Development of a class of multiple time‐stepping schemes for convection–diffusion equations in two dimensions

Abstract

SUMMARYIn this paper we present a class of semi-discretization ÿnite di erence schemes for solving the transient convection-di usion equation in two dimensions. The distinct feature of these scheme developments is to transform the unsteady convection-di usion (CD) equation to the inhomogeneous steady convectiondi usion-reaction (CDR) equation after using di erent time-stepping schemes for the time derivative term. For the sake of saving memory, the alternating direction implicit scheme of Peaceman and Rachford is employed so that all calculations can be carried out within the one-dimensional framework. For the sake of increasing accuracy, the exact solution for the one-dimensional CDR equation is employed in the development of each scheme. Therefore, the numerical error is attributed primarily to the temporal approximation for the one-dimensional problem. Development of the proposed timestepping schemes is rooted in the Taylor series expansion. All higher-order time derivatives are replaced with spatial derivatives through use of the model di erential equation under investigation. Spatial derivatives with orders higher than two are not taken into account for retaining the linear production term in the convection-di usion-reaction di erential system. The proposed schemes with second, third and fourth temporal accuracy orders have been theoretically explored by conducting Fourier and dispersion analyses and numerically validated by solving three test problems with analytic solutions.