An Incremental SVD Method for Non-Fickian Flows in Porous Media: Addressing Storage and Computational Challenges

Abstract: It is well known that the numerical solution of the Non-Fickian flows at the current stage depends on all previous time instances. Consequently, the storage requirement increases linearly, while the computational complexity grows quadratically with the number of time steps. This presents a significant challenge for numerical simulations. While numerous existing methods address this issue, our proposed approach stems from a data science perspective and maintains uniformity. Our method relies solely on the rank of the solution data, dissociating itself from dependency on any specific partial differential equation (PDE). In this paper, we make the assumption that the solution data exhibits approximate low rank. Here, we present a memory-free algorithm, based on the incremental SVD technique, that exhibits only linear growth in computational complexity as the number of time steps increases. We prove that the error between the solutions generated by the conventional algorithm and our innovative approach lies within the scope of machine error. Numerical experiments are showcased to affirm the accuracy and efficiency gains in terms of both memory usage and computational expenses.