| Abstract: | The discretization of differential equations to obtain numerical solutions introduces truncation error (TE). The form of the TE depends on the differential equation, the discretization scheme, and the discretization grid in a fairly complicated fashion. The character and magnitude of the TE determines in part the discrepancy between the approximate and exact solutions to the PDE. In this study, a symbolic manipulator was used to investigate the geometrical dependence of the TE in discrete approximations to Laplace's equation. The effect of various grid irregularities on the leading TE terms was investigated for two discretization schemes. A nine point centered stencil was varied in aspect ratio, skewness, and non-uniformity (irregular spacing). The first scheme was the common centered finite difference (FD) method, applied by transforming to a uniform, orthogonal computational space. The second scheme was the implicit interpolation method ( II ), developed for discretization on irregular grids. Both discretization methods had leading TE that was second order (inversely proportional to the number of grid points squared) and had a sharp rise in TE for skewness angles beyond sixty degrees. The FD method skewness errors were greater than those for the II method, but high aspect ratio errors were less. For non-linear transformations the FD method TE contained second derivatives, the same order as the governing differential equation, while the II method's lowest order TE derivatives were third order. Thus while both methods have the same order of accuracy, the nature of the numerical error in the discrete solution would be different. |
