poisson_1d, a Fortran90 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.
The information on this web page is distributed under the MIT license.
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.
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