A Comparison of Solving the Poisson Equation Using Several Numerical Methods in Matlab and Octave on the Cluster maya

Abstract

Systems of linear equations resulting from partial differential equations arise frequently in many phenomena such as heat, sound, and fluid flow. We apply the finite difference method to the Poisson equation with homogeneous Dirichlet boundary conditions. This yields in a system of linear equations with a large sparse system matrix that is a classical test problem for comparing direct and iterative linear solvers. We compare the performance of Gaussian elimination, three classical iterative methods, and the conjugate gradient method in both Matlab and Octave. Although Gaussian elimination is fastest and can solve large problems, it eventually runs out of memory. If very large problems need to be solved, the conjugate gradient method is available, but preconditioning is vital to keep run times reasonable. Both Matlab and Octave perform well with intermediate mesh resolutions; however, Matlab is eventually able to solve larger problems than Octave and runs moderately faster.