Monotone Finite Volume Schemes of Nonequilibrium Radiation Diffusion Equations on Distorted Meshes

2009 ◽  
Vol 31 (4) ◽  
pp. 2915-2934 ◽  
Author(s):  
Zhiqiang Sheng ◽  
Jingyan Yue ◽  
Guangwei Yuan
Keyword(s):  
Finite Volume ◽  
Diffusion Equations ◽  
Finite Volume Schemes ◽  
Radiation Diffusion ◽  
Nonequilibrium Radiation

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 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.

scholarly journals Finite volume schemes for diffusion equations: Introduction to and review of modern methods

2014 ◽  
Vol 24 (08) ◽  
pp. 1575-1619 ◽  
Author(s):  
Jerome Droniou
Keyword(s):  
Finite Volume ◽  
Approximate Solutions ◽  
Finite Volume Methods ◽  
Diffusion Equations ◽  
Maximum Principles ◽  
Finite Volume Schemes ◽  
Real World Applications ◽  
Continuous Equation ◽  
The Stability ◽  
Main Ideas

We present Finite Volume methods for diffusion equations on generic meshes, that received important coverage in the last decade or so. After introducing the main ideas and construction principles of the methods, we review some literature results, focusing on two important properties of schemes (discrete versions of well-known properties of the continuous equation): coercivity and minimum–maximum principles. Coercivity ensures the stability of the method as well as its convergence under assumptions compatible with real-world applications, whereas minimum–maximum principles are crucial in case of strong anisotropy to obtain physically meaningful approximate solutions.

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