On an optimal diagonal and condiagonal Runge-Kutta simple iterations methods
by
Natalia Andrianova
Department of Mathematics and Mechanics, Ulyanovsk State University

At present DAE's of index 1 seems to be well studied. Theorems of existence and uniqness are proved and some stable numerical methods with fixed and variable step based on the implicit Runge-Kutta methods are offered. By means of RK-methods we replace continuous task with descrete one, which can be solved by iterated process.

Usually this processes are: simple iteration, Newton method or modified Newton method. An approach described above is highly efficient, becouse allow to construct numerical methods of arbitrary convergence order.

A shortage of Newton iteration process are significant requirements for computational resources, growing fast with utilization of l-staged RK-method. The utilization of simple iteration for solving an obtained system after discretization is complicated by a strict boundness condition d<1 on matrix norm of Jacobian of the algebraic part of the system.

The most part of real practical tasks do not satisfy this condition. We propose a modification of Runge-Kutta simple iterations method, permitting to solve tasks with violation of boundness condition via an eqivalent task depending on a parameter.

We found an optimal parameter in class of diagonal and condiagonal matrices, which allow to integrate systems with fulfilled boundness condition, but constant d is close to 1. A corresponding uniquness and existence theorems are proved.

Date received: November 30, 1999