# Numerical Methods for Solving Regional and Mesoscale Problems

DOI: 10.4018/978-1-5225-2636-0.ch009

## Abstract

Regular grids with even steps of the spatial coordinates in the whole computational domain are the most convenient for implementing numerical methods for integration of equations of weather forecasts. However, computing a local numerical weather forecast based on the global general circulation models of the atmosphere will need enormous increase in computation time exceeding reasonable limits. Moreover, as some regional weather details are well localized it is reasonable to apply high-resolution grids locally. In this chapter, we study how to use the high-resolution grids in the numerical methods for solving regional and mesoscale weather forecast problems.
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## Brief Introduction

When using numerical methods for integration of equations for fluid flow, heat and the mass transfer, the regular grid with even steps of the spatial coordinates in the whole computational domain are most convenient conditions. However, computing associated with the numerical weather forecast for the global general circulation models of the atmosphere, even for the most powerful modern computers, limits the size of the bottom of the grid used. Breaking grids to a size suitable for describing the regional and small-scale processes and phenomena in the atmosphere causes an increase in computation time exceeding reasonable limits. As some of the regional weather details are well localized (atmospheric disturbances associated with orographic and thermal heterogeneity of the underlying surface), of course, there appears the idea of a local field of the high-resolution grid.

Separate modeling of regional atmospheric processes, in which the boundary conditions are obtained from the initial to the forecast point of time according to synoptic observations are considered unchanged for the time interval calculation, or extrapolated at each time step to the side edges of the inner part of the solution region, now is only of scientific interest. Stated thus lateral boundary conditions lead to large errors, and sometimes to a false numerical solution because the errors generating fictitious wave with large amplitude at the borders and in the vicinity of them, extends deep into the area of solutions (Asselin, 1972; Davies, 1976; Gustafsson, 1975; Hill, 1968; Kreiss, 1970; Kreiss, 1980; Miyakoda, Rosati, 1977; Oliger, Sundstrom, 1978; Perkey, Kreitzberg, 1976; Shapiro, O'Brien, 1970; Sundstrom, 1986; Williamson, Browning, 1974). It is obvious that to reduce the forecast error in the restricted area, one must add a buffer zone to expand the boundaries of the area to such an extent that generated the disturbance at the boundaries do not reach the computational domain. Increasing the area of solutions entails increasing computer power requirements and, thus, leads to difficulties in the practical realization of such models.

In recent years, regional modeling of atmospheric processes is implemented in view of the fact that the field of meteorological variables in a bounded domain is formed under the influence of macro-scale atmospheric circulation. Therefore, a bounded domain of solutions is seen as part of a whole and non-stationary boundary conditions at its lateral boundaries are formulated on the basis of data obtained from a bordering region. Thus, in practice, when solving numerical weather prediction problem in a limited area, they make the grid more condensed to achieve the desired accuracy for solving the problem in the field of large gradients of dependent functions.

As the use of variable resolution grids in the spectral methods causes considerable inconvenience, we will analyze the problems arising from the application of non-uniform grids to difference schemes of numerical integration of the equations of hydrodynamics and heat and mass transfer, members of regional weather prediction model.

The model with variable resolution can be realized in two ways (Prusov, Doroshenko, 2006). In the first one, you can use a common mathematical model with continuously or stepwise changing density grid (the algorithm of “bilateral interaction”). In this case, there are a number of problematic issues related to the change of the grid step. They belong to the local and global effects of reflection, transmission and conversion of different waves at the boundaries of the variable resolution grid. In addition, most of the difference schemes are the most accurate in the constant grid steps. Even with the gradually changes of grid step in the transition zone, there are errors exceeding the error in the rest of the grid. If this transition occurs abruptly, the error value can be increased by an order of magnitude.

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