# Direct numerical methods dedicated to second-order ordinary differential equations

Applied Mathematics and Computation, Volume 219, Issue 19, 1 June 2013, Pages 10082-10095.
Robert Kostek.

University of Technology and Life Sciences, Faculty of Mechanical Engineering, Al. Prof. S. Kaliskiego 7, 85-796 Bydgoszcz, Poland

## Abstract

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.

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These methods were inspired by Hermite polynomials, which are not very sensitive to the Runge phenomenon and interpolate the acceleration time history with a small error. Thus, simple integration of Hermite polynomials provides an opportunity to avoid numerical phenomena and also easily solve second-order ordinary differential equations. This is the primary idea of these methods.

The most important features of these methods are summarised below.

* These methods are not computationally expensive.

* In addition, the first method is less computationally expensive for the example considered than the fourth-order classical Runge-Kutta method and the fifth-order Dormand-Prince method, whereas the second method is less computationally expensive than the sixth-order Verner method. These results are very important.

* Furthermore, these methods can be easily initialised because these Hermite polynomials are calculated from the values of the acceleration and its derivatives known for two nodes. That, in turn, provides an opportunity to change, if necessary, the time step during simulation.

* Additionally, if iterations are used, then the numerical errors obtained are relatively small for large time steps; this is an advantage.

* Moreover, these methods can even be used if the acting forces are described by non-smooth or discontinuous functions. This feature enables these methods to solve strongly non-linear differential equations. This issue will be discussed in future work.

* An important feature of these methods is that the time histories of acceleration, velocity, and displacement are described by polynomials in the intervals between nodes. This provides an opportunity to precisely find the extrema, roots, and values between nodes, which provides an opportunity to accurately calculate Poincaré maps, bifurcation diagrams, and resonance characteristics. This issue will be investigated in future work. 