Application of dispersion-relation-preserving theory to develop a two-dimensional convection-diffusion scheme

In this paper a finite difference scheme is developed within the nine-point semi-discretization framework for the convection-diffusion equation. The employed Pade approximation renders a fourth-order temporal accuracy and the spatial approximation of convection terms accommodates the dispersion relation. The artificial viscosity introduced in the two-dimensional convection-diffusion-reaction (CDR) equation for stability reasons is analytically derived. Constraints on the mesh size and time interval for rendering a monotonic matrix are also rigorously derived. To validate the proposed method, we investigate several problems that are amenable to the exact solutions. The results with good rates of convergence are obtained for the investigated scalar and Navier-Stokes problems.