The third-order polynomial method for two-dimensional convection and diffusion

Using the upstream polynomial approximation a series of accurate two-dimensional explicit numerical schemes is developed for the solution of the convection-diffusion equation. A third-order polynomial approximation (TOP) of the convection term and a consistent second-order approximation of the diffusion term are combined in a single-step flux-difference algorithm. Stability analysis confirms that the TOP-12 scheme satisfies the CFL condition for two dimensions. Using smaller and narrower flux stencils compared to algorithms of similar accuracy, the TOP-12 scheme is more efficient in terms of computations per single node. Numerical tests and comparison with other well-known algorithms show a high performance of the developed schemes.