jburkardt

FD1D_BVP\ Finite Difference Method\ 1D Boundary Value Problem {#fd1d_bvp-finite-difference-method-1d-boundary-value-problem align=”center”} =========================

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

The boundary value problem (BVP) that is to be solved has the form:

        - d/dx ( a(x) * du/dx ) + c(x) * u(x) = f(x)

in the interval X[0] < x < X[N-1]. The functions a(x), c(x), and f(x) are given functions, and a formula for a’(x) is also available.

Boundary conditions are applied at the endpoints, and in this case, these are assumed to have the form:

        u(X[0]) = 0.0;
        u(X[N-1]) = 0.0.

To compute a finite difference approximation, a set of n nodes is defined over the interval, and, at each interior node, a discretized version of the BVP is written, with u’‘(x) and u’(x) approximated by central differences.

Usage: {#usage align=”center”}

fd1d_bvp ( n, a, aprime, c, f, x, u )

where

• n the number of nodes.
• a the function which evaluates a(x);
• aprime the function which evaluates a’(x);
• c the function which evaluates c(x);
• f the function which evaluates f(x).
• x the vector of n nodes, which may be nonuniformly spaced.
• u the output vector of solution values at the nodes.

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_BVP 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_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_DISPLAY, a MATLAB program which reads a pair of files defining a 1D finite difference model, and plots the data.

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

FEM1D, a C++ program which applies the finite element method to a linear two point boundary value problem in a 1D region.

FEM1D_BVP_LINEAR, a C++ program which applies the finite element method, with piecewise linear elements, to a two point boundary value problem in one spatial dimension.

Reference: {#reference align=”center”}

1. Dianne O’Leary,\ Finite Differences and Finite Elements: Getting to Know You,\ Computing in Science and Engineering,\ Volume 7, Number 3, May/June 2005.
2. Dianne O’Leary,\ Scientific Computing with Case Studies,\ SIAM, 2008,\ ISBN13: 978-0-898716-66-5,\ LC: QA401.O44.
3. Hans Rudolf Schwarz,\ Finite Element Methods,\ Academic Press, 1988,\ ISBN: 0126330107,\ LC: TA347.F5.S3313..
4. Gilbert Strang, George Fix,\ An Analysis of the Finite Element Method,\ Cambridge, 1973,\ ISBN: 096140888X,\ LC: TA335.S77.
5. Olgierd Zienkiewicz,\ The Finite Element Method,\ Sixth Edition,\ Butterworth-Heinemann, 2005,\ ISBN: 0750663200,\ LC: TA640.2.Z54

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

• fd1d_bvp.cpp, the source code.
• fd1d_bvp.hpp, the include file.

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

• fd1d_bvp_prb.cpp, a sample calling program.

• fd1d_bvp_prb_output.txt, the output file.

• fd1d_bvp_test01_nodes.txt, the nodes for test 1.
• fd1d_bvp_test01_values.txt, the computed and exact values for test 1.

• fd1d_bvp_test01.png, a PNG image of a plot of the computed and exact values.

• fd1d_bvp_test02_nodes.txt, the nodes for test 2.
• fd1d_bvp_test02_values.txt, the computed and exact values for test 2.

• fd1d_bvp_test02.png, a PNG image of a plot of the computed and exact values.

• fd1d_bvp_test03_nodes.txt, the nodes for test 3.
• fd1d_bvp_test03_values.txt, the computed and exact values for test 3.

• fd1d_bvp_test03.png, a PNG image of a plot of the computed and exact values.

• fd1d_bvp_test04_nodes.txt, the nodes for test 4
• fd1d_bvp_test04_values.txt, the computed and exact values for test 4.

• fd1d_bvp_test04.png, a PNG image of a plot of the computed and exact values.

• fd1d_bvp_test05_nodes.txt, the nodes for test 5.
• fd1d_bvp_test05_values.txt, the computed and exact values for test 5.
• fd1d_bvp_test05.png, a PNG image of a plot of the computed and exact values.

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

• FD1D_BVP solves a two point boundary value problem.
• R8_ABS returns the absolute value of an R8.
• R83NP_FS factors and solves an R83NP system.
• R8MAT_WRITE writes an R8MAT file.
• R8VEC_EVEN returns N real values, evenly spaced between ALO and AHI.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

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

Last revised on 17 May 2009.