Unconditional stability preserving L2-1σ reduced-dimension finite difference methods for Caputo-type time-fractional diffusion equations

This paper develops unconditional stability preserving reduced-dimension finite difference (RDFD) schemes for solving Caputo-type time-fractional diffusion equations (FDEs) of order α ∈ (0, 1) in one and two spatial dimensions. The Caputo derivative is discretized by the L2-1 σ formula, and spatial derivatives are approximated using the central finite difference (FD), forming the L2-1 σ -FD schemes as the foundational numerical frameworks. To reduce the computational complexity of high-dimensional FDEs, we further develop L2-1 σ -RDFD schemes by integrating proper orthogonal decomposition (POD), achieving efficient dimension reduction while preserving numerical accuracy. We rigorously prove the unconditional stability and optimal convergence order of the proposed RDFD schemes. Numerical experiments confirm the theoretical results and demonstrate that the RDFD schemes significantly reduce computational cost without sacrificing accuracy compared to the original FD schemes. The proposed methods offer a general and efficient framework for numerically solving fractional diffusion models and are readily extensible to more complex fractional partial differential equations arising in interdisciplinary fields such as computational physics, engineering mechanics, and mathematical biology.