Расширенное семейство дважды неявных методов для жестких систем дифференциальных уравнений

The new set of absolutely stable difference schemes for a numerical solution of ODEs stiff systems (1) is submitted: d u(t) f (u), t 0, u(0) u0 dt   . (1) The main feature of the set is the multi-implicit finite differences with the second derivatives of the desired solution. The expanded three-parameter (, , ) set of 2ISD-schemes (2)-(3) is studied in more details in this paper. 2 1 1 1 0 2 2 2 2 0 ( ф ) ф ( ф ) 2ф n n i i n i n i i n n i i n i n i i a E b J f a E b J f                       v v v v (2)     101 128 11 240 240 240 56 128 56 240 240 240 13 40 3 240 240 240 8 8 240 240 3б 2в 4в 3б 2в, 3г 3г б в 4б б в г 4г г ki ki a b                                 (3) At arbitrary (, , ) parameters last difference equation in system (2) has 5th order of accuracy. We found that the set of absolutely stable 2ISD-schemes includes two families: the set of the L-stable schemes and the set of the schemes of heightened accuracy for linear problems. For example: at 1 б  168, в  0, г  0 we have A-stable scheme with 8th order of approximation, at 53 1 6 б   5880, в  148, г  315 we have L1-stable scheme with 7th order of approximation, at 23 1 14 б   360, в  60, г  315 we have L2-stable scheme with 6th order of approximation. The testing of this difference schemes on linear and nonlinear problems with a different stiff power is conducted. The errors of a numerical solution as functions of integration step size are computed in numerical experiments. These results demonstrate high quality of stability and accuracy of the suggested 2ISD-schemes.