A simple Finite volume tool
This code is the result of the efforts of a chemical/petroleum engineer to develop a simple tool to solve the general form of convectiondiffusion equation:
α∂ϕ/∂t+∇.(uϕ)+∇.(−D∇ϕ)+βϕ=γ
on simple uniform mesh over 1D, 1D axisymmetric (radial), 2D, 2D axisymmetric (cylindrical), and 3D domains.
The code accepts Dirichlet, Neumann, and Robin boundary conditions (which can be achieved by changing a, b, and c in the following equation) on a whole or part of a boundary:
a∇ϕ.n+bϕ=c.
It also accepts periodic boundary conditions.
The main purpose of this code is to serve as a handy tool for those who try to play with mathematical models, solve the model numerically in 1D, compare it to analytical solutions, and extend their numerical code to 2D and 3D with the minimum number of modifications in the 1D code.
The discretizaion schemes include
* central difference
* upwind scheme for convective terms
* TVD schemes for convective terms
A short html document is available to get you started. Simply type
showdemo FVTdemo
after downloading and extracting the code.
A few calculus functions (divergence, gradient, etc) and averaging techniques (arithmetic average, harmonic average, etc) are available, which can be helpful specially for solving nonlinear or coupled equations or implementing explicit schemes.
I have used the code to solve coupled nonlinear systems of PDE. You can find some of them in the Examples/advanced folder.
There are a few functions in the 'PhysicalProperties' folder for the calculation of the physical properties of fluids. Some of them are not mine, which is specified inside the file.
I'll try to update the documents regularly, in the github repository. Please give me your feedback/questions by writing a comment in my weblog: <http://fvt.simulkade.com/>
Special thanks: I vastly benefited from the ideas behind Fipy <http://www.ctcms.nist.gov/fipy/>, a pythonbased finite volume solver.
To start the solver, download and extract the zip archive, open and run 'FVToolStartUp' function.
To see the code in action, copy and paste the following in your Matlab command window:
clc; clear;
L = 50; % domain length
Nx = 20; % number of cells
m = createMesh3D(Nx,Nx,Nx, L,L,L);
BC = createBC(m); % all Neumann boundary condition structure
BC.left.a(:) = 0; BC.left.b(:)=1; BC.left.c(:)=1; % Dirichlet for the left boundary
BC.right.a(:) = 0; BC.right.b(:)=1; BC.right.c(:)=0; % Dirichlet for the right boundary
D_val = 1; % value of the diffusion coefficient
D = createCellVariable(m, D_val); % assign the diffusion coefficient to the cells
D_face = harmonicMean(m, D); % calculate harmonic average of the diffusion coef on the cell faces
Mdiff = diffusionTerm(m, D_face); % matrix of coefficients for the diffusion term
[Mbc, RHSbc] = boundaryCondition(m, BC); % matix of coefficients and RHS vector for the BC
M = Mdiff + Mbc; % matrix of cefficients for the PDE
c = solvePDE(m,M, RHSbc); % send M and RHS to the solver
visualizeCells(m, c); % visualize the results
You can find some animated results of this code in my youtube channel:
https://www.youtube.com/user/processsimulation/videos
