Computational Performance Optimization in Solving Singular Boundary Value Problems: A Comparative Study of Finite Difference and Collocation Methods

This paper presents a systematic comparative study of two widely used numerical solvers --- HOFiD_bvp (high-order finite difference scheme) and bvp۴c (collocation-based) --- for solving singular second-order ordinary differential equations (ODEs) with first-kind (regular) boundary singularities. Four representative benchmark problems drawn from fluid dynamics, materials science, and radially symmetric diffusion models are used to evaluate solver performance across key metrics: maximum residual, maximum error, mesh point count, and ODE/BC function call counts. Results show that HOFiD_bvp consistently achieves lower residuals and errors with fewer function evaluations, making it computationally more efficient. Conversely, bvp۴c demonstrates superior robustness for nonlinear singular problems and offers better adaptive mesh refinement capabilities. These findings provide practical guidance for selecting the appropriate numerical technique in applied science and engineering contexts, with implications for optimization of computational simulation workflows.

کلیدواژه ها:

Singular boundary value problem ، Computational performance optimization ، Finite difference method ، Collocation method ، Adaptive mesh refinement