Approximation of a Complex Geometric Domain in Polar Coordinates

Abstract

In this study, a complex geometric domain having a colour picture is approximated through a stair- step representation of the coastal and island boundaries to make it suitable for implementing finite difference method in solving shallow water equations (SWEs) in polar coordinates. As a complex domain, we choose the coastal region of Bangladesh situated at the northern tip of the Bay of Bengal (BOB). To cover the whole coastal region, the pole is selected at the point in the  plane assuming it on the mean sea level (MSL). Along the tangential direction, 265 uniformly distributive straight lines are considered through the pole and 959 circular grid lines centered at  are drawn towards the radial direction covering up to  latitude in the BOB. Firstly, a matrix with 960´265 computational grids is constructed from the colour information of the picture. By representing the grids with suitable notations, a proper stair-step algorithm is employed to the matrix obtained with the 960´265 grids to approximate the coastal and island boundaries to the nearest finite difference grid lines using an Arakawa C-grid system. The whole procedure is done with our developed MATLAB program. The grids representing the coastal stations are also identified closely in the obtained approximated domain. Such a type of presentation of the coastal geometry of the region of interest is found to incorporate its complexities properly with minimum computational grid points.

GANIT J. Bangladesh Math. Soc.Vol. 38 (2018) 105-118