New explicit-implicit predictor-corrector method based on Adams and Rozhkov schemes

The subject of this research is the Adams methods, which have relatively low computational complexity. The focus of the study is on Adams schemes in the "predictor-corrector" formulation, which allow for increased stability in calculations compared to purely explicit schemes while maintaining limited computational cost compared to purely implicit schemes. It is argued that one can further enhance the stability (while preserving reduced computational effort) of the scheme by applying a Rozhkov-type computational approach, known for its stability, to the corrector (the Adams-Moulton scheme). It is noted that attempting to apply the Rozhkov approach directly to the implicit Adams scheme (without a predictor) would introduce time delays in the solution, which can be easily eliminated for a single differential equation but is quite challenging to compensate for in the general case of numerically solving a system of differential equations. Computational mathematics methods are employed, specifically the combination of the "predictor-corrector" explicit Adams-Bashforth scheme and the implicit Adams-Moulton scheme, combined with the Rozhkov-type splitting. A new explicit-implicit "predictor-corrector" scheme based on Adams schemes and the Rozhkov approach is proposed, which does not require solving systems of linear or nonlinear equations and is limited to solving a series of independent linear equations. A corresponding method for numerical integration of systems of ordinary differential equations has been formulated. The error, order of accuracy, and stability of the obtained method have been experimentally investigated. In the case of a high-order method (4th order and above), the application of the proposed scheme is more justified (higher order of computation, greater stability) than the application of the standard "predictor-corrector" Adams scheme; in this case, it is also less computationally intensive than the Runge-Kutta-4 method (with similar other performance indicators). When the order of the proposed method is less than 4, it does not offer any significant advantages over the standard "predictor-corrector" Adams variation and the Runge-Kutta-4 scheme.