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 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，当前阶段非菲克流动的数值解依赖于所有历史时间点，因此存储需求随时间步数线性增长，而计算复杂度呈二次增长。这给数值模拟带来了重大挑战，且据我们所知，该问题至今尚未解决。本文基于增量奇异值分解技术提出一种无记忆算法，其计算复杂度仅随时间步数线性增长。我们证明传统算法与创新方法生成解之间的误差处于机器误差范围内。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优势。