Truncation and Discretization Error for Diffusion Schemes on Unstructured Meshes

Abstract

The accuracy and reliability of CFD simulations depend on the ability to reduce and quantify physical modeling and numerical errors. The fact that numerical errors are at least as large as physical modeling errors was highlighted in the results of the 3rd AIAA Drag Prediction workshop, which also showed the increased severity of this issue for unstructured ow solvers. Furthermore, the varied local shape and connectivity of unstructured meshes make it di cult to quantify numerical error. The 5th AIAA Drag Prediction workshop focussed on reducing grid-related errors even further, where a grid re nement study was performed on a common grid sequence derived from a multiblock structured grid. The study had six di erent levels of grid re nement ranging from 136× 10 cells to 0.64× 10 cells, a much larger range than is typically seen, with structured overset and hexahedral, prismatic, tetrahedral, and hybrid unstructured grid formats. The results of the grid re nement study indicated that there was no clear advantage of any one grid type in terms of a reduced scatter in solution. Moreover, there were no clear breakouts with grid type or turbulence model. The conclusion was that discretization errors and turbulence modeling errors are both still major contributors to error in solution. The impact on solution accuracy by the interactions between mesh quality (cell size, shape, and anisotropy) and discretization schemes is not well understood and demands further investigation. The di erence between the discrete operator and the continuous PDE applied to the solution is referred to as the truncation error, while the di erence between the numerically approximated solution and the exact solution is the discretization error. The truncation error can be expressed in terms of the derivatives of an underlying smooth solution at the points of the discrete domain and can be used to estimate the discretization errors that occur during the approximate numerical solution of PDEs. It can be shown that for the special case of a linear di erential operator L, the truncation error τ can be used to calculate the discretization error e, as