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FD1D_ADVECTION_DIFFUSION_STEADY\ Finite Difference Method\ Steady 1D Advection Diffusion Equation {#fd1d_advection_diffusion_steady-finite-difference-method-steady-1d-advection-diffusion-equation align=”center”} ======================================

FD1D_ADVECTION_DIFFUSION_STEADY is a C++ program 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, creating graphics files for processing by GNUPLOT.

We solve the steady constant-velocity advection diffusion equation in 1D,

        v du/dx - k d^2u/dx^2

over the interval:

        0.0 <= x <= 1.0

with boundary conditions

        u(0) = 0, u(1) = 1.

We do this by discretizing the interval [0,1] into NX nodes. We write the boundary conditions at the first and last nodes. At the i-th interior node, we replace derivatives by differences:

• du/dx = (u(x+dx)-u(x-dx))/2/dx
• d\^2u/dx\^2 = (u(x+dx)-2u(x)+u(x-dx))/dx/dx

This becomes a set of NX linear equations in the NX unknown values of U.

The exact solution to this differential equation is known:

        u = ( 1 - exp ( r * x ) ) / ( 1 - exp ( r ) )

where r = v * l / k;

Licensing: {#licensing align=”center”}

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

Languages: {#languages align=”center”}

FD1D_ADVECTION_DIFFUSION_STEADY 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: {#related-data-and-programs align=”center”}

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_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_ADVECTION_LAX_WENDROFF, 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-Wendroff method to treat the time derivative.

Reference: {#reference align=”center”}

1. Charles Hall, Thomas Porsching,\ Numerical Analysis of Partial Differential Equations,\ Prentice-Hall, 1990,\ ISBN: 013626557X,\ LC: QA374.H29.

Source Code: {#source-code align=”center”}

• fd1d_advection_diffusion_steady.cpp, the source code.

Examples and Tests: {#examples-and-tests align=”center”}

• fd1d_advection_diffusion_steady_output.txt, the output file.
• fd1d_advection_diffusion_steady_commands.txt, Gnuplot commands.
• fd1d_advection_diffusion_steady_data.txt, data for Gnuplot.
• fd1d_advection_diffusion_steady.png, a PNG image of the solution.

List of Routines: {#list-of-routines align=”center”}

• FD1D_ADVECTION_DIFFUSION_STEADY solves steady advection diffusion equation.
• R8VEC_LINSPACE_NEW creates a vector of linearly spaced values.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• TRISOLVE factors and solves a tridiagonal system.

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

Last revised on 03 May 2014.