A superconvergent stencil-adaptive SBP-SAT finite difference scheme

TL;DR

This paper introduces a stencil-adaptive SBP-SAT finite difference scheme that achieves superconvergence, significantly improving accuracy for the linear advection equation compared to conventional methods.

Contribution

The paper demonstrates that the stencil-adaptive SBP-SAT scheme exhibits superconvergence with an $ ext{O}( ext{Δ}x^4)$ rate, surpassing the traditional $ ext{O}( ext{Δ}x^3)$ rate.

Findings

Achieves $ ext{O}( ext{Δ}x^4)$ convergence rate for linear advection.

Outperforms conventional schemes in accuracy.

Validates superconvergent behavior through numerical experiments.

Abstract

A stencil-adaptive SBP-SAT finite difference scheme is shown to display superconvergent behavior. Applied to the linear advection equation, it has a convergence rate $\mathcal{O}(\Delta x^4)$ in contrast to a conventional scheme, which converges at a rate $\mathcal{O}(\Delta x^3)$.