A Study of the Stability of the Numerical Method of RKM-Type for Solving Ordinary Differential Equations of Fifth-Order

Numerical method of RKM-type for solving the ordinary differential equations (ODEs) of fifth-order plays an important role in the accurate and efficient solutions of mathematical models arising in engineering and science. The stability is one of the most important properties of numerical methods for solving ODEs. In this paper, the stability properties of the RKM numerical method for solving ODEs of fifth-order has been introduced and investigated. The stability analysis of this method, polynomial of the stability and the region of absolute stability have been emphasized. However, a comprehensive stability analysis is performed by applying the proposed method to an appropriate linear test problem. The corresponding amplification factor and stability polynomial are derived, and the absolute stability condition is established based on the requirement that the modulus of the amplification factor does not exceed unity. The region of absolute stability is then determined and graphically illustrated in the complex plane. Furthermore, the effects of the step size and problem parameters on the stability of the numerical solution are examined. Numerical experiments are presented to verify the theoretical results and to demonstrate the behavior of the proposed method for representative fifth-order ODEs. The results show that the proposed method produces stable and reliable numerical approximations when the step size lies within the derived stability restrictions.