Numerical Algorithm for Design of Stability Polynomials for the First Order Methods

The algorithm for coefficients determination for stability polynomials of degree up to m = 35 is developed. The coefficients correspond to explicit Runge-Kutta methods of the first accuracy order. Dependence between values of a polynomial at the points of extremum and both size and form of a stability domain is shown. Numerical results are given

Solving Stiff Systems of ODEs by Explicit Methods with Conformed Stability Domains

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 first-order 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