an:05956771
Zbl 1222.65107
Zhang, Xiangxiong; Shu, Chi-Wang
Maximum-principle-satisfying and positivity-preserving high-order schemes for conservation laws: survey and new developments
EN
Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 467, No. 2134, 2752-2776 (2011).
1364-5021 1471-2946
2011
j
65M60 65M12 35L65 35Q35 35B50
hyperbolic conservation laws; discontinuous Galerkin method; weighted essentially non-oscillatory finite-volume scheme; positivity-preserving; maximum-principle-satisfying; high-order accuracy
Summary: In an earlier study of the authors [J. Comput. Phys. 229, No. 9, 3091--3120 (2010; Zbl 1187.65096)], 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.
1187.65096