fd1d_advection_ftcs, a C++ code which applies the finite difference method to solve the time-dependent advection equation ut = - c * ux in one spatial dimension, with a constant velocity, using the FTCS method, forward time difference, centered space difference, writing graphics files for processing by gnuplot.
We solve the constant-velocity advection equation in 1D,
du/dt = - c du/dx
over the interval:
0.0 <= x <= 1.0
with periodic boundary conditions, and
with a given initial condition
u(0,x) = (10x-4)^2 (6-10x)^2 for 0.4 <= x <= 0.6
= 0 elsewhere.
We use a method known as FTCS:
The FTCS method is unstable for the advection problem. One purpose of this example is to demonstrate that fact.
For our simple case, 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. Instead, because of the instabilities, we see that the solution quickly becomes dominated by erroneous oscillations.
There are more sophisticated methods for the advection problem, which do not exhibit this behavior.
The computer code and data files described and made available on this web page are distributed under the MIT license
fd1d_advection_ftcs is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.
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