Efficient and High-Order Numerical Approaches for Variable-Order Fractional Sub-Diffusion Equations

Abstract

This thesis proposes a unified, high-order, and computationally efficient framework for solving multi- dimensional variable-order (VO) fractional sub-diffusion equations. The primary challenge in these problems is the O(N2 t ) computational cost and memory-driven nonlocality of fractional operators, cou- pled with complex accuracy issues. Our framework overcomes these challenges by integrating three key components: (1) a novel high-order temporal discretization based on a VO extension of the L2-1σ scheme, (2) a Shifted Binary Block Partition (SBBP) algorithm for fast computation, and (3) a Weighted Alternating-Direction Implicit (WADI) scheme for multi-dimensional efficiency. The key theoretical contributions of this work are three-fold. First, we establish a rigorous stability analysis for the proposed SBBP-based fast method, defining the necessary conditions on the polynomial approximation’s accuracy to ensure the stability of the full sub-diffusion scheme. Second, we identify and resolve a critical accuracy degradation issue inherent in the VO-L2-1σ method, where its non- uniform nodes can corrupt the global convergence order. We propose a novel approach, restricting this operator to the initial time steps, which is shown to maintain the operator’s high accuracy while restoring the desired global convergence of the full PDE solution. Third, the framework’s accuracy is further advanced to the fourth order in space using a Compact Finite Difference (CFD) scheme, which is efficiently coupled with a corresponding Compact ADI solver. Numerical experiments are conducted to validate the theoretical analysis. The results demonstrate that our proposed integrated framework (e.g., the Fast WADI scheme) achieves significantly higher accu- racy and possesses a superior computational efficiency, with lower CPU times compared to traditional L1-ADI and other benchmark methods. This work establishes a robust, fast, and high-order methodol- ogy, providing a valuable tool for accurately simulating complex physical processes governed by VO fractional dynamics.