The viability of finite difference stencils for higher-order derivatives of low-resolution data, encountered in various applications, is examined. This study’s illustrative example involves the evaluation of the fourth derivative of the digitized free surface radius obtained from pixelated images. Procedures to obtain an optimal approximation to the derivative are made from estimates of truncation and roundoff error, with emphasis on the analysis of roundoff error. A method of successive approximations to evaluate the optimal grid spacing is presented. Higher-order stencils allow for larger optimal grid spacing, thereby reducing the roundoff error that would otherwise dominate. Hence, higher-order stencils are effective for low precision data.

检查了在各种应用中遇到的低分辨率数据的高阶导数的有限差分模板的可行性。本研究的说明性示例涉及评估从像素化图像获得的数字化自由表面半径的四阶导数。根据截断和舍入误差的估计获得导数的最佳近似值的程序，重点是对舍入误差的分析。提出了一种评估最佳网格间距的逐次逼近方法。高阶模板允许更大的最佳网格间距，从而减少原本占主导地位的舍入误差。因此，高阶模板对于低精度数据是有效的。