Generalised summation-by-parts operators and variable coefficients

High-order methods for conservation laws can be highly efficient if their stability is ensured. A suitable means mimicking estimates of the continuous level is provided by summation-by-parts (SBP) operators and the weak enforcement of boundary conditions. Recently, there has been an increasing interest in generalised SBP operators both in the finite difference and the discontinuous Galerkin spectral element framework. However, if generalised SBP operators are used, the treatment of the boundaries becomes more difficult since some properties of the continuous level are no longer mimicked discretely --- interpolating the product of two functions will in general result in a value different from the product of the interpolations. Thus, desired properties such as conservation and stability are more difficult to obtain. Here, new formulations are proposed, allowing the creation of discretisations using general SBP operators that are both conservative and stable. Thus, several shortcomings that might be attributed to generalised SBP operators are overcome (cf. J.~Nordstr\"om and A.~A.~Ruggiu, "On Conservation and Stability Properties for Summation-By-Parts Schemes", \emph{Journal of Computational Physics} 344 (2017), pp. 451--464, and J. Manzanero, G. Rubio, E. Ferrer, E. Valero and D. A. Kopriva, "Insights on aliasing driven instabilities for advection equations with application to Gauss-Lobatto discontinuous Galerkin methods", Journal of Scientific Computing (2017), https://doi.org/10.1007/s10915-017-0585-6).