Real-time aircraft dynamics simulation requires very high accuracy and stability in the numerical integration process. Nonetheless, traditional multistep numerical methods cannot effectively meet the new requirements. Therefore, a novel real-time multistep method based on Predict-Evaluate-Correct scheme of three-step fourth-order method (RTPEC-34) is proposed and developed in this research to address the gap. In addition to the development of a highly accurate algorithm based on predictor-corrector, the contribution of this work also includes the analysis of truncation error for real-time problems. Moreover, the parameters for the RTPEC-34 method are optimized using intelligent optimization algorithms. The application and comparison of the optimization algorithms also lead to general guidelines for their applications in the development of improved multistep methods. Last but not least, theoretical analysis is also conducted on the stability of the proposed RTPEC-34 method, which is corroborated in simulation experiments and thus provides general guidelines for the evaluation of real-time numerical methods. The RTPEC-34 method is compared with other multistep algorithms using both numerical experiments and a real engineering example. As shown in the comparison, it achieves improved performance in terms of accuracy and stability and it is also a viable and efficient algorithm for real-time aircraft dynamics simulation.