Quantitative Analysis of OpenMP Parallelization Efficiency for the Diffusion Equation: An Experimental Study of Scalability and Critical Thresholds

Abstract

Numerical simulation of partial differential equations using finite difference methods presents computational demands that grow rapidly with spatial resolution. This work quantitatively analyzes the efficiency of OpenMP parallelization for the one-dimensional diffusion equation solved with the FTCS scheme. A full factorial experimental design was implemented with mesh sizes from Nx=200 to Nx=20000 and thread configurations from 1 to 16. Results demonstrate the existence of a critical threshold at Nx=20000 where the first speedup greater than unity is achieved (1.05× with 4 threads). For small problems (Nx=200), parallelization is counterproductive with relative overhead up to 118×. Efficiency decreases monotonically from 47.3% (2 threads) to 2.5% (16 threads). Using Amdahl's Law, a parallelizable fraction of 99.8% was estimated. Numerical precision is preserved between serial and parallel versions up to the order of 10⁻¹⁴. This work provides evidence-based practical guidelines for efficient parallel implementations in scientific computing.