Artificial Viscosity for Correction Procedure via Reconstruction Using Summation-by-Parts Operators

Abstract

We focus on spectral viscosity in the framework of correction procedure via reconstruction (CPR, also known as flux reconstruction) using summation-by-parts (SBP) operators. In Ranocha et al. (J Comput Phys 342:13–28, 2017), [10], Ranocha et al. (J Comput Phys 311:299–328, 2016), [9], the authors used SBP operators in the CPR framework and were able to recover and extend some results of Gassner (SIAM J Sci Comput 35(3):A1233–A1253, 2013), [1] and Vincent et al. (Comput Methods Appl Mech Eng 296:248–272, 2015), [12]. In this contribution, we introduce a viscosity term for a scalar conservation law and analyse this new setting in the context of CPR methods using SBP operators. We derive conditions on the viscosity term and the basis, which allow us to prove conservation and stability in the semidiscrete setting. Next, we extend semidiscrete stability results to fully discrete stability by an explicit Euler method. Numerical tests are presented, which verify our results (Ranocha, Enhancing stability of correction procedure via reconstruction using summation-by-parts operators I: artificial dissipation, 2016, [8]). This is an extension of the contribution Correction Procedure via Reconstruction Using Summation-by-parts Operators by Hendrik Ranocha (J Comput Phys 342:13–28, 2017), [10], Ranocha et al. (J Comput Phys 311:299–328, 2016), [9].