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This paper explores a common class of diagonal-norm summation by parts (SBP)operators found in the literature, which can be parameterized by an integertriple $(s,t,r)$ representing the interior order of accuracy ($2s)$, theboundary order of accuracy ($t$), and the dimension of the boundary closure($r$). There is no simple formula for determining whether or not an SBPoperator exists for a given triple of parameters. Instead, one must check thatcertain compatibility conditions are met: namely that a particular linearsystem of equations has a positive solution. Partly because of the complexityinvolved, not much is known about diagonal-norm SBP operators with $2s>10$.  By utilizing a new algorithm for answering the question "Does an SBP operatorexist for the parameters $(s,t,r)$?", it is possible to explore the existenceof SBP operators with high order accuracy, and previously unknown SBP operatorswith interior order of accuracy as large as $2s=30$ are found. Additionally, amethod for optimizing the spectral radius of the SBP derivative is introduced,and the effectiveness of this method is explored through numerical experiment.

An algorithmic exploration of the existence of high-order summation by parts operators with diagonal norm