HYBRID SWEEP TECHNIQUES IN CAPUTO IMPLICIT SCHEMES: A NOVEL FRAMEWORK FOR SPACE-FRACTIONAL PDES STABILITY AND EFFICIENCY

Abstract

This study presents an advanced numerical framework for the efficient and stable solution of Space-Fractional Partial Differential Equations (SFPDEs), which are widely used to model anomalous transport and diffusion processes in complex systems. The proposed method integrates Caputo’s implicit discretization with novel hybrid sweep strategies—namely full-sweep (FS), half-sweep (HS), quarter-sweep (QS), and a newly introduced adaptive quarter-sweep (AQS) scheme. Each sweeping approach is designed to reduce computational burden while maintaining numerical accuracy and stability.To further enhance convergence performance, the framework incorporates preconditioned iterative solvers, specifically the Preconditioned Successive Overrelaxation (PSOR) and Preconditioned Accelerated Overrelaxation (PAOR) methods. These solvers modify the spectral properties of the underlying system matrix, thereby accelerating convergence without compromising solution quality.Comprehensive numerical experiments are conducted on one- and two-dimensional benchmark SFPDEs, spanning matrix sizes from 128 to 2048. Results consistently demonstrate the superiority of the AQS-PAOR combination in terms of convergence speed, computational efficiency, and stability under long-time integration. For large-scale systems, the proposed scheme achieves up to 35% reduction in execution time compared to traditional QS-PSOR methods, with only marginal differences in error norms.The framework offers scalability, robustness, and potential applicability to higher-dimensional or nonlinear SFPDEs, making it suitable for real-time simulation and embedded computing environments. This work bridges the gap between classical integer-order sweeping techniques and fractional-order numerical schemes, setting the stage for further advances in adaptive solvers for complex dynamical systems.