COMPARISON OF EXPLICIT RUNGE-KUTTA METHODS WITH HYBRID METHODS OF MULTISTEP TYPE

As is known, in solving some of applied problems, Runge-Kutta methods are used, considering that these methods are explicit and do not require the use of any other methods. Note that, like other methods, the Runge-Kutta methods also have their own advantages and disadvantages. One of the main disadvantages of Runge-Kutta method consists of repeatedly calculating the function , which is right-hand side of the differential equation, and one of the advantages of Runge-Kutta methods is that they are one step methods. Usually, such methods are considered as the recurrent relationships. A well-known representative of the Runge-Kutta method is the Euler and Midpoint methods. It is known that one of the exact methods is called the hybrid method, which can be taken as the generalization of the Midpoint method. Consequently, the Runge-Kutta and Hybrid methods are generalizations of this method. By using this, here we consider the comparison of some specific Runge-Kutta methods with the corresponding hybrid methods. Since implicit hybrid methods are more exact, it becomes necessary to compare hybrid methods with the semi-implicit Runge-Kutta methods. By considering a specific method, we compare semi-implicit Runge-Kutta methods with implicit hybrid methods.