Numerical Investigation of Solutions of Chaotic Systems Via Domain Decomposition Technique

Chaotic systems, such as the Lorenz and Chen systems, models exhibit high sensitivity to initial conditions. They require fine discretization for accurate long-time integration. Classical time-stepping methods suffer from accumulated truncation errors, whereas, global spectral methods, though highly accurate, yield dense coefficient matrices that are computationally expensive, limiting their efficiency. In this paper we propose an efficient numerical scheme that combines the non-overlapping Domain Decomposition Method (DDM) with the Spectral Relaxation Method (SRM) to solve the Lorenz and Chen systems. The time domain is decomposed into smaller sub intervals, and SRM is applied in each subdomain to linearize the nonlinear terms iteratively, thereby avoiding Taylor series truncation errors while maintaining spectral accuracy. The performance of the proposed Multistage Spectral Relaxation Method (MSRM) is investigated for different step sizes h = 0.01 and h = 0.001 over a long integration time T= 10. Numerical results are compared with the single-domain SRM in terms of accuracy, CPU time and memory usage. The results demonstrate that MSRM achieves comparable accuracy to the single-domain approach while significantly reducing computational time and memory requirements, especially for smaller step sizes. The proposed MSRM provides an effective and efficient framework for the simulation of chaotic systems over a long-time interval.