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The application of the high-order accurate schemes with multi-block domains is essential in problems with complex geometries. Primarily, accurate block-interface treatment is found to be of significant importance for precisely capturing discontinuities in such complex configurations. In the current study, a conservative and accurate multi-block strategy is proposed and implemented for a high-order compact finite-difference solver. For numerical discretization, the Beam-Warming linearization scheme is used and further extended for three-dimensional problems. Moreover, the fourth-order compact finite-difference scheme is employed for spatial discretization. The capability of the high-order multi-block approach is then evaluated for the onedimensional flow inside a Shubin nozzle, two-dimensional flow over a circular bump, and three-dimensional flow around a NACA 0012 airfoil. The results showed a reasonable agreement with the available exact solutions and simulation results in the literature. Further, the proposed block-interface treatment performed quite well in capturing shock waves, even in situations that the location of the shock coincides with block interfaces.