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On An Equal Fourth-Order-Accurate Temporal/Spatial Scheme for the Convection-Diffusion Equation

In this article, the convection-diffusion equation is discretized using the Pade method for the temporal derivative term and the wavenumber-extended method for the spatial derivative term. These temporal and spatial approximations result in two explicit equations and two implicitly coupled equations. To construct an equal-order scheme for the solution obtained at n Δt, both temporal/spatial derivatives are approximated to render fourth-order accuracy without using solutions obtained previously at (n − 2)Δt, (n − 3)Δt, etc. When approximating the first-order derivative term, it is essential to take the upwind nodal points into consideration. For revealing the dispersion and dissipation natures of the proposed scheme, both von Neumann (Fourier) and dispersion analyses were conducted. We validate the proposed method by solving several problems that are amenable to exact solutions. Results with theoretical rates of convergence are obtained for each of the one- and two-dimensional problems investigated.