JFVM

IMPORTANT NOTES

• JFVM is now prepared for Julia package registration and targets Julia 1.10+.
• Once registered in General, install it with:

] add JFVM

• Until registration is completed, install directly from this repository:

] add https://github.com/FiniteVolumeTransportPhenomena/JFVM.jl

• 3D visualization via Mayavi/PyCall is intentionally disabled in this package.
• Visualization support is maintained in JFVMvis.jl, which can be installed 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.

Coordinate systems

Each mesh carries an explicit coordinate-system type, and the discretization kernels dispatch on it (rather than on the legacy numeric dimension tag). The constructors are unchanged; the type is selected automatically:

Constructor	Coordinate system	dimension (legacy)
createMesh1D	Cartesian1D	1.0
createMeshCylindrical1D	Cylindrical1D	1.5
createMesh2D	Cartesian2D	2.0
createMeshCylindrical2D	Cylindrical2D	2.5
createMeshRadial2D	Radial2D	2.8
createMesh3D	Cartesian3D	3.0
createMeshCylindrical3D	Cylindrical3D	3.2

The coordinate system is available as m.coordinatesystem, with helpers is_1d(m), is_2d(m), is_3d(m). The numeric m.dimension field and the coordinatesystem_from_dimension / dimension_from_coordinatesystem bridges are retained for backward compatibility; prefer dispatching on the coordinate-system types in new code.

Linear solvers

solveLinearPDE and solveLinearPDE! accept a pluggable solver strategy (default behavior is unchanged):

solveLinearPDE(m, M, RHS)                              # DirectSolver() — sparse LU via `\`
solveLinearPDE(m, M, RHS; solver = DirectSolver())    # explicit default

# Reuse one factorization across a fixed-matrix transient loop (only RHS changes):
fs = FactorizedSolver(M)
for step in 1:nsteps
    phi = solveLinearPDE(m, M, RHS; solver = fs)      # M is factored once
    # ... update RHS ...
end

solveLinearPDE(m, M, RHS; solver = MUMPSSolver())     # optional MUMPS backend (import MUMPS first)

Installation

General registry (after registration)

Install from the Julia General registry:

] add JFVM

Direct from git (available now)

Install from this repository:

] add https://github.com/FiniteVolumeTransportPhenomena/JFVM.jl

Optional visualization package

If you need plotting/visualization workflows that were moved out of this package:

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

Examples and tutorial

The examples/ folder is organised into:

• examples/tutorial — short .jl scripts introducing the workflow,
• examples/cases — worked engineering problems (heat conduction, plug-flow reactor) that verify against analytical solutions,
• examples/legacy — the original Jupyter notebooks.

A short notebook tutorial is also available.

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
visualizeCells(c_old)

Now change the 4th line to m=createMesh2D(Nx, Nx, Lx, Lx) and see this:

More examples

TO DO

IJulia notebooks (legacy)

• 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}
}