Proper Selection of Obreshkov-Like Numerical Integrators Used as Numerical Differentiators to Avoid Numerical Oscillation and Bias

Misuse of Obreshkov-like numerical integrators as numerical differentiators may lead to numerical oscillation or bias. Criteria for Obreshkov-like numerical integrators to be used as numerical differentiators are proposed in this paper to avoid these misleading phenomena. The coefficients of a numerical integrator for the highest order derivative turn out to determine its suitability. Some existing Obreshkov-like numerical integrators are examined under the proposed criteria. It is revealed that the notorious numerical oscillations induced by the implicit trapezoidal method cannot always be eliminated by using the backward Euler method for a few time steps. Guided by the proposed criteria, a frequency response optimized integrator considering second order derivative is put forward which is suitable to be used as a numerical differentiator. Theoretical observations are demonstrated in time domain via case studies.