Summation-by-parts operators for correction procedure via reconstruction

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• 作者：Hendrik Ranocha ; ^{h.ranocha@tu-bs.de" class="auth_mail" title="E-mail the corresponding author} ; Philipp Ö ; ffner ; Thomas Sonar
• 关键词：Hyperbolic conservation laws ; High order methods ; Summation-by-parts ; Flux reconstruction ; Lifting collocation penalty ; Correction procedure via reconstruction
• 刊名：Journal of Computational Physics
• 出版年：2016
• 出版时间：15 April 2016
• 年：2016
• 卷：311
• 期：Complete
• 页码：299-328
• 全文大小：1985 K

文摘

The correction procedure via reconstruction (CPR, formerly known as flux reconstruction) is a framework of high order methods for conservation laws, unifying some discontinuous Galerkin, spectral difference and spectral volume methods. Linearly stable schemes were presented by Vincent et al. (2011, 2015), but proofs of non-linear (entropy) stability in this framework have not been published yet (to the knowledge of the authors). We reformulate CPR methods using summation-by-parts (SBP) operators with simultaneous approximation terms (SATs), a framework popular for finite difference methods, extending the results obtained by Gassner (2013) for a special discontinuous Galerkin spectral element method. This reformulation leads to proofs of conservation and stability in discrete norms associated with the method, recovering the linearly stable CPR schemes of Vincent et al. (2011, 2015). Additionally, extending the skew-symmetric formulation of conservation laws by additional correction terms, entropy stability for Burgers' equation is proved for general SBP CPR methods not including boundary nodes.