The extended family of 2ISD-methods for differential stiff systems

The new set of absolutely stable difference schemes for a numerical solution of ODEs stiff systems (1) is submitted:
\begin{equation} \frac{d}{dt} u(t) = f (u), \quad t>0, \quad u(0) = u_0. \end{equation}
The main feature of the set is the multi-implicit finite differences with the second derivatives of the desired solution. The expanded three-parameter $(\alpha, \beta, \gamma)$ set of 2ISD-schemes (2)–(3) is studied in more details in this paper.
\begin{equation} \left\{ \begin{aligned} \frac{{\nu}_{n+1} - {\nu}_{n}}{ф} = \sum_{i=0}^2 (a_{1i} E + ф b_{1i} J_{n+i}) f_{n+i},\\ \frac{{\nu}_{n+2} - {\nu}_{n}}{2ф} = \sum_{i=0}^2 (a_{2i} E + ф b_{2i} J_{n+i}) f_{n+i},\end{aligned} \right. \end{equation}

\begin{equation} (a_{ki})= \begin{pmatrix} \frac{101}{240}+3б-2в & \frac{128}{240}+4в & \frac{11}{240}-3б-2в\\ \frac{56}{240}-3г & \frac{128}{240} & \frac{56}{240}+3г \end{pmatrix} , \\ (b_{ki})= \begin{pmatrix} \frac{13}{240}+б-в & -\frac{40}{240}+4б & -\frac{3}{240}+б+в\\ \frac{8}{240}-г & -4г & -\frac{8}{240}-г \end{pmatrix} . \end{equation}
At arbitrary $(\alpha, \beta, \gamma)$ 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 $б = 1/168, в = 0, г = 0$ we have $A$-stable scheme with 8th order of approximation,
at $б = -53/5880, в = 1/148 , г = 6/315$ we have $L_1$-stable scheme with 7th order of approximation,
at $б = -23/360, в = 1/60, г = 14/315$ we have $L_2$-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.