Enhancing stability of correction procedure via reconstruction using summation-by-parts operators II: Modal filtering

TL;DR

This paper extends the correction procedure via reconstruction framework with summation-by-parts operators by incorporating modal filters, leading to stable, fully discrete schemes for hyperbolic conservation laws and introducing an adaptive filter strength strategy.

Contribution

It introduces modal filtering into the CPR/SBP framework, extending previous artificial dissipation methods, and proposes an adaptive strategy for filter strength with theoretical and numerical comparisons.

Findings

Stable schemes for linear advection and Burgers' equation achieved.

The adaptive filter strength strategy improves stability and accuracy.

Numerical experiments demonstrate effectiveness of modal filters.

Abstract

A recently introduced framework of semidiscretisations for hyperbolic conservation laws known as correction procedure via reconstruction (CPR, also known as flux reconstruction) is considered in the extended setting of summation-by-parts (SBP) operators using simultaneous approximation terms (SATs). This reformulation can yield stable semidiscretisations for linear advection and Burgers' equation as model problems. In order to enhance these properties, modal filters are introduced to this framework. As a second part of a series, the results of Ranocha, Glaubitz, \"Offner, and Sonar ("Enhancing stability of correction procedure via reconstruction using summation-by-parts operators I: Artificial dissipation", 2016) concerning artificial dissipation / spectral viscosity are extended, yielding fully discrete stable schemes. Additionally, a new adaptive strategy to compute the filter strength is introduced and different possible applications of modal filters are compared both theoretically and numerically.