Solving the Two-Dimensional Poisson Equation by the Finite-Difference Method

(A comparative study of some simple methods in Python)

Abstract

Simulating incompressible lid-driven cavity flow requires solving the Navier–Stokes equations, where the pressure Poisson solve is often the dominant computational cost in pressure–velocity coupling schemes. This motivates Poisson solvers that are straightforward to implement, fast, and sufficiently accurate. In this work, we solve the two-dimensional Poisson equation on a
square grid using a second-order five-point finite-difference discretization with homogeneous Neumann boundary conditions. We benchmark representative direct methods (dense-matrix LU factorization, sparse-matrix LU factorization, and a discrete cosine transform (DCT) approach) against simple stationary iterative methods (Jacobi, Gauss–Seidel, and successive overrelaxation).
All solvers are implemented in Python 3.12 using NumPy, SciPy, and Numba in a Jupyter Notebook/Lab workflow on an Intel i7 quad-core laptop (4 CPUs, 8 threads, 2.3GHz base) with 16GB RAM under Windows. In our tests, the DCT-based solver is the fastest and most memory-efficient among the direct methods, followed by sparse LU. Among the iterative methods, Numba-accelerated SOR is the fastest when the relaxation parameter is chosen near its optimal value, but it remains slower and less accurate than the direct solvers considered here. The novelty of this work is a transparent, tool-specific benchmark of these standard methods on clearly documented commodity hardware, yielding concrete runtime and memory comparisons that can be reused in similar studies. Our focus is the practical selection of an easy-to-use Poisson solver that remains performant;
integrating the selected solver into full incompressible cavity-flow simulations is left for future work.