FD1D_ADVECTION_FTCS
Finite Difference Method
1D Advection Equation
Forward Time Difference, Centered Space Difference

FD1D_ADVECTION_FTCS is a C++ 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 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
      

        0.0 <= x <= 1.0
      

        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.

Licensing:

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

Languages:

FD1D_ADVECTION_FTCS is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

FD1D_ADVECTION_LAX, a C++ 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 method to treat the time derivative.

FD1D_BURGERS_LAX, a C++ program 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_BURGERS_LEAP, a C++ program which applies the finite difference method and the leapfrog approach to solve the non-viscous time-dependent Burgers equation in one spatial dimension.

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

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

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

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

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

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

GNUPLOT, C++ programs 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:

• fd1d_advection_ftcs.cpp, the source code.
• fd1d_advection_ftcs.sh, BASH commands to compile the source code.

Examples and Tests:

• fd1d_advection_ftcs_output.txt, the output file.

Graphical output for this program was created using GNUPLOT. Data at selected time steps was written to a "data" file, and the appropriate GNUPLOT commands were written to a "command" file. The plot can be created by the command

        gnuplot < advection_commands.txt
      

• advection_data.txt, the solution data.
• advection_commands.txt, gnuplot commands to plot the data.
• advection.png, a (not very satisfactory) image of the solution.

List of Routines:

• FD1D_ADVECTION_FTCS solves the advection equation using the FTCS method.
• GET_UNIT returns a free FORTRAN unit number.
• I4_MODP returns the nonnegative remainder of I4 division.
• I4_WRAP forces an I4 to lie between given limits by wrapping.
• INITIAL_CONDITION sets the initial condition.
• R8VEC_LINSPACE creates a vector of linearly spaced values.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the C++ source codes.