FD1D_ADVECTION_LAX_WENDROFF is a C program 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-Wendroff method for the time derivative, writing graphics files for processing by gnuplot.
The Lax-Wendroff method is a modification to the Lax method with improved accuracy.
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.
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. While the Lax method produces an artificial smearing of the solution because of an artificial viscosity effect, this behavior is much reduced for the Lax-Wendroff method.
The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.
FD1D_ADVECTION_LAX_WENDROFF is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version and a Python version.
FD1D_ADVECTION_FTCS, a C program 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 forward time, centered space (FTCS) difference method.
FD1D_ADVECTION_LAX, a C program 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.
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GNUPLOT, C programs which illustrate how a program can write data and command files so that gnuplot can create plots of the program results.
Graphical output for this program was created using GNUPLOT. Data at selected time steps was written to a "data" file, and the appropriate GNUPLOT commands were written to a "command" file. The plot can be created by the command
gnuplot < advection_commands.txt
You can go up one level to the C source codes.