We numerically model key building blocks of a phononic integrated circuit that enable phonon routing in high-acoustic-index waveguides. Our particular focus is on Gallium Nitride-on-sapphire phononic platform which has recently demonstrated high acoustic confinement in its top layer without the use of suspended structures. We start with systematic simulation of various transverse phonon modes supported in strip waveguides and ring resonators with sub-wavelength cross-section. Mode confinement and quality factors of phonon modes are numerically investigated with respect to geometric parameters. Quality factor up to $10^{8}$ is predicted in optimized ring resonators. We next study the design of the phononic directional couplers, and present key design parameters for achieving strong evanescent couplings between modes propagating in parallel waveguides. Last, interdigitated transducer electrodes are included in the simulation for direct excitation of a ring resonator and critical coupling between microwave input and phononic dissipation. Our work provides comprehensive numerical characterization of phonon modes and functional phononic components in high-acoustic-index phononic circuits, which supplements previous theories and contributes to the emerging field of phononic integrated circuits.
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