J. Thomas Beale, Professor

Contact Info:

Office Location:	217 Physics
Office Phone:	(919) 660-2839
Email Address:

Teaching (Fall 2013):

• MATH 356.01, ELEM DIFFERENTIAL EQUAT Synopsis

  Physics 235, MWF 08:45 AM-09:35 AM

• MATH 653.01, ELLIPTIC PDE Synopsis

  Physics 205, MWF 10:20 AM-11:10 AM

Office Hours:

by appointment

Education:

B.S., Caltech, 1967
M.S., Stanford University, 1969
Ph.D., Stanford University, 1973

Specialties:

Analysis
Applied Math

Research Interests: Partial Differential Equations and Fluid Mechanics

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.

Curriculum Vitae

Current Ph.D. Students   (Former Students)

Postdocs Mentored

• Nathan Totz (August, 2011 - present)  

Representative Publications   (More Publications)

1. S. Tlupova and J. T. Beale, Nearly singular integrals in 3D Stokes flow, Commun. Comput. Phys. (Accepted, 2013) [pdf]
2. W. Ying and J. T. Beale, A fast accurate boundary integral method for potentials on closely packed cells, Commun. Comput. Phys. (Accepted, 2013) [pdf]
3. J. T. Beale, Partially implicit motion of a sharp interface in Navier-Stokes flow, J. Comput. Phys., vol. 231 (2012), pp. 6159-72 [pdf]
4. A. T. Layton and J. T. Beale, A partially implicit hybrid method for computing interface motion in Stokes flow, Discrete and Continuous Dynamical Systems B, vol. 17 (2012), pp. 1139-53 [pdf]
5. J. T. Beale, Smoothing properties of implicit finite difference methods for a diffusion equation in maximum norm, SIAM J. Numer. Anal., vol. 47 (2009), pp. 2476-95 [pdf]
6. J. T. Beale and A. T. Layton, A velocity decomposition approach for moving interfaces in viscous fluids, J. Comput. Phys. 228, 3358-67 (2009) [pdf]
7. J. T. Beale, A proof that a discrete delta function is second-order accurate, J. Comput. Phys., vol. 227 (2008), pp. 2195-97 [pdf]
8. J. T. Beale and J. Strain, Locally corrected semi-Lagrangian methods for Stokes flow with moving elastic interfaces, J. Comput. Phys., vol. 227 (2008), pp. 3896-3920 [pdf]
9. J. T. Beale and A. T. Layton, On the accuracy of finite difference methods for elliptic problems with interfaces, Commun. Appl. Math. Comput. Sci., vol. 1 (2006), pp. 91-119 [pdf]
10. G. R. Baker and J. T. Beale, Vortex blob methods applied to interfacial motion, J. Comput. Phys., vol. 196 (2004), pp. 233-58 [pdf]
11. J. T. Beale, A grid-based boundary integral method for elliptic problems in three dimensions, SIAM J. Numer. Anal., vol. 42 (2004), pp. 599-620 [pdf]
12. J. T. Beale, M.-C. Lai, A Method for Computing Nearly Singular Integrals, SIAM J. Numer. Anal., 38 (2001), 1902-25 [ps]
13. J. T. Beale, A Convergent Boundary Integral Method for Three-Dimensional Water Waves, Math. Comp. 70 (2001), 977-1029 [ps]

Recent Grant Support

• Numerical Methods for Moving Interfaces in Fluids, N.S.F., DMS-0806482, 2008/07-2012/06.