Piecewise Jacobi–Gauss spectral collocation simulations for a multi-particle model involving processing delay

In this work, we propose a piecewise Jacobi–Gauss spectral collocation (JGSC) method for simulating a multi-particle system involving processing delay. Through the use of Jacobi orthogonal approximation and simple Picard iteration, the method obtains the Jacobi series solution of the multi-particle model, allowing us to derive the numerical solution of the processing delay directly. The matrix–vector form of the method helps to obtain the solution parallelly and improves the efficiency significantly. Additionally, the eigenvalues of the iteration’s coefficient matrix are evaluated in order to analyze the convergence of the JGSC method numerically. Numerical experiments illustrate that the JGSC method can keep the high accuracy and efficiency. Furthermore, simulation results of the model indicate that the flocking behavior can only achieved with small enough processing time delays and large enough sizes of the neighborhood.