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

From: A specialized ODE integrator for the efficient computation of parameter sensitivities

Integrator Nodes, explicit1 Nodes, implicit
Euler x(t), x ̇ (t) x(t), x ̇ (t+h)
Backward Differentiation Formula (BDF) x ̇ (t), x(th k ) x(t), x ̇ (t+h), x(th k )
Adams-Moulton (AM) x ̇ (t), x ̇ (t h k ) x(t), x ̇ (t+h), x ̇ (t h k )
Second-derivative rule (this work) x(t), x ̇ (t), x ̈ (t), x ̇ (t+h), x ̈ (t+h)
  1. 1For multi-step methods, the index k = 0,−1,…,−n.