Optimal Difference Formulas for the Approximate Solution of the Second-Order Cauchy Problem

It is well established that the Cauchy problem for second-order differential equations serves as a canonical model of gradient conservative or weakly damped dynamical systems, widely applied in mechanics, astronomy, molecular and structural dynamics, acoustics, and radio-frequency systems. Exact solutions are usually attainable only for linear or simple functions, whereas in most other cases approximate solution methods are employed. In this work, we make use of the Sobolev method to develop an optimal explicit difference formula of the Störmer type for the Cauchy problem of second-order differential equations.

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