论文标题
估计离散误差，具有预设的准确性和分数优化比率
Estimating Discretization Error with Preset Orders of Accuracy and Fractional Refinement Ratios
论文作者
论文摘要
解决方案的验证对于建立模拟的可靠性至关重要。一个核心挑战是找到离散误差的准确可靠的估计。当前的此估计方法取决于观察到的准确性顺序；但是，研究表明，它可能会不规则地改变或不确定。因此，我们提出了一种网格完善方法，该方法采用用户给出的恒定订单，称为预设订单扩展方法（Poem）。确保用户通过迭代获得最佳订单集，从而准确估计离散误差。该方法通过评估扩展项的收敛性来评估估计的可靠性，这对于所有网格细化方法至关重要。我们使用沿不同改进路径的对流和扩散问题证明了这些功能。当改进比较高时，诗需要较低的计算成本。但是，由于共享网格点的数量减少，估计的误差遭受了较高的不确定性。我们通过使用分数改进比和近似解决方案（MIDAS）之间插值差异的方法来阐明这一点。结果，我们可以在降低的计算成本下获得较低不确定性的离散误差的全球估计。
Verification of solutions is crucial for establishing the reliability of simulations. A central challenge is to find an accurate and reliable estimate of the discretization error. Current approaches to this estimation rely on the observed order of accuracy; however, studies have shown that it may alter irregularly or become undefined. Therefore, we propose a grid refinement method which adopts constant orders given by the user, called the Preset Orders Expansion Method (POEM). The user is guaranteed to obtain the optimal set of orders through iterations and hence an accurate estimate of the discretization error. This method evaluates the reliability of the estimation by assessing the convergence of the expansion terms, which is fundamental for all grid refinement methods. We demonstrate these capabilities using advection and diffusion problems along different refinement paths. POEM requires a lower computational cost when the refinement ratio is higher. However, the estimated error suffers from higher uncertainty due to the reduced number of shared grid points. We circumvent this by using fractional refinement ratios and the Method of Interpolating Differences between Approximate Solutions (MIDAS). As a result, we can obtain a global estimate of the discretization error of lower uncertainty at a reduced computational cost.
