Numerical solution of conservative form of 2-D compressible and nonhydrostatic equations of atmosphere using the compact MacCormack scheme

Increasing the accuracy of numerical methods used for simulation of fluid dynamics problems, particularly the geophysical fluid dynamics problems (atmospheric and oceanic), has been the subject of many research works. Recently, due to the increasing computing power of computers, the advantage of high-resolution numerical methods for numerical simulation of the governing equations of fluid flow is further emphasized. The idea of compact finite difference methods goes back to some works conducted in 1920s and 1940s. However, the pioneering works of Kreiss and Oliger (1972), Hirsh (1975) and Lele (1992) made these methdos popular and showd that compact finite difference methdos can be used as a powerful tool for numerical simulation of fluid dynamics problems appearing in different branches of science. These methods have also been used in numerical simulation of geophysical fluid dynamics problems. Due to the promising performance of compact finite difference methods, application of these schemes to numerical simulations of atmospheric and oceanic flows has increased. The compact finite difference schemes have shown that are able to provide a simple way to reach one of the main objectives in the development of numerical algorithms, i.e., having in our disposal a low-cost and highly-accurate computational method. The compact methods have been used extensively for numerical solution of various fluid dynamics problems. These methods have also been applied to numerical solution of some prototype geophysical fluid dynamics problems (e.g., shallow water equations). Most of the compact finite difference methods are symmetric (usually with a 3- or 5-point stencil) and finding each derivative requires a matrix inversion. Recently, a new class of highly-accurate explicit MacCormack type methods has been introduced for computational fluid dynamic. The compact MacCormack type methods were developed by Hixon and Turkel (2000) to split the derivative operator of the central compact method into two one-sided forward and backward operators. This study is devoted to application of the fourth-order compact MacCormack method for numerical solution of the conservative form of two-dimensional non-hydrostatic and fully compressible Navier Stokes equations governing an inviscid and adiabatic atmosphere. Moreover, the second-order MacCormack method is used to compare the performance of the computations. This enables us to measure some aspects of the computational results (such as efficiency and accuracy). Various aspects of the computations such as discretization of the equations for the interior and boundary points, the details of implementation of boundary conditions for different boundary types (e.g., rigid and open boundaries), time step, grid resolution and dissipation are presented. Since, unlike the second-order MacCormack method, the forward operator in the fourth-order compact MacCormack method for approximation of the first derivative at an arbitrary grid point (e.g, j) is not equal to the backward operator at its adjacent point (i.e., j 1), the application of the fourthorder compact MacCormack method for spatial discretization of the source term in vertical momentum equation in non-hydrostatic models needs special treatment. In this work we have used the conventional second-order MacCormack method (MC2), the standard fourth-order compact MacCormack method (MC4) developed by Hixon and Turkel (2000) and a fourth-order compact MacCormack method with a four-stage Runge-Kutta for time advancing (MCRK4) in our numerical simulations. To evaluate the performance of these methods, two test cases including evolution of a warm bubble, and evolution of a cold bubble in a netural atmosphere were simulated. To simulate cold bubble, the test case presented by Straka et al. (1993) and for simulation of warm bubble, the test case of Mendea-Nunez and carrol (1993) are used. Qualitative and quantitative assessment of the results for different test cases showed the superiority of the MCRK4 and MC4 methods over the MC2 method.