An Algorithmic Exploration of the Existence of High-Order Summation by Parts Operators with Diagonal Norm

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.