On the efficiency and accuracy of interpolation methods for spectral codes

TL;DR

This paper introduces a B-spline based interpolation method for spectral codes that offers higher continuity, fewer FFTs, and comparable accuracy to Hermite interpolation, supported by a general theory linking interpolation order and spectral properties.

Contribution

The paper develops a general theory for interpolation on rectangular grids and presents a fast, efficient B-spline based interpolation method with superior spectral properties.

Findings

Higher order of continuity than other methods

Requires only one FFT for implementation

Error nearly matches Hermite interpolation

Abstract

In this paper a general theory for interpolation methods on a rectangular grid is introduced. By the use of this theory an efficient B-spline based interpolation method for spectral codes is presented. The theory links the order of the interpolation method with its spectral properties. In this way many properties like order of continuity, order of convergence and magnitude of errors can be explained. Furthermore, a fast implementation of the interpolation methods is given. We show that the B-spline based interpolation method has several advantages compared to other methods. First, the order of continuity of the interpolated field is higher than for other methods. Second, only one FFT is needed whereas e.g. Hermite interpolation needs multiple FFTs for computing the derivatives. Third, the interpolation error almost matches the one of Hermite interpolation, a property not reached by other methods investigated.