Joshua Klarmann


Education: Bachelor of Arts in mathematics and secondary education (May 2015)

Currently pursuing a Doctor of Philosophy in mathematics from Michigan State University

McNair Project: Summation by Parts Operators of High Order with Diagonal H Norm (2013)

Mentor: Nathin Albin, Ph.D.

Finite difference operators have been evaluated previously up to order 8 with positive diagonal norm. There are also operators of 10th, 12th, and 14th order, developed from an algorithm that proves the existence of such operators for varying block dimension and block order utilizing the Fredholm Alternative Theorem. Results were computed using a combination of Python evaluations and Matlab mathematical software. Operators of order 10 with 5th, 4th and 3rd order blocks are found as well as 12th order operators with 6th, 4th and 3rd order closures, and a 14th order operator with 7th order closure. Furthermore, lower order operators (4th, 6th, 8th) are also investigated to establish trends for lesser closure orders. All approximations are for firrst derivative only and dependent on several free variables. Operators satisfy positive diagonal norm in order to mind stability issues on curvilinear grids. Operators found were not optimized for spectral radii.