Convection–diffusion equation

Summary

The convection–diffusion equation is a combination of the diffusion and convection (advection) equations, and describes physical phenomena where particles, energy, or other physical quantities are transferred inside a physical system due to two processes: diffusion and convection. Depending on context, the same equation can be called the advection–diffusion equation, drift–diffusion equation, or (generic) scalar transport equation. Equation General The general equation is \frac{\partial c}{\partial t} = \mathbf{\nabla} \cdot (D \mathbf{\nabla} c) - \mathbf{\nabla} \cdot (\mathbf{v} c) + R where *c is the variable of interest (species concentration for mass transfer, temperature for heat transfer), *D is the diffusivity (also called diffusion coefficient), such as mass diffusivity for particle motion or thermal diffusivity for heat transport, *v is the velocity field that the quantity

Official source

https://en.wikipedia.org/wiki/Convection%E2%80%93diffusion_equation

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Iterative substructuring methods for advection-diffusion problems in heterogeneous media

Paolo Zunino

Springer2004

Exploring stabilisation techniques for the reduced basis approximation of avection-diffusion PDEs

Christèle Marie Zbinden

In this thesis, we explore possible stabilisation methods for the reduce basis approximation of advection-diffusion problems, for which the advection term is dominating. The options we consider are mainly inspired by the Variational Multiscale method (VMS), which decomposes the solution of a variational problem into its coarse scale component, from a coarse scale space, and a fine scale component, from a fine scale space. Our stabilisation proposals are divided into three classes. The first one groups methods that rely on a stabilisation parameter. The second class uses VMS at the algebraic level to attempt stabilisation. Finally the third class is also inspired by VMS at the algebraic level, but with the additional constraint that the fine scale space is orthogonal to the coarse scale space. Numericals tests reported in this thesis show that the methods of the first class is not viable options as the best stabilisation parameter among those tested is the stabilisation parameter that is used at the high fidelity level. Although the stabilisation methods of the second class give accurate results when applied to stable problems, they were also dismissed by the numerical tests, as they did not improve the accuracy of the already stabilised problem. The third class also performs well when applied to stable problems. It has been shown in [7] one of those methods can improve accuracy. However in the current implementation, this result was not achieved here.

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This thesis is about the numerical simulation and optimization of the alumina repartition in the bath of an aluminium electrolysis pot. A mathematical model is set up which contains the feeding of alumina particles to the bath, the dissolution of the particles, the motion of the alumina in the bath, and the consumption of the alumina by electrolysis. The arising convection-diffusion equations are solved using stabilized finite elements. Different weak formulations of the convection-diffusion equation are compared and we find a formulation which ensures mass conservation even if the velocity field is not divergence free. A comparison of the complete numerical simulation with real world experiments is carried out, showing that the numerical results are in very good agreement with the measured values. In a second part we optimize the feeding of the particles, with the aim of getting a more uniform distribution of the alumina concentration in the bath. A mathematical model of the optimization problem is proposed and subsequently solved using particle swarm optimization and Newton's method.

EPFL2011