High‐Efficiency Semi‐Analytical Method for Initial Value Problem Under Dense Initial Conditions

ABSTRACT The initial value problem under dense initial conditions (D‐IVP) is a critical challenge in aerospace engineering, such as debris tracking and orbital uncertainty propagation. Finite‐difference‐based methods are the most commonly used methods for solving this type of problem; however, they suffer from low computational efficiency when dealing with large‐scale D‐IVPs due to their reliance on small integration step sizes to maintain accuracy. To address this issue, this paper proposes an efficient semi‐analytical method that combines polynomial approximation with parallel large‐step computation. Using Taylor expansion in the spatial domain, the proposed method expresses the solutions of the D‐IVP as a polynomial with temporally varying coefficients, whose computational cost is independent of the problem's scale. In particular, these coefficients are efficiently derived through parallel integral iteration in large steps, bypassing the limitations of finite‐difference methods. The method's performance is validated through three classical dynamic problems, and the computational results demonstrate that its efficiency advantage over conventional methods increases with the scale of the D‐IVP.