SCHOLARLY INTERESTS

I am primarily interested in numerical analysis and applied mathematics. The title of the volume my first published paper appears in says it best: Math Is For Solving Problems (Editors L. Pamela Cook and Victor Roytburd, SIAM, 1996).

My thesis work (under Julian D. Cole and Don W. Schwendeman) was in theoretical aerodynamics/ scientific computing. There I showed that it was possible to compute solutions of the Transonic Small Disturbance equations that in the case of flow around a slender body of revolution without fore-aft symmetry possessed vanishingly small shocks. This was a very tricky calculation because it involves trying to obtain the solution accurately as near as one can get to a rather nasty singularity. In 1998, I was invited to present these results at the 2nd Theoretical Fluid Mechanics meeting at the annual American Institute of Aeronautics and Astronautics conference. Along the way we stumbled upon an interesting way to discretize the radial derivatives in our partial differential equation. I later discovered that Professor Ron Mickens of Clark Atlanta University had been writing about these same kinds of "nonstandard finite difference" schemes for years.

I have always been interested in applying mathematics to unusual phenomena so when I had a sabbatical coming up in Spring 2000 I suggested to my colleague David Edwards that we come up with a new mathematical model for "how movies make money."

My latest work has been a return to nonstandard finite difference schemes, particularly Mickens discretizations of the radial operators in the cylindrical and spherical Laplacian operators.

A famous old problem, called the Bratu problem, is a nonlinear version of Helmholtz equation where the right hand side of
Laplacian u = - f is exponential. In cylindrical and planar coordinates there are known exact solutions to the Bratu problem, making it ideal for numerical benchmarking of my nonstandard finite difference scheme(s).

NONSTANDARD FINITE DIFFERENCES

MATHEMATICAL MODELLING

CLASSROOM VOTING / PEER INSTRUCTION

 

COMPUTATIONAL AERODYNAMICS

GENDER, SEXUALITY and THE LAW

VISUALIZATION OF CONCEPT MAPS

UNDERREPRESENTATION IN MATHEMATICS

 

Ron Buckmire

Last modified: 09/06/13