fd1d_advection_lax_wendroff, 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-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/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.

### Licensing:

The computer code and data files described and made available on this web page are distributed under the MIT license

### 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_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 forward time, centered space (FTCS) difference method.

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.

FD1D_BURGERS_LAX, a C code which applies the finite difference method and the Lax-Wendroff method to solve the non-viscous time-dependent Burgers equation in one spatial dimension.

FD1D_BVP, a C code which applies the finite difference method to a two point boundary value problem in one spatial dimension.

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

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

FD1D_HEAT_STEADY, a C code which uses the finite difference method to solve the steady (time independent) heat equation in 1D.

FD1D_PREDATOR_PREY, a C code which implements a finite difference algorithm for predator-prey system with spatial variation in 1D.

FD1D_WAVE, a C code which applies the finite difference method to solve the time-dependent wave equation utt = c * uxx in one spatial dimension.

gnuplot_test, examples which illustrate the use of the gnuplot graphics program.

### 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:

Last revised on 20 June 2019.