Strict estimate of the error of numerical solution to the Cauchy problem for a second-order nonlinear differential equation (Q5954764)

A nonlinear differential equation of the form \[ y''= f(x, y), \quad 0\leq x \leq X_0, \quad y(0)=y_0, \quad y'(0)=y'_0 \] in the case when \(\frac{\partial f}{\partial y}<0\) is considered. It is noted that in the case mentioned there an effect of exponential increasing of the error estimation with increasing \(x\) occurs while if the solution has a vibrating character then the error is increasing basically slow. A method of finding of a correct error estimation of the numerical solution of the differential equation based on a difference scheme is proposed. The numerical value \(y_m\) can be found by Noumerov's method in the form \[ y_m = 2y_{m-1} - y_{m-2} + \frac{h^2}{12}(f(x_m,y_m) +10f(x_{m-1}, y_{m-1} +f(x_{m-2}, y_{m-2})) + \omega_m, \] where \( m \geq 2\) and \(\omega_m\) is the rounding off error in the \(m\)-th step. The goal of the paper is to estimate the numerical solution \(z_m=y_m(x_m)-y_m\) where \[ \begin{multlined} y(x_m)= \\ =2y(x_{m-1})-y(x_{m-2}) +\frac{h^2}{12}(f(x_m,y(x_m)) + 10f(x_{m-1},y(x_{m-1}))+ f(x_{m-2}, y(x_{m-2}))) + N_m,\end{multlined} \] where \(N_m\) is the local error of Noumerov's method in the \(m\)-th step. An exact expression for \(\mid z_m\mid\) is obtained and two examples of its application are presented.