An Implicit Scheme for Solving the Convection–Diffusion–Reaction Equation in Two Dimensions

In this paper we consider a passive scalar transported in two-dimensional flow. The governing equation is that of the convection–diffusion–reaction equation. For purposes of computational efficiency, we apply an alternating-direction implicit scheme akin to that proposed by Polezhaev. Use of this implicit operator-splitting scheme allows the application of a tridiagonal Thomas solver to obtain the solution. Within each solution step, a semidiscretization scheme is applied to discretize the differential equation in one dimension. We approximate the time derivative term using a forward time-stepping scheme. The resulting inhomogeneous differential equation has only the spatial derivative terms and is solved using a newly proposed nodally exact steady-state convection–diffusion–reaction scheme. Details on the development of the flux discretization scheme are provided. Modified equation analysis, Fourier stability analysis, and a study on scheme monotonicity are also performed to shed further light on the proposed transient scheme. To validate the proposed scheme, we first consider test problems which are amenable to analytic solutions. Good agreement is obtained with both one- and two-dimensional steady/unsteady problems, thus demonstrating the validity of the method.