Investigations in high-order accurate finite difference schemes for non-uniform grids
University of New Brunswick
In this work we develop fourth-order accurate, forward and backward finite difference schemes for the first derivative. Both uniform and non-uniform finite difference expressions are developed and applied in the study of a two-dimensional viscous fluid flow through an irregular domain. The von Mises transformation is used to transform the governing equations, and map the irregular domain onto a rectangular computational domain. Vorticity on the solid boundary (given in terms of the first partial derivative of the tangential velocity component) is expressed in terms of the first partial derivative of the square of the speed of the flow in the computational domain, and the derived finite difference schemes are used to calculate the vorticity at the computational boundary grid points using up to five computational domain grid points. This work extends previous work in which first-order schemes were devised for the first derivative. The aim here is to shed further light onto the use of first- and higher- order accurate non-uniform finite difference schemes that are essential when the von Mises transformation is used. Furthermore, a fourth-order accurate finite difference scheme is developed in this work using five computational grid points and compared with the lower-order accurate schemes. In all schemes developed, we study the effect of coordinate clustering on the computed results.