fd1d_advection_lax, a MATLAB 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.

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/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.
```

The Lax method modifies the FTCS approximation to the time derivative:

• du/dt = (u(t+dt,x)-0.5*u(t,x-dx)-0.5*u(t,x+dx))/dt
• du/dx = (u(t,x+dx)-u(t,x-dx))/2/dx

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.

### Languages:

fd1d_advection_lax is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.

### Related Data and Programs:

fd1d_advection_diffusion_steady, a MATLAB code which applies the finite difference method to solve the steady advection diffusion equation v*ux-k*uxx=0 in one spatial dimension, with constant velocity v and diffusivity k.

fd1d_advection_ftcs, a MATLAB 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_wendroff, a MATLAB 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 to treat the time derivative.

fd1d_burgers_lax, a MATLAB 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 MATLAB code which applies the finite difference method to a two point boundary value problem in one spatial dimension.

fd1d_heat_explicit, a MATLAB code which uses the finite difference method and explicit time stepping to solve the time dependent heat equation in 1D.

fd1d_heat_implicit, a MATLAB code which uses the finite difference method and implicit time stepping to solve the time dependent heat equation in 1D.

fd1d_heat_steady, a MATLAB code which uses the finite difference method to solve the steady (time independent) heat equation in 1D.

fd1d_predator_prey, a MATLAB code which implements a finite difference algorithm for predator-prey system with spatial variation in 1D.

fd1d_wave, a MATLAB code which applies the finite difference method to solve the time-dependent wave equation utt = c * uxx in one spatial dimension.

### 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 12 January 2019.