GPU-ACCELERATED SOLUTION OF THE 2D POISSON EQUATION: A COMPARATIVE STUDY OF JACOBI AND RED-BLACK GAUSS–SEIDEL METHODS WITH GLOBAL AND SHARED MEMORY

Аннотация

The Poisson equation is a canonical elliptic partial differential equation (PDE) that arises in diverse fields such as electrostatics, fluid dynamics, and heat transfer. Analytical solutions exist only for simple geometries and boundary conditions, making numerical approaches indispensable for realistic problems. This study presents a comparative analysis of four GPU-accelerated CUDA solvers for the two-dimensional Poisson equation: the Jacobi method with global memory (Jacobi-Global), the Jacobi method with shared memory tiling (Jacobi-Shared), the Red-Black Gauss–Seidel method with global memory (RBGS-Global), and RBGS with shared memory tiling (RBGS-Shared). A finite difference discretization reduces the PDE to a large sparse system of linear equations, which is then solved iteratively. All GPU implementations are compared against a single-threaded CPU baseline across grid sizes from 64×64 to 4096×4096. Results show that GPU acceleration is ineffective for small grids due to kernel launch latency and memory transfer overheads, but becomes highly advantageous for larger problems. Jacobi-Shared achieves a peak speedup of 77.32× at 1024×1024, while RBGS-Shared converges in orders of magnitude fewer iterations and reaches 47.24× speedup at the same grid size. Shared memory consistently outperforms global memory, with execution time reductions of up to 2.4× for narrow block configurations at large grid sizes. These findings provide practical guidance for solver and memory strategy selection in GPU-based PDE computations.