The rigorous analysis of circular patch radiators based on metamaterial substrates is presented in this paper via a stencil-controllable set of time-domain methods. Constructing a family of curvilinear 3-D ensembles, the new framework assigns the appropriate coefficient to each optimal spatial increment and introduces extra nodes in terms of a convex combination process. So, instead of employing the typical derivative approximators, accurate interpolating polynomials, weighted by particular accuracy parameters, are derived. Also, to evade the inevitably lengthened simulations and accomplish trustworthy predictions, the generalized pencil of matrix technique is incorporated. Numerical outcomes substantiate these qualities along with the significant system savings though various real-world metamaterial structures.