diffusion_ftcs_pde

diffusion_ftcs_pde, a MATLAB code which solves the diffusion PDE dudt - mu * d2udx2 = 0 in one spatial dimension, with a constant diffusion coefficient mu, and periodic boundary conditions, using FTCS, the forward time difference, centered space 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
      

        c = 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 computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

References:

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

Source Code:

• diffusion_ftcs_pde.m, the source code.
• diffusion_ftcs_pde.sh, runs all the tests.
• diffusion_ftcs_pde.txt, the output file.

Plots:

• diffusion_ftcs_pde_initial.png, the initial condition.
• diffusion_ftcs_pde_final.png, the approximated solution at the final time.
• diffusion_ftcs_pde_conserved.png, the conserved quantity over time.
• diffusion_ftcs_pde.avi, an animation of the sequence of solutions.