FD1D_ADVECTION_LAX
Finite Difference Method
1D Advection Equation
Lax Method

FD1D_ADVECTION_LAX 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 Lax method for the time derivative, writing graphics files for processing by gnuplot.

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
      

        0.0 <= x <= 1.0
      

        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.

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

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_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_lax.cpp, the source code.

Examples and Tests:

• fd1d_advection_lax_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_lax.png, a (not very satisfactory) image of the solution.

List of Routines:

• FD1D_ADVECTION_LAX solves the advection equation using the Lax method.
• 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.