In this article some fundamental properties of existentially and algebraically closed lattices are investigated. We also define the notion of strongly algebraically closed lattices and we show that a q ′ -compact complete boolean lattice is strongly algebraically closed.

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The Cauchy problem for a stiff system of ODEs is considered. The explicit m-stage first order methods of the Runge-Kutta type are designed with stability domains of intermediate numerical schemes conformed with the stability domain of the basic scheme. Inequalities for accuracy and stability control are obtained. A numerical algorithm based on the firstorder method and the five-stage fourth order Merson method is developed. The algorithm is aimed at solving large-scale systems of ODEs of moderate stiffness with low accuracy. It has been included in the library of solvers of the ISMA simulation environment. Numerical results showing growth of the efficiency are given.

Stability control of Runge-Kutta numerical schemes is studied to increase efficiency of in- tegrating stiff problems. The implementation of the algorithm to determine coefficients of stability polynomials with the use of the GMP library is presented. Shape and size of the stability region of a method can be preassigned using proposed algorithm. Sets of first-order methods with extended stability domains are built. The results of electrical circuits simulation show the increase of the efficiency of the constructed first-order methods in comparison with methods of higher order