An efficient and memory free algorithm for subdiffusion equation using incremental singular value decomposition

Abstract

In this paper, we address the well-known challenge in the numerical solution of time-fractional partial differential equations (TFPDEs), namely, that the dependence on all previous time levels leads to storage requirements that grow linearly with the number of time steps. To overcome this difficulty, we develop an efficient algorithm based on incremental singular value decomposition (ISVD), which avoids the excessive memory demands associated with storing the full solution history. A rigorous error analysis is established, and numerical experiments are presented to validate the theoretical results. Comparisons with the direct method and a representative fast evaluation method show that the proposed ISVD approach dramatically reduces memory usage relative to the direct method and remains competitive with the fast method over the tested parameter regimes.