A new class of high-order summation by parts finite-difference schemes

TL;DR

This paper introduces a new class of high-order finite-difference schemes based on summation by parts, achieving up to 12th order accuracy with improved stability and accuracy near boundaries.

Contribution

The authors develop novel SBP-based finite-difference schemes up to 12th order, optimizing boundary coefficients and avoiding spurious solutions, with detailed coefficients for practical use.

Findings

Identified 8th and 10th order schemes with superior accuracy.

Proposed boundary coefficient adjustments improve stability.

Validated schemes through test computations.

Abstract

We develop summation by parts (SBP) approach for generating high-order finite-difference schemes on the interval and propose new sets of schemes up to the 12th order. The coefficients of the schemes are governed by values of grid spacing near the ends of the interval: we shift one or two or three nodes at the ends of originally equidistant grid. The new finite-difference operators use forward and backward differences to avoid saw-tooth spurious solutions. Test computations point out two schemes (of 8th and 10th orders) that have the best accuracy among others; we yield the coefficients of new schemes.