Numerical methods based on multipoint Hermite interpolating polynomials for solving the Cauchy problem for stiff systems of ordinary differential equations

Families of $A$-, $L$- and $L(\delta)$-stable methods are constructed for solving the Cauchy problem for a system of ordinary differential equations (ODEs). The $L(\delta)$-stability of a method with a parameter $\delta\in(0,1)$ is defined. The methods are based on the representation of the right-hand sides of an ODE system at the step $h$ in terms of two-or three-point Hermite interpolating polynomials. Comparative results are reported for some test problems. The multipoint Hermite interpolating polynomials are used to derive formulas for evaluating definite integrals. Error estimates are given.