JFVM

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 convection-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 and mayavi installed.

Linux

In Ubuntu-based systems, try

sudo apt-get install python-matplotlib mayavi2

Then install JFVM by the following commands. The second line pulls the latest (and recommended) version of JFVM:

Pkg.add("JFVM")
Pkg.checkout("JFVM")

Windows

There are a few issues with 3D visualization in windows right now. This is the workflow if you want to give it a try:

• Download and install Anaconda
• Run anaconda command prompt (as administrator) and install mayavi and wxpython:
  • conda install mayavi
  • conda install wxpython (Not necessary if you clone the latest version of JFVM)
• Install github for windows
• open Julia and type

Pkg.add("JFVM")
Pkg.checkout("JFVM")

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
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
    visualizeCells(c_old)
end

Now change the 4th line to m=createMesh2D(Nx,2*Nx, Lx,2*Lx) and see what happens.

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...