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We discuss a new nine-point fourth-order and five-point second-order accurate finite-difference scheme for the numerical solution of two-space dimensional convection-diffusion problems. The compact operators are defined on a quasi-variable mesh network with the same order and accuracy as obtained by the central difference and averaging operators on uniform meshes. Subsequently, a high-order difference scheme is developed to get the numerical accuracy of order four on quasi-variable meshes as well as on uniform meshes. The error analysis of the fourth-order compact scheme is described in detail by means of matrix analysis. Some examples related with convection-diffusion equations are provided to present performance and robustness of the proposed scheme.

A fourth-order accurate quasi-variable mesh compact finite-difference scheme for two-space dimensional convection-diffusion problems