A high-order local maximum principle for abnormal extremals - examples

Over thirty years ago Baker & Kippenhahn (1962) demonstrated that an instability driven by the opacity mechanism is the cause of Cepheid pulsations. Recently it has been shown that the same mechanism is responsible for oscillations observed in β Cephei, SPB and perhaps in other variable B-type stars. The search for the driving mechanism in hot stars began in the late sixties with no success until the opacities calculated with the OPAL code (Iglesias, Rogers and Wilson 1990, 1992) became available. The crucial new feature in the opacity is the local maximum at T ≈ 2 × 105 K caused by iron lines which was ignored in earlier calculations. Recently, stellar opacity data from an independent project (OP) became available (Seaton et al., 1993). The agreement between the two opacity data is satisfactory.In B stars the opacity mechanism drives two distinct categories of normal modes. The one relevant to β Cep stars encompasses low order p- and g-modes with periods 0.1–0.3 d. The other includes high-order g-modes with periods ranging up above 4 d. Excitation of such modes may explain most of the slow variability observed in B stars. The theoretical instability domain in the H-R diagram is very sensitive to metal abundance. For the standard value of Z = 0.02, the total instability domain in the main sequence band extends from spectral type O9.5 to B9. In types later than B2 only high-order g-modes are unstable.

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High order parametrized maximum-principle-preserving and positivity-preserving WENO schemes on unstructured meshes

2014 IEEE 41st International Conference on Plasma Sciences (ICOPS) held with 2014 IEEE International Conference on High-Power Particle Beams (BEAMS) ◽

10.1109/plasma.2014.7012313 ◽

2014 ◽

Author(s):

Andrew Christlieb ◽

Yuan Liu ◽

Qi Tang ◽

Zhengfu Xu

Keyword(s):

Maximum Principle ◽

Unstructured Meshes ◽

High Order ◽

Positivity Preserving ◽

Weno Schemes

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Maximum-principle-satisfying and positivity-preserving high-order schemes for conservation laws: survey and new developments

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2011.0153 ◽

2011 ◽

Vol 467 (2134) ◽

pp. 2752-2776 ◽

Cited By ~ 133

Author(s):

Xiangxiong Zhang ◽

Chi-Wang Shu

Keyword(s):

Maximum Principle ◽

Conservation Laws ◽

Finite Volume ◽

Computational Cost ◽

High Order ◽

High Order Accuracy ◽

Boltzmann Transport ◽

Finite Volume Schemes ◽

Positivity Preserving ◽

High Order Schemes

In an earlier study (Zhang & Shu 2010 b J. Comput. Phys. 229 , 3091–3120 ( doi:10.1016/j.jcp.2009.12.030 )), genuinely high-order accurate finite volume and discontinuous Galerkin schemes satisfying a strict maximum principle for scalar conservation laws were developed. The main advantages of such schemes are their provable high-order accuracy and their easiness for generalization to multi-dimensions for arbitrarily high-order schemes on structured and unstructured meshes. The same idea can be used to construct high-order schemes preserving the positivity of certain physical quantities, such as density and pressure for compressible Euler equations, water height for shallow water equations and density for Vlasov–Boltzmann transport equations. These schemes have been applied in computational fluid dynamics, computational astronomy and astrophysics, plasma simulation, population models and traffic flow models. In this paper, we first review the main ideas of these maximum-principle-satisfying and positivity-preserving high-order schemes, then present a simpler implementation which will result in a significant reduction of computational cost especially for weighted essentially non-oscillatory finite-volume schemes.

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