diffusion_pde

diffusion_pde, a Python code which solves the diffusion partial differential equation (PDE) dudt = mu * d2udx2 in one spatial dimension and time, with a constant diffusion coefficient mu, and periodic boundary conditions, using the forward time centered space (FTCS) difference method.

We solve for u(x,t), the solution of the constant-velocity diffusion equation in 1D,

        du/dt - mu d2u/dx2 = 0
      

        0.0 <= x <= 1.0
      

        mu = 0.5
      

        u(0,t) = u(1,t) for all t
      

        u(x,0) = (10x-6)^2 (8-10x)^2 for 0.6 <= x <= 0.8
               = 0                   elsewhere.
      

We use a method known as FTCS (forward time, centered space):

• FT: du/dt approximated at (x,t) by (u(x,t+dt)-u(x,t))/dt
• CS: d2u/dx2 approximated at (x,t) by (u(x+dx,t)-2u(x,t)+u(x-dx,t))/2/dx^2

Licensing:

The information on this web page is distributed under the MIT license.

Languages:

diffusion_pde is available in a MATLAB version and an Octave version and a Python version.

Related Data and codes:

allen_cahn_pde, a Python code which sets up and solves the Allen-Cahn reaction-diffusion system of partial differential equations (PDE) in 1 space dimension and time.

References:

1. Willem Hundsdorfer, Jan Verwer,
   Numerical solution of time-dependent advection-diffusion-reaction equations,
   Springer, 2003
   ISBN: 978-3-662-09017-6

Source Code:

• diffusion_pde.py, the source code.
• diffusion_pde.sh, runs all the tests.
• diffusion_pde.txt, the output file.

• diffusion_pde_ftcs_initial.png, the initial condition.
• diffusion_pde_ftcs_final.png, the approximated solution at the final time.
• diffusion_pde_ftcs_conservation.png, the conserved quantity over time.