Development of irregular-grid finite difference method (IFDM) for governing equations in strong form

Abstract

In this paper, an irregular-grid finite difference method (IFDM) with the use of Green-Gauss theorem was developed to numerically deal with problems subject to arbitrary geometrical boundaries. The focus is to elucidate the principle of IFDM and the numerical procedure to solve partial differential equations. Attention is paid to the discretization of spatial terms in partial differential format. Totally six types of discretization schemes are proposed and assessed from points of views of efficiency and accuracy. Theoretical analyses are conducted with regard to the compactness of stencil and positivity of the coefficients of supporting nodes. Upon the analyses, two schemes, II and VI, are selected for further study. Scheme II is based on one-point quadrature rule while scheme VI corresponds to 2-point quadrature rule. Numerical excises by using the two schemes are carried out to predict the solutions in a square domain that is governed by a Poisson equation. The effects of irregularity of grids are also studied. Numerical results indicate that the both schemes give satisfactory prediction. In addition, the scheme VI gives better accuracy, especially on irregular grids, but at slightly higher cost in computation. Efforts here demonstrate that the proposed IFDM method is good to be used in numerical simulations with arbitrarily geometrical bounds.