Direct numerical methods dedicated to second-order ordinary differential equations

This article presents numerical methods for solving second-order ordinary differential equations. These methods are based on Hermite polynomials, which makes them more computationally effective than, for example, the classical fourth-order Runge–Kutta method. In addition, the presented algorithms were modified to reduce the CPU time required. Hermite polynomials are not very sensitive to the Runge phenomenon; moreover, the numerical errors of interpolation are relatively small for large time steps, which is an advantage. These methods are presented in the form of pseudo-code for easier application. The presented approach to numerical methods is a result of simulated, strongly non-linear vibrations with contact phenomena such as Coulomb friction and impact.