A Study of Numerical Methods for Solving the Nonlinear Energy Resources Supply-Demand System

In this study, we implement and estimate various numerical methods for solving a nonlinear differential equation system modeling energy resources supply-demand dynamics. Both single-step methods (Taylor series, Runge-Kutta) and multi-step methods (Adams - Bashforth, Adams Predictor-Corrector) are employed. In addition to standard fourth-order methods, higher-order techniques such as the fifth-order Runge-Kutta method and the sixth-order Taylor series method are also applied. Furthermore, along with fixed-step numerical methods, we implement and assess adaptive step-size methods, including the explicit Runge-Kutta method of order 5(4) (that is RK45), the explicit Runge-Kutta method of order 8(5, 3) (or DOP853), the implicit Runge-Kutta method from the Radau IIA family of order 5 (Radau), the implicit method based on backward differentiation formulas (BDF), and the Adams/BDF method with automatic switching (LSODA). The results indicate that, in the cases we considered, single-step methods are more effective than multi-step ones in capturing and tracking rapid variations of the system, while multi-step methods require less computation time. Adaptive step-size numerical methods demonstrate both flexibility and stability. Through the evaluation and analysis of numerical solutions obtained by various methods, the behaviour and dynamic characteristics of the system are explored.