J. Thomas Beale, Professor of Mathematics

J. Thomas Beale
Office Location:  217 Physics
Office Phone:  (919) 660-2839
Email Address: send me a message
Web Page:  http://www.math.duke.edu/cncs/~beale

Teaching (Spring 2015):

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. Recently I have developed a general method for the numerical computation of singular integrals, such as layer potentials on a curve or surface, evaluated at a point on or near the layer (partly with M.-C. Lai). A standard quadrature is used for a regularized integral on overlapping coordinate grids with analytical corrections. In work with J. Strain we have applied this approach to Stokes flow with a moving elastic interface. A recent 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). 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.

Current Ph.D. Students  

    Representative Publications

    1. S. Tlupova and J. T. Beale, Nearly singular integrals in 3D Stokes flow, Commun. Comput. Phys., vol. 14 (2013), pp. 1207-27 [pdf]
    2. W. Ying and J. T. Beale, A fast accurate boundary integral method for potentials on closely packed cells, Commun. Comput. Phys., vol. 14 (2013), pp. 1073-93 [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]