scholarly journals A finite volume scheme for convection–diffusion equations with nonlinear diffusion derived from the Scharfetter–Gummel scheme

2012 ◽  
Vol 121 (4) ◽  
pp. 637-670 ◽  
Author(s):  
Marianne Bessemoulin-Chatard
Keyword(s):  
Finite Volume ◽  
Nonlinear Diffusion ◽  
Diffusion Equations ◽  
Finite Volume Scheme ◽  
Convection Diffusion ◽  
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.

scholarly journals A multilevel Monte Carlo finite difference method for random scalar degenerate convection–diffusion equations

2017 ◽  
Vol 14 (03) ◽  
pp. 415-454 ◽  
Author(s):  
Ujjwal Koley ◽  
Nils Henrik Risebro ◽  
Christoph Schwab ◽  
Franziska Weber
Keyword(s):  
Monte Carlo ◽  
Finite Difference ◽  
Convergence Rate ◽  
Finite Volume ◽  
Monte Carlo Algorithm ◽  
Diffusion Equations ◽  
Entropy Solutions ◽  
Finite Volume Scheme ◽  
Multilevel Monte Carlo ◽  
Convection Diffusion

This paper proposes a finite difference multilevel Monte Carlo algorithm for degenerate parabolic convection–diffusion equations where the convective and diffusive fluxes are allowed to be random. We establish a notion of stochastic entropy solutions to these equations. Our chief goal is to efficiently compute approximations to statistical moments of these stochastic entropy solutions. To this end, we design a multilevel Monte Carlo method based on a finite volume scheme for each sample. We present a novel convergence rate analysis of the combined multilevel Monte Carlo finite volume method, allowing in particular for low [Formula: see text]-integrability of the random solution with [Formula: see text], and low deterministic convergence rates (here, the theoretical rate is [Formula: see text]). We analyze the design and error versus work of the multilevel estimators. We obtain that the maximal rate (based on optimizing possibly the pessimistic upper bounds on the discretization error) is obtained for [Formula: see text], for finite volume convergence rate of [Formula: see text]. We conclude with numerical experiments.

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