Table 1 Common ODE integration schemes and the values that are used to approximate the polynomial in Eq. (5)

Euler

x(t), $\dot{\mathbf{x}}\left(t\right)$

x(t), $\dot{\mathbf{x}}(t+h)$

Backward Differentiation Formula (BDF)

$\dot{\mathbf{x}}\left(t\right)$, x(t−h _k )

x(t), $\dot{\mathbf{x}}(t+h)$, x(t−h _k )

Adams-Moulton (AM)

$\dot{\mathbf{x}}\left(t\right)$, $\dot{\mathbf{x}}(t-h_k)$

x(t), $\dot{\mathbf{x}}(t+h)$, $\dot{\mathbf{x}}(t-h_k)$

Second-derivative rule (this work)

—

x(t), $\dot{\mathbf{x}}\left(t\right)$, $\ddot{\mathbf{x}}\left(t\right)$, $\dot{\mathbf{x}}(t+h)$, $\ddot{\mathbf{x}}(t+h)$