FD1D_ADVECTION_LAX_WENDROFF is a Python 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, creating a graphics file using matplotlib.

Note that, because the graphics program needs the 3D option, this program will NOT run with older versions of Python...like the one we have installed on our common system!

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/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.
```

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.

### Languages:

FD1D_ADVECTION_LAX_WENDROFF is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python version.

### Related Data and Programs:

FD1D_HEAT_EXPLICIT, a Python program which uses the finite difference method and explicit time stepping to solve the time dependent heat equation in 1D.

FD1D_HEAT_IMPLICIT, a Python program which uses the finite difference method and implicit time stepping to solve the time dependent heat equation in 1D.

FD2D_HEAT_STEADY, a Python program which uses the finite difference method (FDM) to solve the steady (time independent) heat equation in 2D.

### Reference:

1. George Lindfield, John Penny,
Numerical Methods Using MATLAB,
Second Edition,
Prentice Hall, 1999,
ISBN: 0-13-012641-1,
LC: QA297.P45.

### Source Code:

You can go up one level to the Python source codes.

Last revised on 07 January 2015.