An algorithmic exploration of the existence of high-order summation by parts operators with diagonal norm
For researchers in numerical methods for PDEs, this work provides a tool to explore high-order SBP operators, though the contribution is incremental as it builds on existing theory.
This paper presents a new algorithm to determine the existence of diagonal-norm summation by parts (SBP) operators, discovering previously unknown operators with interior order of accuracy up to 2s=30. It also introduces a method to optimize the spectral radius of the SBP derivative, validated through numerical experiments.
This paper explores a common class of diagonal-norm summation by parts (SBP) operators found in the literature, which can be parameterized by an integer triple $(s,t,r)$ representing the interior order of accuracy ($2s)$, the boundary order of accuracy ($t$), and the dimension of the boundary closure ($r$). There is no simple formula for determining whether or not an SBP operator exists for a given triple of parameters. Instead, one must check that certain compatibility conditions are met: namely that a particular linear system of equations has a positive solution. Partly because of the complexity involved, not much is known about diagonal-norm SBP operators with $2s>10$. By utilizing a new algorithm for answering the question "Does an SBP operator exist for the parameters $(s,t,r)$?", it is possible to explore the existence of SBP operators with high order accuracy, and previously unknown SBP operators with interior order of accuracy as large as $2s=30$ are found. Additionally, a method for optimizing the spectral radius of the SBP derivative is introduced, and the effectiveness of this method is explored through numerical experiment.