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 FTCS method, forward time difference, centered space difference.

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.
```

We use a method known as FTCS:

• FT: Forward Time : du/dt = (u(t+dt,x)-u(t,x))/dt
• CS: Centered Space: du/dx = (u(t,x+dx)-u(t,x-dx))/2/dx

The FTCS method is unstable for the advection problem. One purpose of this example is to demonstrate that fact.

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. Instead, because of the instabilities, we see that the solution quickly becomes dominated by erroneous oscillations.

There are more sophisticated methods for the advection problem, which do not exhibit this behavior.

### Languages:

fd1d_advection_ftcs is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.

### Related Data and Programs:

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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.

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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.

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### 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.