Accurate long-time simulation of open quantum systems is a central challenge in studying non-equilibrium phenomena, which has gained increasing attention in light of recent experimental advances. In this talk, we focus on algorithmic strategies for overcoming the long-time bottleneck in non-Markovian simulations. We introduce the coupled Lindblad pseudomode dynamics, a framework that achieves both physicality (complete positivity and trace preserving) and computational efficiency for long time evolution. We further discuss how this framework naturally leads to new algorithmic questions in the classical problem of sum-of-exponentials (SOE) approximation. These connections suggest new directions both in classical numerical analysis and the study of non-Markovian open quantum systems.