Order Preserving Interpolation for Summation-by-Parts Operators at Non-Conforming Grid Interfaces

TL;DR

This paper introduces order preserving interpolation operators for summation-by-parts finite difference methods at non-conforming grid interfaces, maintaining stability and accuracy without reducing convergence order.

Contribution

It generalizes interface treatment to avoid order reduction, proving existence of suitable interpolation operators for any grid distribution, thus preserving stability and accuracy.

Findings

Order preserving interpolation operators exist for any grid distribution.

The new methods maintain stability and accuracy of underlying schemes.

Order reduction is avoided at non-conforming interfaces.

Abstract

We study non-conforming grid interfaces for summation-by-parts finite difference methods applied to partial differential equations with second derivatives in space. To maintain energy stability, previous efforts have been forced to accept a reduction of the global convergence rate by one order, due to large truncation errors at the non-conforming interface. We avoid the order reduction by generalizing the interface treatment and introducing order preserving interpolation operators. We prove that, given two diagonal-norm summation-by-parts schemes, order preserving interpolation operators with the necessary properties are guaranteed to exist, regardless of the grid-point distributions along the interface. The new methods retain the stability and global accuracy properties of the underlying schemes for conforming interfaces.