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In current research, we analyse dissipation and dispersion characteristics ofmost accurate two and three stage Gauss-Legendre implicit Runge-Kutta (R-K)methods. These methods, known for their $A$-stability and immense accuracy, areobserved to carry minimum dissipation error along with highest possibledispersive order in their respective classes. We investigate to reveal thatthese schemes are inherently optimized to carry low phase error only at smallwavenumber. As larger temporal step size is imperative in conjunction withimplicit R-K methods for physical problems, we interpret to derive a class ofminimum dissipation and optimally low dispersion implicit R-K schemes. Schemesthus obtained by cutting down amplification error and maximum reduction ofweighted phase error, suggest better accuracy for relatively bigger CFL number.Significantly, we are able to outline an algorithm that can be used to designstable implicit R-K methods for suitable time step with better accuracyvirtues. The algorithm is potentially generalizable for implicit R-K class ofmethods. As we focus on two and three stage schemes a comprehensive comparisonis carried out using numerical test cases.

Phase error analysis of implicit Runge-Kutta methods: Introducing new classes of minimal dissipation low dispersion high order schemes