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In a previous paper, based on the black hole perturbation approach, weformulated a new analytical method for regularizing the self-force acting on aparticle of small mass $\mu$ orbiting a Schwarzschild black hole of mass $M$,where $\mu\ll M$. In our method, we divide the self-force into the $\tildeS$-part and $\tilde R$-part. All the singular behaviors are contained in the$\tilde S$-part, and hence the $\tilde R$-part is guaranteed to be regular. Inthis paper, focusing on the case of a scalar-charged particle for simplicity,we investigate the precision of both the regularized $\tilde S$-part and the$\tilde R$-part required for the construction of sufficiently accuratewaveforms for almost circular inspiral orbits. For the regularized $\tildeS$-part, we calculate it for circular orbits to 18 post-Newtonian (PN) orderand investigate the convergence of the post-Newtonian expansion. We also studythe convergence of the remaining $\tilde{R}$-part in the spherical harmonicexpansion. We find that a sufficiently accurate Green function can be obtainedby keeping the terms up to $\ell=13$.Comment: 21pages, 12 figure

A New Analytical Method for Self-Force Regularization. II: -- Testing the Efficiency for Circular Orbits --