Approximation of the Jacobian matrix in $(m,2)$-methods for solving stiff problems

An initial value problem for stiff systems of first-order ordinary differential equations is considered. In the class of $(m,k)$-methods, two integration algorithms with a variable step size based on second $(m=k=2)$ and third $(k=2,m=3)$ order-accurate schemes are constructed in which both analytical and numerical Jacobian matrices can be frozen. A theorem on the maximum order of accuracy of $(m,2)$-methods with a certain approximation of the Jacobian matrix is proved. Numerical results are presented.