SBP operators for CPR methods : Master's thesis

Summation-by-parts (SBP) operators have been used in the finite difference framework, providing means to prove conservation and discrete stability by the energy method, predominantly for linear (or linearised) equations. Recently, there have been some approaches to generalise the notion of SBP operators and to apply these ideas to other methods. The correction procedure via reconstruction (CPR), also known as flux reconstruction (FR) or lifting collocation penalty (LCP), is a unifying framework of high order methods for conservation laws, recovering some discontinuous Galerkin, spectral difference and spectral volume methods. Using a reformulation of CPR methods relying on SBP operators and simultaneous approximation terms (SATs), conservation and stability are investigated, recovering the linearly stable CPR schemes of Vincent et al. (2011, 2015). Extensions of SBP methods with diagonal-norm operators to Burgers’ equation are possible by a skew-symmetric form and the introduction of additional correction terms. An analytical setting allowing a generalised notion of SBP methods including modal bases is described and applied to Burgers’ equation, resulting in an extension of the previously mentioned skew-symmetric form. Finally, an extension of the results to multiple space dimensions is presented.