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, and to the best of our knowledge, it remains an unresolved issue. 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.

翻译：众所周知，当前阶段的非菲克流动数值解依赖于所有先前时间实例。因此，存储需求线性增长，而计算复杂度随时间步数的增加呈二次方增长。这给数值模拟带来了重大挑战，据我们所知，这一问题至今尚未解决。本文假设解数据具有近似低秩性。我们提出了一种基于增量SVD技术的无存储算法，该算法仅随时间步数增加呈现计算复杂度的线性增长。我们证明了传统算法与本文创新方法生成的解之间的误差在机器误差范围内。数值实验验证了该方法在内存占用和计算开销方面的精度与效率提升。