In this work, we systematically investigate linear multi-step methods for differential equations with memory. In particular, we focus on the numerical stability for multi-step methods. According to this investigation, we give some sufficient conditions for the stability and convergence of some common multi-step methods, and accordingly, a notion of A-stability for differential equations with memory. Finally, we carry out the computational performance of our theory through numerical examples.

翻译：本文系统地研究了用于带记忆微分方程的线性多步法。特别地，我们聚焦于多步法的数值稳定性。基于此研究，我们给出了一些常见多步法稳定性和收敛性的充分条件，并据此提出了适用于带记忆微分方程的A-稳定性概念。最后，我们通过数值算例展示了所提理论的计算性能。