poisson_openmp, a FORTRAN90 code which computes an approximate solution to the Poisson equation in a rectangular region, using OpenMP to carry out the Jacobi iteration in parallel.
The version of Poisson's equation being solved here is
- ( d/dx d/dx + d/dy d/dy ) U(x,y) = F(x,y)over the rectangle 0 <= X <= 1, 0 <= Y <= 1, with exact solution
U(x,y) = sin ( pi * x * y )so that
F(x,y) = pi^2 * ( x^2 + y^2 ) * sin ( pi * x * y )and with Dirichlet boundary conditions along the lines x = 0, x = 1, y = 0 and y = 1. (The boundary conditions will actually be zero in this case, but we write up the problem as though we didn't know that, which makes it easy to change the problem later.)
We compute an approximate solution by discretizing the geometry, assuming that DX = DY, and approximating the Poisson operator by
( U(i-1,j) + U(i+1,j) + U(i,j-1) + U(i,j+1) - 4*U(i,j) ) / dx /dyAlong with the boundary conditions at the boundary nodes, we have a linear system for U. We can apply the Jacobi iteration to estimate the solution to the linear system.
OpenMP is used in this example to carry out the Jacobi iteration in parallel. Note that the Jacobi iteration can converge very slowly, and the slowness increases as the matrix gets bigger. Thus, if you must use the Jacobi iteration, parallelism can help you. But you might also find, at some point, that getting a better linear solver (even a non-parallel one!) would help you more.
The computer code and data files described and made available on this web page are distributed under the MIT license
poisson_openmp is available in a C version and a C++ version and a FORTRAN90 version.
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