High-order Maximum Principle for Abnormal Extremals

2012 ◽  
Keyword(s):  
Maximum Principle ◽  
High Order ◽  
Abnormal Extremals

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

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.

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