Relationship between the truncation errors of centered finite-difference approximations on uniform and nonuniform meshes

It is pointed out that two of the major problems facing the numerical analyst when constructing the numerical solution of partial differential equations are: (1) the numerical implementation of the boundary conditions along the boundaries of the physical space, and (2) the selection of the finite-difference mesh to represent the continuous physical space. The careful implementation of boundary conditions is essential. When higher-order accuracy is desired, serious problems are encountered. This has led to the extensive use of coordinate transformations to map nonuniform meshes in the physical space into uniform meshes in the transformed space. The present investigation is concerned with the relationship between the truncation errors of centered finite-difference approximations applied directly on the nonuniform physical mesh and the truncation errors of the same centered finite-difference approximations applied on the corresponding transformed uniform mesh. An analysis illustrates the effect of coordinate transformations on the accuracy of centered finite-difference approximation.