advection_ftcs_pde

advection_ftcs_pde, a MATLAB code which solves the advection PDE dudt + c * dudx = 0 in one spatial dimension, with a constant velocity c, and periodic boundary conditions, using the FTCS method, forward time difference, centered space difference.

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

        du/dt + c du/dx = 0
      

        0.0 <= x <= 1.0
      

        c = 1
      

        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: du/dx approximated at (x,t) by (u(x+dx,t)-u(x-dx,t))/2/dx

The advection velocity is constant in time and space. Therefore, (given our periodic boundary conditions), the solution should simply move smoothly from left to right, returning on the left again. However, the FTCS numerical method is unstable for this advection problem. Hence, the numerical solution will include erroneous oscillations which spread and blow up over time. There are more sophisticated methods for the advection problem, which do not exhibit this instability.

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 advection-diffusion-reaction equations,
   Springer, 2003
   ISBN: 978-3-662-09017-6

Source Code:

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

Plots:

• advection_ftcs_pde_initial.png, the initial condition.
• advection_ftcs_pde_final.png, the approximated solution at the final time.
• advection_ftcs_pde_conserved.png, the conserved quantity over time.
• advection_ftcs_pde.avi, an animation of the sequence of solutions.