This research addresses a key aspect of finite-difference schemes on curvilinear grids, ensuring exact free-stream preservation and accurate flow simulation over boundary walls. A major issue is the loss of stream preservation due to geometrically induced errors, which can compromise computational accuracy. One solution involves solving the full conservation form of the Euler equations and modifying flux functions so the Jacobian and grid metrics are consistently shared at the points where flux derivatives are computed. Proper treatment of geometric parameters like metrics and the Jacobian, appearing as derivatives in the equations, is essential to maintaining solution integrity. The paper adopts a novel method for coordinate transformation, aiming to eliminate errors caused by switching from Cartesian to curvilinear systems in finite-difference methods. This method, called the Discrete Equivalence Equation and its Discrete-Rule (DEER), is tested to validate its effectiveness. Results and analyses demonstrate its ability to manage geometry-induced inaccuracies in numerical simulations.