Local maximum principle satisfying high-order non-oscillatory schemes

2015 ◽  
Vol 81 (11) ◽  
pp. 689-715 ◽  
Author(s):  
Ritesh Kumar Dubey ◽  
Biswarup Biswas ◽  
Vikas Gupta
Keyword(s):  
Maximum Principle ◽  
Local Maximum ◽  
High Order

A High-Order Generalized Local Maximum Principle

2000 ◽  
Vol 38 (3) ◽  
pp. 823-854 ◽  
Author(s):  
Urszula Ledzewicz ◽  
Heinz Schättler
Keyword(s):  
Maximum Principle ◽  
Local Maximum ◽  
High Order

scholarly journals The New Opacities and B-Star Pulsations

1994 ◽  
Vol 162 ◽  
pp. 55-66
Author(s):  
W.A. Dziembowski
Keyword(s):  
Normal Modes ◽  
Local Maximum ◽  
High Order ◽  
Driving Mechanism ◽  
Instability Domain ◽  
B Stars ◽  
Sequence Band ◽  
Metal Abundance ◽  
New Feature ◽  
The One

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.

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

2011 ◽  
Vol 467 (2134) ◽  
pp. 2752-2776 ◽  
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.

scholarly journals Harnack Inequalities and ABP Estimates for Nonlinear Second-Order Elliptic Equations in Unbounded Domains

2008 ◽  
Vol 2008 ◽  
pp. 1-19 ◽  
Author(s):  
M. E. Amendola ◽  
L. Rossi ◽  
A. Vitolo
Keyword(s):  
Maximum Principle ◽  
Type Theorem ◽  
Unbounded Domains ◽  
Local Maximum ◽  
Maximum Principles ◽  
Gradient Term ◽  
Harnack Inequalities ◽  
General Unbounded Domains ◽  
Second Order Elliptic Equations ◽  
Weak Harnack Inequality

We are concerned with fully nonlinear uniformly elliptic operators with a superlinear gradient term. We look for local estimates, such as weak Harnack inequality and local maximum principle, and their extension up to the boundary. As applications, we deduce ABP-type estimates and weak maximum principles in general unbounded domains, a strong maximum principle, and a Liouville-type theorem.

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