An accurate integral‐based scheme for advection–diffusion equation

Abstract

AbstractThis paper proposes an accurate integral‐based scheme for solving the advection–diffusion equation. In the proposed scheme the advection–diffusion equation is integrated over a computational element using the quadratic polynomial interpolation function. Then elements are connected by the continuity of first derivative at boundary points of adjacent elements. The proposed scheme is unconditionally stable and results in a tridiagonal system of equations which can be solved efficiently by the Thomas algorithm. Using the method of fractional steps, the proposed scheme can be extended straightforwardly from one‐dimensional to multi‐dimensional problems without much difficulty and complication. To investigate the computational performances of the proposed scheme five numerical examples are considered: (i) dispersion of Gaussian concentration distribution in one‐dimensional uniform flow; (ii) one‐dimensional viscous Burgers equation; (iii) pure advection of Gaussian concentration distribution in two‐dimensional uniform flow; (iv) pure advection of Gaussian concentration distribution in two‐dimensional rigid‐body rotating flow; and (v) three‐dimensional diffusion in a shear flow. In comparison not only with the QUICKEST scheme, the fully time‐centred implicit QUICK scheme and the fully time‐centred implicit TCSD scheme for one‐dimensional problem but also with the ADI‐QUICK scheme, the ADI‐TCSD scheme and the MOSQUITO scheme for two‐dimensional problems, the proposed scheme shows convincing computational performances. Copyright © 2001 John Wiley & Sons, Ltd.