Numerical Schemes for the Convection-Diffusion Equation Using a Meshless Finite-Difference Method

Abstract

A meshless finite-difference method is developed for solving the steady convection-diffusion equation. A weighted least-squares procedure is used to compute a local polynomial fit which is used to find the derivatives appearing in the governing partial differential equation. A number of upwind-weighted discretization schemes for the convective operator are developed, analogous to traditional finite-difference schemes. These include a first-order upwind and a second-order upwind scheme, as well as a new scheme, the minimum gradient scheme, analogous to the essentially nonoscillatory (ENO) scheme. The order of accuracy of these methods is studied by applying them to two test problems. The first is the convection and diffusion of a scalar in a uniform velocity field, and the second pertains to scalar transport in a vortical flow field. The first-order upwind scheme results in an oscillation-free solution across the range of Peclet numbers and displays an order of accuracy close to unity. The second-order upwind scheme results in spurious oscillations for coarse nonuniform point distributions for Peclet numbers greater than 2.0, but monotonic solutions are obtained for finer point distributions across the range of Peclet numbers. The minimum gradient scheme yields nonoscillatory solutions for all Peclet numbers and point distributions explored in this work.