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
      
over the interval:
        0.0 <= x <= 1.0
      
with constant diffusion coefficient:
        mu = 0.5
      
with periodic boundary conditions:
        u(0,t) = u(1,t) for all t
      
and initial condition
        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):

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.

gray_scott_pde, a Python code which solves the partial differential equation (PDE) known as the Gray-Scott reaction diffusion equation, in two spatal dimension and time, displaying a sequence of solutions as time progresses.

spiral_pde, a Python code which solves a pair of reaction-diffusion partial differential equations (PDE) over a rectangular domain with periodic boundary condition, whose solution is known to evolve into a pair of spirals.

wave_pde, a Python code which uses the finite difference method (FDM) in space, and the method of lines in time, to set up and solve the partial differential equations (PDE) known as the wave equations, utt = c uxx, in one spatial 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:


Last revised on 05 September 2024.