A High-Accuracy Finite Difference Scheme for Solving Reaction-Convection-Diffusion Problems with a Small Diffusivity (Advances in Applied Mathematics and Mechanics, 2014)
This paper is devoted to a new high-accuracy finite differencescheme for solving reaction-convection-diffusion problemswith a small diffusivity $ε$.With a novel treatment for the reaction term, we first derive a difference schemeof accuracy $\mathcal{O}(ε^2 h + ε h^2 + h^3)$ for the 1-D case.Using the alternating direction technique, we then extend the schemeto the 2-D case on a nine-point stencil.We apply the high-accuracy finite difference scheme to solve the 2-D steadyincompressible Navier-Stokes equations in the stream function-vorticity formulation.Numerical examples are given to illustrate the effectivenessof the proposed difference scheme.Comparisons made with some high-order compact difference schemesshow that the newly proposed scheme can achieve good accuracywith better stability.