# JFVM

## IMPORTANT NOTES

- The code now works with Julia 1.0. All you need to do is to check out the master branch:

```
] add https://github.com/simulkade/JFVM.jl
```

- 3D visualization requires calling Mayavi via PyCall. It made too many problems recently, so I have decided to disable it until I find a better solution for 3D visualization. Suggestions/PRs are very welcome.
- I have decided to move the visualization to a new package JFVMvis.jl, that ypu need to install by:

```
] add https://github.com/simulkade/JFVMvis.jl
```

## Equations

You can solve the following PDE (or a subset of it):

with the following boundary conditions:

Believe it or not, the above equations describe the majority of the transport phenomena in chemical and petroleum engineering and similar fields.

## A simple finite volume tool written in Julia

This code is a Matlabesque implementation of my Matlab finite volume tool. The code is not in its most beautiful form, but it works if you believe my words. Please remember that the code is written by a chemical/petroleum engineer. Petroleum engineers are known for being simple-minded folks and chemical engineers have only one rule: "any answer is better than no answer". You can expect to easily discretize a linear transient advection-diffusion PDE into the matrix of coefficients and RHS vectors. Domain shape is limited to rectangles, circles (or a section of a circle), cylinders, and soon spheres. The mesh can be uniform or nonuniform:

- Cartesian (1D, 2D, 3D)
- Cylindrical (1D, 2D, 3D)
- Radial (2D r and \theta)

You can have the following boundary conditions or a combination of them on each boundary:

- Dirichlet (constant value)
- Neumann (constant flux)
- Robin (a linear combination of the above)
- Periodic (so funny when visualize)

It is relatively easy to use the code to solve a system of coupled linear PDE's and not too difficult to solve nonlinear PDE's.

## Installation

You need to have matplotlib (only for visualization)

### Linux

In Ubuntu-based systems, try

```
sudo apt-get install python-matplotlib
```

Then install `JFVM`

by the following commands:

```
] add https://github.com/simulkade/JFVM.jl
```

### Windows

- open
`Julia`

and type

```
] add https://github.com/simulkade/JFVM.jl
```

- For visualization, download and install Anaconda

Run`anaconda command prompt`

(as administrator) and install`matplotlib`

by

```
conda install matplotlib
```

Please let me know if it does not work on your windows machines.

## Tutorial

I have written a short tutorial, which will be extended gradually.

## In action

Copy and paste the following code to solve a transient diffusion equation:

```
using JFVM, JFVMvis
Nx = 10
Lx = 1.0
m = createMesh1D(Nx, Lx)
BC = createBC(m)
BC.left.a[:]=BC.right.a[:]=0.0
BC.left.b[:]=BC.right.b[:]=1.0
BC.left.c[:]=1.0
BC.right.c[:]=0.0
c_init = 0.0 # initial value of the variable
c_old = createCellVariable(m, 0.0, BC)
D_val = 1.0 # value of the diffusion coefficient
D_cell = createCellVariable(m, D_val) # assigned to cells
# Harmonic average
D_face = harmonicMean(D_cell)
N_steps = 20 # number of time steps
dt= sqrt(Lx^2/D_val)/N_steps # time step
M_diff = diffusionTerm(D_face) # matrix of coefficient for diffusion term
(M_bc, RHS_bc)=boundaryConditionTerm(BC) # matrix of coefficient and RHS for the BC
for i =1:5
(M_t, RHS_t)=transientTerm(c_old, dt, 1.0)
M=M_t-M_diff+M_bc # add all the [sparse] matrices of coefficient
RHS=RHS_bc+RHS_t # add all the RHS's together
c_old = solveLinearPDE(m, M, RHS) # solve the PDE
end
figure(figsize=(5,6))
visualizeCells(c_old)
colorbar()
```

Now change the 4th line to `m=createMesh2D(Nx, Nx, Lx, Lx)`

and see this:

## More examples

TO DO

## IJulia notebooks

- Introduction
- tutorial
- compare analytical solution of a diffusion equation with uniform and nonuniform grids
- Nonlinear PDE
- Foam flow in porous media
- New notebooks soon...

# How to cite

If you have used the code in your research, please cite it as

Ali A Eftekhari. (2017, August 23). JFVM.jl: A Finite Volume Tool for Solving Advection-Diffusion Equations. Zenodo. http://doi.org/10.5281/zenodo.847056

```
@misc{ali_a_eftekhari_2017_847056,
author = {Ali A Eftekhari},
title = {{JFVM.jl: A Finite Volume Tool for Solving
Advection-Diffusion Equations}},
month = aug,
year = 2017,
doi = {10.5281/zenodo.847056},
url = {https://doi.org/10.5281/zenodo.847056}
}
```