  In current research, we analyse dissipation and dispersion characteristics of
most accurate two and three stage Gauss-Legendre implicit Runge-Kutta (R-K)
methods. These methods, known for their $A$-stability and immense accuracy, are
observed to carry minimum dissipation error along with highest possible
dispersive order in their respective classes. We investigate to reveal that
these schemes are inherently optimized to carry low phase error only at small
wavenumber. As larger temporal step size is imperative in conjunction with
implicit R-K methods for physical problems, we interpret to derive a class of
minimum dissipation and optimally low dispersion implicit R-K schemes. Schemes
thus obtained by cutting down amplification error and maximum reduction of
weighted phase error, suggest better accuracy for relatively bigger CFL number.
Significantly, we are able to outline an algorithm that can be used to design
stable implicit R-K methods for suitable time step with better accuracy
virtues. The algorithm is potentially generalizable for implicit R-K class of
methods. As we focus on two and three stage schemes a comprehensive comparison
is carried out using numerical test cases.
