A locally analytic technique applied to grid generation by elliptic equations

One technique for obtaining grids for irregular geometries is to solve sets of elliptic partial differential equations. The solution of the partial differential equations yields a grid which discretizes the physical solution domain and also a transformation for the irregular physical domain to a regular computational domain. Expressing the governing equation of interest in the computational domain requires the derivatives of the physical to computational domain transformation, i.e., the metrics. These metrics are typically determined by numerical differentiation, which is a potential source of error. The locally analytic method uses the analytic solution of the locally linearized equation to develop numerical stencils. Thus, the locally analytic method allows numerical differentiation with no loss of accuracy. In this paper, the locally analytic method is applied to the solution of the Poisson and Brackbill–Saltzman equations. Comparison with an exact solution shows the locally analytic method to be more accurate than the finite difference method, both in solving the partial differential equation and evaluating the metrics. However, it is more computationally expensive.