A Convergent BDF2-SAV Scheme for Fractional Cahn-Hilliard Equations with High-Dimensional Integral Fractional Laplacian

Numerical simulation of the multi-dimensional space-fractional Cahn-Hilliard equation faces two main computational challenges: the inherent temporal accuracy limitations of standard scalar auxiliary variable (SAV) methods and the escalating computational cost in high-dimensional domains. To address these issues, this study constructs a fully discrete algorithmic framework integrating a second-order backward differentiation formula (SAV-BDF2) with a sixth-order centered difference scheme. Under this formulation, we rigorously prove unconditional energy stability and establish the theoretical validity of the dual temporal and spatial accuracy. To solve the resulting indefinite algebraic systems, a minimal residual solver is paired with a sine-transform block diagonal preconditioner. Additionally, a hardware-level Vectorized Tensor Processing (VTP) architecture is deployed to resolve cache thrashing caused by non-contiguous memory access during multidimensional tensor evaluations. Numerical experiments in 3D to 8D domains demonstrate that the framework improves memory throughput and reduces execution time. By avoiding standard hardware execution inefficiencies, this integrated strategy provides an efficient numerical solution for large-scale simulations of high-dimensional fractional systems.