As First-Order System

Many numerical time-integration techniques are usually applied to first order ODE systems in the literature. It is simple enough to expand the definition of the simple harmonic oscillator, from (3.1), to a first order system in two variables, i.e., in matrix form,

$$\frac{d}{dt}\begin{bmatrix} u\\ v\\ \end{bmatrix} = \begin{b... ... -\omega_{0}^2 & 0\\ \end{bmatrix}\begin{bmatrix} u\\ v\\ \end{bmatrix}$$ (3.15)

All frequency domain and energy analysis of course applied equally to this equivalent system. Though first-order systems will only rarely appear in this book, it is worth being aware of such systems, as many time-integration techniques, including the ubiquitous Runge-Kutta family of methods, are indeed usually presented with reference to first order systems. In the distributed case, the important ``finite-difference time domain" family of methods for hyperbolic PDEs[215], such as the defining equations of electromagnetics, are also usually applied to first order systems.