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 FTCS method, forward time difference, centered space difference, writing graphics files for processing by gnuplot.

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:

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 to treat the time derivative.

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.

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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, C++ codes which illustrate how a program can write data and command files so that gnuplot can create plots of the program results.

### 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 28 February 2020.