High-order compact MacCormack scheme for two-dimensional compressible and non-hydrostatic equations of the atmosphere

•4th-order compact MC method is used to spatial differencing of 2D equations of atmosphere.•A four-stage Runge–Kutta method is employed to advance the solution in time.•The accuracy of solution is studied and verified.•To perform the simulations two well-known test cases are used. This study is devoted to application of the fourth-order compact MacCormack scheme to spatial differencing of the conservative form of two-dimensional and non-hydrostatic equation of a dry atmosphere. To advance the solution in time a four-stage Runge–Kutta method is used. To perform the simulations, two test cases including evolution of a warm bubble and a cold bubble in a neutral atmosphere with open and rigid boundaries are employed. In addition, the second-order MacCormack and the standard fourth-order compact MacCormack schemes are used to perform the simulations. Qualitative and quantitative assessment of the numerical results for different test cases exhibit the superiority of the fourth-order compact MacCormack scheme on the second-order method.