Enhancing stability of correction procedure via reconstruction using summation-by-parts operators II: Modal filtering
For researchers in numerical methods for hyperbolic PDEs, this provides a stable and adaptive filtering approach within the CPR-SBP framework, though it is an incremental extension of existing work.
This paper extends the correction procedure via reconstruction (CPR) framework with summation-by-parts operators by incorporating modal filters to enhance stability for hyperbolic conservation laws. It introduces an adaptive strategy for filter strength and demonstrates fully discrete stable schemes, extending prior work on artificial dissipation.
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, Öffner, 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.