This contribution proposes a third-order numerical scheme for solving time-dependent partial differential equations (PDEs). This third-order scheme is further modified, and the new scheme is obtained with second-order accuracy in time and is unconditionally stable. The unconditional stability of the new scheme is proved by employing von Neumann stability analysis. For spatial discretization, a compact fourth-order accurate scheme is adopted. Moreover, a mathematical model for heat transfer of Darcy–Forchheimer flow of micropolar fluid is modified with an oscillatory sheet, nonlinear mixed convection, thermal radiation, and viscous dissipation. Later on, the dimensionless model is solved by the proposed second-order scheme. The results show that velocity and angular velocity have dual behaviors by incrementing coupling parameters. The proposed second-order accurate in-time scheme is compared with an existing Crank–Nicolson scheme and backward in-time and central in space (BTCS) scheme. The proposed scheme is shown to have faster convergence than the existing Crank–Nicolson scheme with the same order of accuracy in time and space. Also, the proposed scheme produces better order of convergence than an existing Crank–Nicolson scheme.