poisson_1d


poisson_1d, an Octave code which solves a discretized version of the Poisson equation -uxx = f(x) on the interval a ≤ x ≤ b, with Dirichlet boundary conditions u(a) = ua, u(b) = ub. The linear system is solved using Gauss-Seidel iteration.

The 1D Poisson equation is assumed to have the form

        -u''(x) = f(x), for a < x < b
         u(a) = ua, u(b) = ub
      

Let K be a small positive integer called the mesh index, and let N = 2^K be the corresponding number of uniform subintervals into which [A,B] is divided. Assigning a value U(I) to each of the N+1 equally spaced nodes with coordinate X(I), we approximate the equation by

        -U(i-1) + 2 U(i) - U(i+1)
        -------------------------   = f( X(i) ), 1 < I < N+1
                  h^2

        U(1) = ua, U(N+1) = ub.
      

It remains to solve the linear system for the desired values of U. This could be done directly, or iteratively. An iterative method such as Jacobi, Gauss-Seidel or SOR might be suitable, but experience shows that the convergence rate of these iterative methods decreases drastically as the value of K is increased - that is, as a more refined and accurate answer is sought.

Licensing:

The information on this web page is distributed under the MIT license.

Languages:

poisson_1d is available in a C version and a C++ version and a Fortran77 version and a Fortran90 version and a FreeFem++ version and a MATLAB version and an Octave version and a Python version.

Related Data and Programs:

poisson_1d_test

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fd1d_bvp, an Octave code which applies the finite difference method to a two point boundary value problem in one spatial dimension.

gauss_seidel, an Octave code which uses the Gauss-Seidel iteration to solve a linear system with a symmetric positive definite (SPD) matrix.

jacobi, an Octave code which uses the Jacobi iteration to solving a system of linear equations with a symmetric positive definite (SPD) matrix.

mgmres, an Octave code which applies the restarted GMRES algorithm to solve a sparse linear system, by Lili Ju.

poisson_1d_multigrid, an Octave code which applies the multigrid method to a discretized version of the 1D Poisson equation.

poisson_2d, an Octave code which computes an approximate solution to the Poisson equation in a rectangle, using the finite difference method (FDM) and Jacobi iteration.

sor, an Octave code which implements the successive over-relaxation (SOR) method for the iterative solution of a linear system of equations.

Reference:

  1. William Hager,
    Applied Numerical Linear Algebra,
    Prentice-Hall, 1988,
    ISBN13: 978-0130412942,
    LC: QA184.H33.

Source Code:


Last revised on 02 October 2024.