*(English)*
Zbl 1142.65073

Summary: This paper addresses entropy generation and the corresponding dissipation of kinetic energy associated with high-resolution, shock-capturing (Godunov) methods. Analytical formulae are derived for the rate of increase of entropy given arbitrary jumps in primitive variables at a cell interface. It is demonstrated that for general continuously varying flows the inherent numerical entropy increase of Godunov methods is not proportional to the velocity jump cubed as is commonly assumed, but it is proportional to the velocity jump squared.

Furthermore, the dissipation of kinetic energy is directly linked to temperature multiplied by change in entropy at low Mach numbers. The kinetic energy dissipation rate is shown to be proportional to the velocity jump squared and the speed of sound. The leading order dissipation rate associated with jumps in pressure, density and shear waves is detailed and further shown that at low Mach number it is the dissipation due to the perpendicular velocity jumps which dominates. This explains directly the poor performance of Godunov methods at low Mach numbers. The analysis is also applied to high-order accurate methods in space and time and all analytical results are validated with simple numerical experiments.

Furthermore, the dissipation of kinetic energy is directly linked to temperature multiplied by change in entropy at low Mach numbers. The kinetic energy dissipation rate is shown to be proportional to the velocity jump squared and the speed of sound. The leading order dissipation rate associated with jumps in pressure, density and shear waves is detailed and further shown that at low Mach number it is the dissipation due to the perpendicular velocity jumps which dominates. This explains directly the poor performance of Godunov methods at low Mach numbers. The analysis is also applied to high-order accurate methods in space and time and all analytical results are validated with simple numerical experiments.

### MSC:

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

35L45 | Initial value problems for first-order hyperbolic systems |

76N15 | Gas dynamics (general theory) |

76M12 | Finite volume methods applied to problems in fluid mechanics |

### Keywords:

high-resolution methods; Godunov methods; dissipation; kinetic energy; entropy; large eddy simulation; low Mach number; finite volume method; Euler equation; shock-capturing methods; numerical experiments
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\textit{B. Thornber} et al., J. Comput. Phys. 227, No. 10, 4853--4872 (2008; Zbl 1142.65073)

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