Summation-by-parts operators for general function spaces: optimal nodes
This work provides a theoretical and algorithmic foundation for constructing optimal SBP operators in non-polynomial settings, benefiting numerical simulations of PDEs.
The authors extend the optimality of Gauss-Lobatto quadrature for summation-by-parts (SBP) operators from polynomial to general function spaces, showing that generalized Gauss-Lobatto quadrature yields minimal-dimension SBP operators. They provide an algorithm and demonstrate accuracy and efficiency across various function spaces.
Gauss-Lobatto quadrature nodes and weights are optimal for closed summation-by-parts (SBP) formulations based on polynomial approximation spaces in the sense that for a prescribed function space they yield an SBP operator of minimal dimension. We show that the same principle extends to general (possibly non-polynomial) function spaces: an associated generalised Gauss-Lobatto quadrature provides the optimal nodes and weights for the SBP formulation. We present an algorithm for computing these quadrature rules, demonstrate their accuracy and efficiency across a range of function spaces, and illustrate their use in solving initial boundary value problems.