J. Thomas Beale

  • Professor Emeritus of Mathematics
External address: 231 Physics Bldg, Durham, NC 27708-0320
Internal office address: Box 90320, Durham, NC 27708-0320
Phone: (919) 660-2814
Office Hours: 

by appointment.

Research Areas and Keywords

Analysis
computation of singular and nearly singular integrals, motion of fluid interfaces, equations of incompressible flow, maximum norm estimates for finite difference methods, convergence of numerical methods for fluid flow
Computational Mathematics
boundary integral methods, computation of singular and nearly singular integrals, maximum norm estimates for finite difference methods, convergence of numerical methods for fluid flow
PDE & Dynamical Systems
boundary integral methods, motion of fluid interfaces, equations of incompressible flow, maximum norm estimates for finite difference methods, convergence of numerical methods for fluid flow
Physical Modeling
boundary integral methods, motion of fluid interfaces, equations of incompressible flow, convergence of numerical methods for fluid flow

Here are two recent papers:
J. T. Beale, W. Ying, and J. R. Wilson, A simple method for computing singular or nearly singular integrals on closed surfaces,  Comm. Comput. Phys. 20 (2016), 733-753 or  arxiv.org/abs/1508.00265
J. T. Beale, Uniform error estimates for Navier-Stokes flow with an exact moving boundary using the immersed interface method, SIAM J. Numer. Anal. 53 (2015), 2097-2111 or  arxiv.org/abs/1503.05810

Much of my work has to do with incompressible fluid flow, especially qualitative properties of solutions and behavior of numerical methods, using analytical tools of partial differential equations. My research of the last few years has the dual goals of designing numerical methods for problems with interfaces, especially moving interfaces in fluid flow, and the analysis of errors in computational methods of this type. We have developed a general method for the numerical computation of singular or nearly singular integrals, such as layer potentials on a curve or surface, evaluated at a point on the curve or surface or nearby, in work with M.-C. Lai, A. Layton, S. Tlupova, and W. Ying. After regularizing the integrand, a standard quadrature is used, and corrections are added which are determined analytically. Current work with coworkers is intended to make these methods more practical, especially in three dimensional simulations. Some projects (partly with Anita Layton) concern the design of numerical methods which combine finite difference methods with separate computations on interfaces. We developed a relatively simple approach for computing Navier-Stokes flow with an elastic interface. In analytical work we have derived estimates in maximum norm for elliptic (steady-state) and parabolic (diffusive) partial differential equations. For problems with interfaces, maximum norm estimates are more informative than the usual ones in the L^2 sense. More general estimates were proved by Michael Pruitt in his Ph.D. thesis.

Education & Training
  • Ph.D., Stanford University 1973

  • M.S., Stanford University 1969

  • B.S., California Institute of Technology 1967

Selected Grants

Development and Analysis of Numerical Methods for Fluid Interfaces awarded by National Science Foundation (Principal Investigator). 2013 to 2017

Beale, JT, Ying, W, and Wilson, JR. "A Simple Method for Computing Singular or Nearly Singular Integrals on Closed Surfaces." Communications in Computational Physics 20.03 (September 2016): 733-753. Full Text

Ying, W, and Beale, JT. "A fast accurate boundary integral method for potentials on closely packed cells." Communications in Computational Physics 14.4 (2013): 1073-1093. Full Text

Tlupova, S, and Beale, JT. "Nearly singular integrals in 3D stokes flow." Communications in Computational Physics 14.5 (2013): 1207-1227. Full Text

Layton, AT, and Beale, JT. "A partially implicit hybrid method for computing interface motion in stokes flow." Discrete and Continuous Dynamical Systems - Series B 17.4 (2012): 1139-1153. Full Text

Beale, JT. "Partially implicit motion of a sharp interface in Navier-Stokes flow." J. Comput. Phys. 231 (2012): 6159-6172. (Academic Article)

Beale, JT, and Layton, AT. "A velocity decomposition approach for moving interfaces in viscous fluids." Journal of Computational Physics 228.9 (2009): 3358-3367. Full Text

Beale, JT. "A proof that a discrete delta function is second-order accurate." Journal of Computational Physics 227.4 (2008): 2195-2197. Full Text

Pages

Professor J Thomas Beale will retire from the mathematics Department in August after 33 years at Duke. A native of Georgia, he was an undergraduate at Caltech and earned his Ph.D. in mathematics at Stanford with the supervision of Ralph Phillips. He... read more »