Numerical schemes for advection-diffusion problems are oftenused with non-uniform grids. Non-uniform grids are known to greatlycomplicate the convergence analysis and their use thereforeis much less straightforward than for uniform grids. For example,it is possible that a scheme which is inconsistent at the level ofthe local truncation error truly converges with order two.The purpose of this paper is to contribute to the theory ofspatial discretizations on non-uniform grids.We shall present spatial convergence results for a number ofvertex and cell centered schemes for the linear 1D time-dependent advection-diffusion problem. The focus hereby lies on the discrepancy between local and global order.

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Modelling, Analysis and Simulation [MAS]
Computational Dynamics

Hundsdorfer, W., & Verwer, J. (2001). On the convergence analysis of advection-diffusion schemes on non-uniform grids. Modelling, Analysis and Simulation [MAS]. CWI.