Combined with simultaneous approximation terms, summation-by-parts (SBP) operators offer a versatile and efficient methodology that leads to consistent, conservative, and provably stable discretizations. However, diagonal-norm operators with a repeating interior-point operator that have thus far been constructed suffer from a loss of accuracy. While on the interior, these operators are of degree 2 p , at a number of nodes near the boundaries, they are of degree  p , and therefore of global degree  p  — meaning the highest degree monomial for which the operators are exact at all nodes. This implies that for hyperbolic problems and operators of degree greater than unity they lead to solutions with a global order of accuracy lower than the degree of the interior-point operator. In this paper, we develop a procedure to construct diagonal-norm first-derivative SBP operators that are of degree 2 p    at all nodes and therefore can lead to solutions of hyperbolic problems of order   2p+1     2  p  +  1       . This is accomplished by adding nonzero entries in the upper–right and lower–left corners of SBP operator matrices with a repeating interior-point operator. This modification necessitates treating these new operators as elements, where mesh refinement is accomplished by increasing the number of elements in the mesh rather than increasing the number of nodes. The significant improvements in accuracy of this new family, for the same repeating interior-point operator, are demonstrated in the context of the linear convection equation.
['David C. Del Rey Fernández', 'Pieter D. Boom', 'David W. Zingg']
Corner-corrected diagonal-norm summation-by-parts operators for the first derivative with increased order of accuracy