The cubic interpolation spline for functions with boundary layer on a Bakhvalov mesh

Abstract

The problem of cubic spline interpolation on the Bakhvalov mesh of functions with region of large gradients is considered. Asymptotically accurate error estimates O(N −4) are obtained for a class of functions with an exponential boundary layer in case 1/N ≤ ε, where N is number of nodes, ε is small parameter. In case ε ≤ 1/N we have experimentally shown that the error estimates of traditional spline interpolation are not uniform in a small parameter, and the error itself can increase indefinitely when the small parameter tends to zero at a fixed number of nodes N. A modified cubic spline is proposed for which uniform estimates of the order O(N −4) have been experimentally confirmed.

Analysis of approaches to spline interpolation of functions with large gradients in the boundary layer

“…Table 5 shows the error and the order of accuracy of the exponential spline on a uniform grid. The calculation results show that as ε decreases, the order of accuracy decreases from the fourth to the third, which is consistent with the estimate (14).…”

“…In accordance with the estimate (14), for ε = 1, the error of the spline is of the order of O(h 4 ). This is consistent with the fact that as ε/h increases, the spline becomes cubic [17].…”

A Scheme for Estimating Time-Varying Wind Stress Drag Coefficient in the Ekman Model with Adjoint Assimilation

“…However, the Cressman that was interpolation traditionally used in the IP scheme can result in unsmooth curves. Meanwhile, we found that the cubic spline interpolation (CSI) scheme was often used in practice because of its favorable smoothness and continuity [25][26][27][28]. Therefore, we try to use the CSI instead of linear interpolation to invert the time-varying WSDC in an Ekman model.…”