# 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

##### Computational Mathematics

##### PDE & Dynamical Systems

##### Physical Modeling

Here are four recent papers:

J. T. Beale and S. Tlupova, Regularized single and double layer integrals in 3D Stokes flow, submitted to J. Comput. Phys., arxiv.org/abs/1808.02177

J. T. Beale and W. Ying, Solution of the Dirichlet problem by a finite difference analog of the boundary integral equation, submitted to Numer. Math., arxiv.org/abs/1803.08532

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.

Beale, JT. "Large-time behavior of the Broadwell model of a discrete velocity gas." *Communications in Mathematical Physics* 102.2 (1985): 217-235.
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Beale, JT, and Majda, A. "High order accurate vortex methods with explicit velocity kernels." *Journal of Computational Physics* 58.2 (1985): 188-208.
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Beale, JT, and Nishida, T. "Large-Time Behavior of Viscous Surface Waves." *North-Holland Mathematics Studies* 128.C (1985): 1-14.
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Beale, JT. "Large-time regularity of viscous surface waves." *Archive for Rational Mechanics and Analysis* 84.4 (December 1984): 307-352.
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Beale, JT, Kato, T, and Majda, A. "Remarks on the breakdown of smooth solutions for the 3-D Euler equations." *Communications in Mathematical Physics* 94.1 (March 1984): 61-66.
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Beale, JT, and Majda, AJ. "Explicit smooth velocity kernels for vortex methods." (January 1, 1983).

Beale, JT, and Majda, A. "Vortex methods. II. Higher order accuracy in two and three dimensions." *Mathematics of Computation* 39.159 (September 1, 1982): 29-29.
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Beale, JT, and MAJDA, A. "Vortex Methods 1: Convergence in 3 Dimensions." *Mathematics of Computation* 39.159 (1982): 1-27.

Beale, JT, and MAJDA, A. "Vortex Methods 2: Higher-Order Accuracy in 2 and 3 Dimensions." *MATHEMATICS OF COMPUTATION* 39.159 (1982): 29-52.
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Beale, JT. "The initial value problem for the navier-stokes equations with a free surface." *Communications on Pure and Applied Mathematics* 34.3 (May 1981): 359-392.
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