<p>Time-sensitive missions in Low Earth Orbit (LEO) satellite networks require deterministic end-to-end delay and jitter. However, the dynamic topology of LEO satellite networks, especially the time-varying behavior of inter-satellite links (ISLs), makes deterministic transmission difficult to guarantee. In addition, the expansion of LEO constellations and mission flows increases scheduling complexity and calls for efficient optimization methods with parallel computing capability. To address this issue, we propose an adaptive time-slot-correction (ATSC)-based scheduling mechanism for deterministic transmission. Specifically, adaptive transmission-time correction and queuing-delay adjustment mechanisms are developed to compensate for propagation-delay variations caused by dynamic ISLs. Based on this design, the deterministic scheduling problem is formulated as a mixed nonlinear optimization problem under delay and jitter constraints, with the objective of minimizing the maximum completion time of all missions. The problem is NP-hard. We further develop an ATSC-PSO scheduling algorithm by combining ATSC with Particle Swarm Optimization (PSO) for time-slot allocation. Since particle fitness evaluations in PSO are independent, the algorithm has the potential for parallel acceleration in large-scale scheduling scenarios. Finally, simulations in Network Simulator EXATA show that the proposed mechanism controls end-to-end delay at the millisecond level while maintaining jitter at the microsecond level. Moreover, under the evaluated scenarios, it reduces delay by 36.36%-47.44% and 28.57%-30.51% compared with DTS and CoLT-TSA, respectively.</p>