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UBC Theses and Dissertations

Numerical estimation of discretization error on unstructured meshes Yan, Kai Kin Gary


A numerical estimation of discretization error is performed for solutions to steady and unsteady models for compressible flow. An accurate and reliable estimate of discretization error is useful in obtaining a more accurate defect corrected solution, as well as a tight uncertainty bound as error bars. The error estimation procedure is performed by solving an auxiliary problem, known as the error transport equation (ETE), solved on the same mesh as the original model equations. Unlike unsteady adjoint methods for error in functionals, the ETE requires only one other set of equations to be solved, agnostic to the choice and number of output functionals, including common aerodynamic quantities such as lift or drag. Furthermore, co-advancing the ETE in time only requires the storage of local solutions in time and not the entire history, reducing memory requirements. This method of error estimation is performed in the context of higher order finite-volume methods on unstructured meshes. Approaches based on solving the ETE can be found in the literature for uniform or smooth meshes, but this has not been well studied for unstructured meshes. Such meshes necessarily have nonsmooth geometric features, which create many difficulties in accurate error estimation. These difficulties in accurate discretizations of the ETE are investigated, including the discretization of the ETE source term, which is critical to error estimate accuracy. It was found that the proposed schemes by the ETE approach can be more efficient and robust compared to solving the higher order problem. The choices of discretization schemes need to be made carefully, and these results demonstrate how it is possible, along with justification, to obtain asymptotically accurate, efficient, and robust error estimates that can be used with vast possibilities of model equations in practice.

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