Abstract
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.
Highlights
The two-dimensional elliptic equations–ε∇ U + a(x, y)Ux + b(x, y)Uy + c(x, y)U + d(x, y) =, (x, y) ∈, ( . )will be considered to develop numerical algorithms for computing the concentration U(x, y) of mass transfer
The solution of singular perturbation problems approaches a discontinuous limit as a small positive quantity (ε), known as perturbation parameter, approaches zero
In Section, we describe two-dimensional quasi-variable meshes to deal with parallel and normal layer by means of mesh parameters in the x- and y-directions
Summary
) describe the diffusion process, while the first-order partial derivatives are associated with the convection phenomenon. The boundary layer may occur at x = and x = , known as normal layer, and/or at y = and y = , known as parallel layer, while the one-dimensional convection-diffusion problems exhibit only normal boundary layer [ ]. Such type of a differential equation is said to be singularly perturbed. The analysis and numerical solution of singular perturbation problems are significant
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