Summation-by-parts operators for general function spaces: Flux differencing, entropy stability, and accuracy
Summation-by-parts (SBP) operators are a cornerstone of provably stable high-order methods for hyperbolic conservation laws. For nonlinear problems, the flux-differencing framework of Fisher and Carpenter combines such SBP operators with entropy-conservative two-point fluxes to obtain semi-discretizations that are conservative in the sense of Lax--Wendroff and discretely entropy-conservative or stable. So far, this framework has been developed exclusively for operators that are exact for polynomials. Function-space SBP (FSBP) operators lift this restriction by being exact on a general finite-dimensional function space that can be tailored to the problem at hand. In this work, we extend flux differencing to FSBP operators for general (potentially non-polynomial) function spaces. We prove that, whenever the operator has a diagonal norm and its function space contains the constants, the resulting semi-discretization is conservative in the sense of Lax--Wendroff, semi-discretely entropy-conservative within a single element, and entropy-stable across element interfaces. We also provide the first theoretical accuracy analysis of FSBP operators: We characterize their design order through an algebraic criterion on the underlying function space, prove that suitable non-polynomial spaces achieve the same design order as polynomial spaces of equal dimension, and show that the flux-differencing semi-discretization retains this order. Notably, the entropy-conservative two-point flux is reused unchanged; only the underlying function space differs from the polynomial case. Numerical experiments for the Burgers' equation and the compressible Euler equations in one and two dimensions confirm these properties.