J. Thomas Beale, Professor and Director of Undergraduate Studies

Contact Info:

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

Teaching (Fall 2009):

• MATH 231.01, ORDINARY DIFF EQUATIONS Synopsis

  Physics 227, MWF 10:20 AM-11:10 AM

Office Hours:

to be announced

Education:

B.S., California Institute of Technology, 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. Recently I 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 (partly with M.-C. Lai). After regularizing the integrand, a standard quadrature is used on overlapping coordinate grids, and analytical corrections are added. In work with J. Strain we have applied this approach to Stokes flow (viscosity-dominated fluid flow) with a moving elastic interface. Anita Layton and I have developed a relatively simple approach for the more general problem of Navier-Stokes flow with an interface. Jumps in velocity gradient and pressure at the interface have to be accounted for, but we avoid extra work near the interface. Another paper with A. Layton gives an analytical explanation, with applications, of the observed gain in order of accuracy in certain finite difference methods for computing boundary value problems with irregular boundaries using only regular grids (A. Mayo's method or the immersed interface method of R. LeVeque and Z. Li). In a recent paper I proved that such a gain in accuracy is also possible for (time-dependent) diffusion equations, with appropriate choice of time-stepping, as a consequence of regularity estimates for the discrete problem analogous to those for the exact differential equation. Related projects include computation of unstable fluid interfaces (with G. Baker) and the design of a boundary integral method for 3-D water waves which is numerically stable and proved to converge; the discretization of potentials on a moving surface is a central issue.

Curriculum Vitae

Current Ph.D. Students   (Former Students)

• Jason Wilson  

Representative Publications   (More Publications)

1. 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]
2. J. T. Beale, Smoothing properties of implicit finite difference methods for a diffusion equation in maximum norm, SIAM J. Numer. Anal. (Accepted, April, 2009) [pdf]
3. J. T. Beale, A proof that a discrete delta function is second-order accurate, J. Comput. Phys., vol. 227 (2008), pp. 2195-97 [pdf]
4. 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]
5. 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]
6. G. R. Baker and J. T. Beale, Vortex blob methods applied to interfacial motion, J. Comput. Phys., vol. 196 (2004), pp. 233-58 [pdf]
7. 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]
8. J. T. Beale, M.-C. Lai, A Method for Computing Nearly Singular Integrals, SIAM J. Numer. Anal., 38 (2001), 1902-25 [ps]
9. 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, 2008/07-2011/06.      
• Supplement to "Computational Methods for Singular..." for graduate student support, N. S. F., DMS-0404765-001, 2005/07-2007/06.      
• Computational Methods for Singular and Nearly Singular Integrals with Applications to Fluid Dynamics, N.S.F., 2004/07-2007/06.