New class of post-Newtonian approximants to the waveform templates of   inspiralling compact binaries: Test-mass in the Schwarzschild spacetime
  (Abridged): The standard adiabatic approximation to phasing of gravitational waves from inspiralling compact binaries uses the post-Newtonian expansions of the binding energy and gravitational wave flux both truncated at the same relative post-Newtonian order. Motivated by the eventual need to go beyond the adiabatic approximation we must view the problem as the dynamics of the binary under conservative post-Newtonian forces and gravitational radiation damping.   From the viewpoint of the dynamics of the binary, the standard approximation at leading order is equivalent to retaining the 0PN and 2.5PN terms in the acceleration and neglecting the intervening 1PN and 2PN terms. A complete mathematically consistent treatment of the acceleration at leading order should include all PN terms up to 2.5PN without any gaps. These define the 'standard' and 'complete' non-adiabatic approximants respectively. We propose a new and simple complete adiabatic approximant constructed from the energy and flux functions. At the leading order it uses the 2PN energy function rather than the 0PN one in the standard approximation so that in spirit it corresponds to the dynamics where there are no missing post-Newtonian terms in the acceleration. We compare the overlaps of the standard and complete adiabatic approximants with the exact waveforms for a test-particle orbiting a Schwarzschild black hole. The complete adiabatic approximants lead to a remarkable improvement in the effectualness at lower PN (< 3PN) orders. However, standard adiabatic approximants of order $\geq$ 3PN are nearly as good as the complete adiabatic approximants for the construction of effectual templates. Standard and complete approximants beyond the adiabatic approximation are next studied using the Lagrangian models of Buonanno, Chen and Vallisneri. 
P. Ajith, Bala R. Iyer, C. A. K. Robinson and B. S. Sathyaprakash