Finite volume and finite difference discretization on nonuniform grids

Abstract

Nonuniform grids are often used to obtain accuracy in regious where the solution varies rapidly. We will see that on nonuniform grids, finite volume and finite difference discretization are not equivalent. On arbitrary nonuniform grids the local truncation error is usually larger than on uniform grids, or grids on which the mesh size varies smoothly. This has sometimes led to confusion. Cell-centered finite volume discretization is sometimes advised against, because the local truncation error is larger than for vertex-centered finite volumes, and is in fact of the same order even as the term that is approximated. Nevertheless, this is a good discretization method that is popular in reservoir engineering and porous media flow computation, and in computational fluid dynamics in general. The source of the confusion is that the relation between the local and global truncation error is complicated. Surprisingly, the global truncation error is small, as we will see. Of course, it is the global truncation error that counts.