Modifications of Runge-Kutta methods for nonlinear differential equations

Abstract

This article studies various modifications of Runge–Kutta methods, which are widely used in the numerical solution of nonlinear differential equations. The problems of accuracy, stability, and computational efficiency of classical Runge–Kutta methods in nonlinear problems are analyzed. On this basis, adaptive stepwise Runge–Kutta methods, implicit Runge–Kutta schemes, as well as modifications with higher order and modified coefficients are considered. The advantages and disadvantages of each modification are compared and evaluated on the example of nonlinear differential equations. The results of the study allow us to choose the appropriate modification of Runge–Kutta methods in modeling complex nonlinear systems and serve to increase the accuracy and stability of calculations in solving practical problems.