Bound-Preserving High Order Finite Volume Schemes for Conservation Laws and Convection-Diffusion Equations

2017 ◽  
pp. 3-14 ◽  
Author(s):  
Chi-Wang Shu
Keyword(s):  
Conservation Laws ◽  
Finite Volume ◽  
High Order ◽  
Diffusion Equations ◽  
Finite Volume Schemes ◽  
Convection Diffusion

scholarly journals Large Time Behavior of Nonlinear Finite Volume Schemes for Convection-Diffusion Equations

2020 ◽  
Vol 58 (5) ◽  
pp. 2544-2571
Author(s):  
Clément Cancès ◽  
Claire Chainais-Hillairet ◽  
Maxime Herda ◽  
Stella Krell
Keyword(s):  
Finite Volume ◽  
Large Time ◽  
Large Time Behavior ◽  
Diffusion Equations ◽  
Time Behavior ◽  
Finite Volume Schemes ◽  
Convection Diffusion

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 Large-time behaviour of a family of finite volume schemes for boundary-driven convection–diffusion equations

2019 ◽  
Vol 40 (4) ◽  
pp. 2473-2504 ◽  
Author(s):  
Claire Chainais-Hillairet ◽  
Maxime Herda
Keyword(s):  
Finite Volume ◽  
Exponential Decay ◽  
Large Time ◽  
Nonlinear Models ◽  
Numerical Test ◽  
Diffusion Equations ◽  
Time Behaviour ◽  
Finite Volume Schemes ◽  
Large Time Behaviour ◽  
Convection Diffusion

Abstract We are interested in the large-time behaviour of solutions to finite volume discretizations of convection–diffusion equations or systems endowed with nonhomogeneous Dirichlet- and Neumann-type boundary conditions. Our results concern various linear and nonlinear models such as Fokker–Planck equations, porous media equations or drift–diffusion systems for semiconductors. For all of these models, some relative entropy principle is satisfied and implies exponential decay to the stationary state. In this paper we show that in the framework of finite volume schemes on orthogonal meshes, a large class of two-point monotone fluxes preserves this exponential decay of the discrete solution to the discrete steady state of the scheme. This includes for instance upwind and centred convections or Scharfetter–Gummel discretizations. We illustrate our theoretical results on several numerical test cases.

Convergence of finite volume schemes for semilinear convection diffusion equations

1999 ◽  
Vol 82 (1) ◽  
pp. 91-116 ◽  
Author(s):  
Robert Eymard ◽  
Thierry Gallouët ◽  
Raphaèle Herbin
Keyword(s):  
Finite Volume ◽  
Diffusion Equations ◽  
Finite Volume Schemes ◽  
Convection Diffusion
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