  This paper presents comprehensive studies on frequency response optimized
integrators considering second order derivative regarding their numerical
error, numerical stability and transient performance. Frequency domain error
analysis is conducted on these numerical integrators to reveal their accuracy.
Numerical stability of the numerical integrators is investigated. Interesting
new types of numerical stability are recognized. Transient performance of the
numerical integrators is defined to qualitatively characterize their ability to
track fast decaying transients. This property is related to unsatisfactory
phenomena such as numerical oscillation which frequently appear in time domain
simulation of circuits and systems. Transient performance analysis of the
numerical integrators is provided. Theoretical observations from the analysis
of the numerical integrators are verified via time domain case studies.
