An Optimization-Based Construction Procedure for Function Space-Based Summation-by-Parts Operators on Arbitrary Grids

Abstract We introduce a novel construction procedure for one-dimensional function space summation-by-parts (FSBP) operators. Existing construction procedures for FSBP operators of the form $$D = P^{-1} Q$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>P</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mi>Q</mml:mi> </mml:mrow> </mml:math> proceed as follows: Given a boundary operator B , the norm matrix P is first determined and then in a second step the complementary matrix Q is calculated to finally get the FSBP operator D . In contrast, the approach proposed here determines the norm and complementary matrices, P and Q , simultaneously by solving an optimization problem. The proposed construction procedure applies to classical summation-by-parts (SBP) operators based on polynomial approximation and the broader class of FSBP operators. According to our experiments, the presented approach yields a numerically stable construction procedure and FSBP operators with higher accuracy for diagonal norm difference operators at the boundaries than the traditional approach. Through numerical simulations, we highlight the advantages of our proposed technique.