Computational Mathematics and Mathematical Physics, 40(7) (2000), 1027-1045

Special features of the use of implicit Runge-Kutta methods for solving large systems of differential-algebraic equations of index 1 are examined. Simple iteration can be especially useful in this case since its computer implementation is straightforward irrespective of the dimension of the problem and the number of stages in the Runge-Kutta method. It has been shown earlier, however, that, first, combined Runge-Kutta simple iteration methods are of restricted applicability and, second, the number of iterations to ensure the maximum convergence order, i.e., the order of the Runge-Kutta method used, can be very large. An approach is proposed that makes it possible to construct Runge-Kutta simple iteration methods that are optimal in a certain sense. These methods fully share the property of having straightforward computer implementation; on the other hand, their advantages include wider applicability and a relatively small number of iterations. Existence and uniqueness theorems are proved for the optimal Runge-Kutta simple iteration methods, and practical algorithms for their computer implementation are proposed. The efficiency of the new class of methods is shown theoretically and by a number of numerical examples.