Two approaches to the construction of optimal second-order numerical methods and their application to the analysis of oscillatory nonlinear systems - Zayats, V.M. (Reports of the National Academy of Sciences of Ukraine, 2014)
Iterative and direct approaches to the minimization of errors at a discretization of second-order numerical methods are proposed. The iterative approach is based on a modification of the method of trapezoids and setting the time when the explicit and implicit Euler methods give the same contribution to the amendment to the next discretization point of a dynamical system. Combining the derived formula with the method of trapezoids, the possibility of constructing the optimal precision numerical method is shown. The direct approach is based on determining a time when the tangents drawn to the nearby points of discretization of the continuous system intersect, which provides the zero error of a discretization. The expediency of their application to the analysis of nonlinear dynamical oscillatory systems with a low coefficient of attenuation, long transients, and high power is confirmed.