Application of the fourth-order compact MacCormack scheme with a four-stage Runge- Kutta time marching for numerical solution of unsteady and non-linear Rossby adjustment problem

By increasing the computing power of computers, the advantage of high-resolution numerical methods for numerical simulation of the governing equations of fluid flow is further emphasized. Recently, increasing the accuracy of numerical methods used for simulation of fluid dynamics problems, particularly the geophysical fluid dynamics problems (e.g., shallow water equations) has been the subject of many research works.
The compact finite difference schemes can provide a simple way to reach the main objectives in the development of numerical algorithms, i.e., having a low cost on the one hand and a highly accurate computational method on the other hand. These methods have also been used for numerical simulation of some geophysical fluid dynamics problems.
However, by splitting the derivative operator of a l compact centra method into one-sided forward and backward operators, a family of compact MacCormack-type schemes can be derived (Hixon and Turkel, 2000). While these classes of compact methods are as accurate as the original compact central methods used to derive the one-sided forward and backward operators, they need less computational work per grid point.
The present work is devoted to the assessment of the accuracy of different methods. The one-dimensional advection equation with the known analytical solution is employed as a prototype model. Also, the truncation error of the traditional second-order MacCormack scheme, the standard fourth-order compact Mac-Cormack scheme, and a fourth-order compact MacCormack scheme with a four-stage Runge–Kutta time marching method are studied. Furthermore, to be able to examine the accuracy, the Lax–Wendroff, the leap-frog and the Beam–Warming methods combined with the second-order and fourth-order compact finite difference methods for spatial differencing are also used. In addition, the convergence rates of different methods are studied. It can be seen that the convergence rates are in agreement with the theoretical order of convergence.
In this work, the traditional second-order MacCormack scheme (MC2), the standard fourth-order compact Mac-Cormack scheme (MC4) developed by Hixon and Turkel (2000) and a fourth-order compact MacCormack scheme with a four-stage Runge–Kutta time marching method (MCRK4) are used for numerical solution of the unsteady and non-linear Rossby adjustment problem (one- and two-dimensional cases). In the one-dimensional case, a single layer shallow water model is used to study the unsteady and nonlinear Rossby adjustment problem. The conservative form of the two-dimensional shallow water equations is used to study the unsteady and nonlinear Rossby adjustment problem in the two-dimensional case. For both cases, the time evolution of a fluid layer initially at rest with a discontinuity in height filed is considered for numerical simulations.