A type of high order schemes for steady convection-diffusion problems

Based on the viewpoint that the flow direction can be accounted for by adding more weigh of the contribution of the upstream grid point in evaluating the transport fluxes at the cell faces for the convection-dominated flows, a type of high order schemes are proposed for solving convection-diffusion equations. These schemes share the same main algorithm which is developed by integrating the convection-diffusion equation expressed as the first order partial differential equations over selected regions and applying the numerical quadrature with a weighting parameter for approximating the resulting integrals. Interestingly, the error analysis allows us to determine the expressions of the weighting parameter for a series of schemes with the third order scheme as the lowest and the infinite order as the highest order scheme for the source free convection-diffusion problems. Numerical results show that the highest order scheme achieves almost the same accuracy as the exact solution, and does not induce any unphysical oscillation for the convection-dominated flows. Even the third order scheme shows obvious advantages over the traditional finite volume method (FVM) and QUICK scheme in dealing with the multi-dimensional convection-diffusion problems.