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The inner doubly iterative efficient algorithm (IDEAL), an advanced fully implicit algorithm for solving the pressure-velocity coupling problem, lacks a comprehensive implementation procedure on commonly used collocated unstructured grids. Therefore, this study provides a thorough demonstration of the fundamental principles, solving procedures, discretizations of momentum and pressure equations, Rhie-Chow interpolation implementation, mass continuity processing, and boundary condition treatment associated with the IDEAL algorithm on a collocated unstructured grid. Additionally, a comparative analysis between the semi-implicit method for pressure-linked equations (SIMPLE) and the IDEAL algorithm is conducted concerning computational efficiency under different grid types and flow conditions. The results indicate a significantly higher computational efficiency achieved by the IDEAL algorithm across all tested cases, attributed to its ability to overcome two fundamental assumptions inherent in the SIMPLE algorithm. The average acceleration ratio for these cases is approximately 3.13 (tSIMPLE/tIDEAL). This study serves as a valuable reference for the further development and application of the IDEAL algorithm in solving flow problems with complex irregular domains discretized by unstructured grids. [ABSTRACT FROM AUTHOR]