Implicit Runge–Kutta methods with explicit internal stages

The main computational costs of implicit Runge–Kutta methods are caused by solving a system of algebraic equations at every step. By introducing explicit stages, it is possible to increase the stage (or pseudo-stage) order of the method, which makes it possible to increase the accuracy and avoid reducing the order in solving stiff problems, without additional costs of solving algebraic equations. The paper presents implicit methods with an explicit first stage and one or two explicit internal stages. The results of solving test problems are compared with similar methods having no explicit internal stages.