On an implicit optimal difference formula

Differential equations are used in many fields of physics and engineering, as they are convenient tools for mathematical modelling of various processes in science and technology. Since finding exact solutions to differential equations is not always feasible when solving practical problems, numerical methods are employed. The theory of finite differences is one of the principal methods of approximate calculations. It is widely utilized in the numerical computation of sums and integrals, solving interpolation and extrapolation problems, and solving differential equations numerically. Therefore, this work considers the problem of constructing an implicit optimal difference formula in the Sobolev space for the approximate solution of the Cauchy problem for a second-order ordinary differential equation. In addition, an algorithm for constructing the implicit optimal difference formula is presented. The implicit optimal difference formula of the St¨ormer type is also constructed using the Sobolev method. For this purpose, the implicit optimal difference formula’s analytical form is derived using the second-order differential operator’s discrete analogue. Sharp estimation of the error functional norm is obtained using optimal coefficients.