Estimating Discretization Error with Preset Orders of Accuracy and Fractional Refinement Ratios
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In solution verification, the primary goal is finding an accurate and reliable estimate of the discretization error. A commonly used approach, however, has potential problems due to the use of the observed order of accuracy. Therefore, we propose a grid refinement method called the Preset Orders Expansion Method (POEM) which employs constant orders given by the user. With the scheme outlined in this paper, the user is guaranteed to obtain the optimal set of orders through iterations and subsequently an accurate estimate of the discretization error. Regarding the reliability of the estimation, the proposed method targets on the asymptotic convergence of numerical solutions, which is fundamental to all grid refinement methods. The above capabilities are demonstrated with problems in which multiple dimensions are refined. Moreover, POEM can be applied with using a fractional refinement ratio greater than 0.5. Although this can lower the computational demand, the estimated error will become more uncertain due to the reduction in the number of shared grid points. We circumvent this with the Method of Interpolating Differences between Approximate Solutions (MIDAS) which introduces additional shared grid points during refinement. As a result, the proposed grid refinement method is lifted to a practical level.
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