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https://github.com/louis-finegan/convection-diffusion-models-finite-difference-method-python3

Applying the finite-difference method to the Convection Diffusion equation in python3. Examples included: One dimensional Heat equation, Transport equation, Fokker-Plank equation and some two dimensional examples.
https://github.com/louis-finegan/convection-diffusion-models-finite-difference-method-python3

convection-diffusion finite-difference-method fokker-planck-equation heat-equation jupyter-notebook mathematical-modelling matplotlib matplotlib-animation numpy python3 transport-equation

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Applying the finite-difference method to the Convection Diffusion equation in python3. Examples included: One dimensional Heat equation, Transport equation, Fokker-Plank equation and some two dimensional examples.

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README 

          
# Diffusion Convection Equation



## Introduction



The Diffusion Convection Equation is a Partial Differential Equation writen in the form:



$$\frac{\partial u}{\partial t} = \nabla ( D \nabla u) + \nabla \cdot (\mathbf{c} u)$$



This Equation can model most physical phenomena involving the transfer of a quantity by 'Diffusion' and 'Convection' (Advection).



1. $\nabla(D \nabla u)$ is the diffusion term. Diffusion describes the net movement of a quantity $u$, generally from a region of higher concentration to lower concentraction.



2. $\nabla \cdot (\mathbf{c} u)$ is the convection term. Convection (Advection) describes the bulk motion of a quantity $u$ under the a velocity field $\mathbf{c}$.



Click [here](https://en.wikipedia.org/wiki/Convection%E2%80%93diffusion_equation) for more information on the Diffusion Convection Equation.



## Simplifications



There are a number of simplifications made to this model.



1. Let the diffusion coefficient $D$ be constant, this means it can come outside of the gradient operator resulting in the dot product of the gradients becoming the Laplacian. Hence the diffusion is written as $D \nabla^2 u$.



2. Let the convection vector field be constant, this means the convection term siplifies to $\mathbf{c} \cdot \nabla u$.



## Diffusion Equation



Suppose a model where $\mathbf{c} = 0$. This can be interpreted as a quantity $u$ under going diffusion only, so $u$ is not flowing with any vector field. instead it is moving into the less concentrated areas. This model is governed by the following Partial Differential Equation:



$$\frac{\partial u}{\partial t} = D\nabla^2 u$$



## Convection Equation



Suppose a model where $D = 0$. this can be interpreted as a under going convection only, so $u$ is being transported with the same density distribution by a constant vector field $\mathbf{c}$. This model is governed by the following Partial Differential Equation:



$$\frac{\partial u}{\partial t} = \mathbf{c} \cdot \nabla u$$



## Examples



Consider a Diffusion Convection model in 2 dimension with the vector $\mathbf{c}$ having equal vector components and $D$ is constant.



Let $\mathbf{c} = [-0.1, -0.1]$ and $D = 0.009$. (Neumann boundary condition and 3d animations can be found in  [Two dimensional examples](examples/Two_Dimensional_Models.ipynb))



### Diffusion Convection Model



Solution:






### Diffusion Model



Solution:






### Convection Model



Solution:






## Applications



Some applications of these models include:



1. Heat Transfer or cooling of a system.



2. Tranportation of a fluids density distribution that is flowing uniformly.



3. Fokker Plank Equation of a Stochastic Differential Equation with uniform drift and standard deviation.