fd1d_advection_lax, 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 Lax method for the time derivative, writing graphics files for processing by gnuplot.
The Lax method is an improvement to the FTCS method. The FTCS method is always unstable; nonphysical oscillations appear and grow. The Lax method is stable, if the time step is small enough; however, it does cause the wave to gradually spread out and flatten.
We solve the constant-velocity advection equation in 1D,
du/dt = - c du/dxover the interval:
0.0 <= x <= 1.0with 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.
The Lax method modifies the FTCS approximation to the time derivative:
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. Unlike in the case of the FTCS approach, we do not see unstable oscillations. However, instead, we definitely find the wave gradually flattening out.
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_lax is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.
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