advection_pde, a Python code which solves the advection partial differential equation (PDE) dudt + c * dudx = 0 in one spatial dimension and time, 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
over the interval:
0.0 <= x <= 1.0
with constant velocity:
c = 1
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):
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.
The information on this web page is distributed under the MIT license.
advection_pde is available in a MATLAB version and an Octave version and a Python version.
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