A new nonlinear finite volume scheme preserving positivity for diffusion equations

2016 ◽  
Vol 315 ◽  
pp. 182-193 ◽  
Author(s):  
Zhiqiang Sheng ◽  
Guangwei Yuan
Keyword(s):  
Finite Volume ◽  
Diffusion Equations ◽  
Finite Volume Scheme ◽  
Volume Scheme

scholarly journals Numerical Analysis of a Nonlinear Free-Energy Diminishing Discrete Duality Finite Volume Scheme for Convection Diffusion Equations

2018 ◽  
Vol 18 (3) ◽  
pp. 407-432 ◽  
Author(s):  
Clément Cancès ◽  
Claire Chainais-Hillairet ◽  
Stella Krell
Keyword(s):  
Finite Volume ◽  
Diffusion Equations ◽  
Finite Volume Scheme ◽  
Time Behavior ◽  
Convection Diffusion ◽  
Volume Scheme ◽  
Existence Of Positive Solutions ◽  
Long Time ◽  
Nonlinear Form ◽  
Compactness Arguments

AbstractWe propose a nonlinear Discrete Duality Finite Volume scheme to approximate the solutions of drift diffusion equations. The scheme is built to preserve at the discrete level even on severely distorted meshes the energy/energy dissipation relation. This relation is of paramount importance to capture the long-time behavior of the problem in an accurate way. To enforce it, the linear convection diffusion equation is rewritten in a nonlinear form before being discretized. We establish the existence of positive solutions to the scheme. Based on compactness arguments, the convergence of the approximate solution towards a weak solution is established. Finally, we provide numerical evidences of the good behavior of the scheme when the discretization parameters tend to 0 and when time goes to infinity.

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