Analysis of coupled grid methods for solving convection-diffusion equations

Abstract

The error behavior of finite difference discretization schemes has been researched. Starting from the onedimensional case, where two particles of a fluidized bed were taken, the diffusion equation was numerically solved on a single grid as reference. Then the diffusion equation, a model equation for the real problem, was solved on a separated grids setup, where the grid spacing of the grid adjacent to the particles is relatively fine and the other grid relatively coarse. The main problem is the communication/coupling between these two grids. The research question of this report is: Is it possible to couple the two grids in a way that the numerical solution on the coupled grids has the same order of error behavior as the numerical solution on a single grid? If this is possible what are the minimal conditions for the coupling strategy? Different strategies of this coupling are proposed. The order of the error of the numerical solution was first estimated by Richarson error estimation and then computed by the means of the L2-norm, where the numerical solution is compared with the analytical solution. Coupling by using an interpolation polynomial of at least second order in both directions resulted in an error behavior of second order of the numerical solution, which is the same error behavior as the numerical solution on a single grid. Hereafter the problem is extended to two dimensions where also a convective term is taken into account. The results of the coupling from the one-dimensional problem were used for the two dimensional problem. For the two-dimensional problem the L2-error could also be determined. The conclusion is the same as for the one dimensional case: At least a second order interpolation polynomial is required to get second order error behavior.