The difference schemes of higher order accuracy for numerical solution of the stiff first order ordinary differential equation with linear coefficients

The family of a new implicit economic one-step difference schemes from the second to fifth order of accuracy for numerical solution of the first order stiff ordinary differential equation with linear coefficients are proposed. The construction of schemes is based on using the increased accuracy Taylor expansion of desired function in the vicinity of the solution and the direct integration of the differential equation. The simplified variants and the asymptotic of schemes are considered. Good practical convergence of numerical results to exact solutions is shown on test examples at a rough step of integration, including a small parameter at a derivative. Comparison of different schemes efficiency with other known one-step methods is carried out.