Mathematics & Physics 2016, 9(2), 209–219 УДК 519.622 Application of Explicit Methods with Extended Stability Regions for Solving Stiff Problems Eugeny A.Novikov∗ Institute of Computational Modeling SB RAS Akademgorodok, 50/44, Krasnoyarsk, 660036 Russia Mikhail V.Rybkov† Institute of Mathematics and Computer Science Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041 Russia Received 10.11.2015, received in revised form 15.01.2016, accepted 20.02.2016 An algorithm is developed to determine coefficients of the stability polynomials such that the explicit Runge-Kutta methods have a predetermined shape and size of the stability region. <...> Inequalities for accuracy and stability control are obtained. <...> The impact of the stability control on efficiency of explicit methods to solving stiff problems is shown. <...> Numerical calculations confirm that the three-step method of the first order with extended stability region is more efficient than the traditional three-stage method of the third order. <...> Keywords: stiff problem, explicit methods, stability region, accuracy and stability control. <...> Introduction In some cases large-scale stiff problems need to be solved with the algorithms based on explicit methods as shown in [1, 2]. <...> At the present time, algorithms of variable step and order are developed [1, 2, 6]. <...> They use inequalities for stability control as a criterion for choosing between methods of high and low order of accuracy at the integration step. <...> In a settling region there is no point in using high-order methods because the integration step is restricted by the condition of stability. <...> Further raising of efficiency may be achieved by developing not only variable order and step algorithms but algorithms with variable number of stages. <...> The stability polynomials of degree up to m = 13 are constructed in [1, 8]. <...> Here an algorithm to determine the coefficients of the stability polynomial coefficients is developed such that the ∗novikov@icm.krasn.ru †mixailrybkov@yandex.ru ⃝ Siberian Federal University. <...> All rights reserved c – 209 – Eugeny A.Novikov, Mikhail V.Rybkov Application of Explicit Methods with Extended Stability . . . corresponding explicit Runge-Kutta methods have a predetermined shape and size of the stability region. <...> Formulas for the coefficients of Runge-Kutta methods with extended stability regions are obtained. <...> The three-stage explicit Runge-Kutta method is considered <...>