Scholars@Duke
J. Thomas Beale

J. Thomas Beale

Professor Emeritus of Mathematics
Mathematics
beale@math.duke.edu
Box 90320, Durham, NC 27708-0320
120 Science Drive, Durham, NC 27708
Office hours by appointment.  
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Selected Publications

Extrapolated regularization of nearly singular integrals on surfaces

Journal Article Advances in Computational Mathematics · August 1, 2024 We present a method for computing nearly singular integrals that occur when single or double layer surface integrals, for harmonic potentials or Stokes flow, are evaluated at points nearby. Such values could be needed in solving an integral equation when o ... Full text Cite

Solving partial differential equations on closed surfaces with planar cartesian grids

Journal Article SIAM Journal on Scientific Computing · January 1, 2020 We present a general purpose method for solving partial differential equations on a closed surface, based on a technique for discretizing the surface introduced by Wenjun Ying and Wei-Cheng Wang [J. Comput. Phys., 252 (2013), pp. 606{624] which uses projec ... Full text Cite

Regularized single and double layer integrals in 3D Stokes flow

Journal Article Journal of Computational Physics · June 1, 2019 We present a numerical method for computing the single layer (Stokeslet) and double layer (stresslet) integrals in Stokes flow. The method applies to smooth, closed surfaces in three dimensions, and achieves high accuracy both on and near the surface. The ... Full text Cite

Solution of the Dirichlet problem by a finite difference analog of the boundary integral equation

Journal Article Numerische Mathematik · March 13, 2019 Several important problems in partial differential equations can be formulated as integral equations. Often the integral operator defines the solution of an elliptic problem with specified jump conditions at an interface. In principle the integral equation ... Full text Cite

A Simple Method for Computing Singular or Nearly Singular Integrals on Closed Surfaces

Journal Article Communications in Computational Physics · September 1, 2016 We present a simple, accurate method for computing singular or nearly singular integrals on a smooth, closed surface, such as layer potentials for harmonic functions evaluated at points on or near the surface. The integral is computed with a regularized ke ... Full text Cite

Uniform error estimates for Navier-Stokes flow with an exact moving boundary using the immersed interface method

Journal Article SIAM Journal on Numerical Analysis · January 1, 2015 We prove that uniform accuracy of almost second order can be achieved with a finite difference method applied to Navier-Stokes flow at low Reynolds number with a moving boundary, or interface, creating jumps in the velocity gradient and pressure. Differenc ... Full text Cite

A fast accurate boundary integral method for potentials on closely packed cells

Journal Article Communications in Computational Physics · 2013 Boundary integral methods are naturally suited for the computation of harmonic functions on a region having inclusions or cells with different material properties. However, accuracy deteriorates when the cell boundaries are close to each other. We present ... Full text Cite

Nearly singular integrals in 3D stokes flow

Journal Article Communications in Computational Physics · 2013 A straightforward method is presented for computing three-dimensional Stokes flow, due to forces on a surface, with high accuracy at points near the surface. The flowquantities arewritten as boundary integrals using the free-spaceGreen's function. To evalu ... Full text Cite

A partially implicit hybrid method for computing interface motion in stokes flow

Journal Article Discrete and Continuous Dynamical Systems - Series B · June 1, 2012 We present a partially implicit hybrid method for simulating the motion of a stiff interface immersed in Stokes flow, in free space or in a rectangular domain with boundary conditions. We assume the interface is a closed curve which remains in the interior ... Full text Cite

Smoothing properties of implicit finite difference methods for a diffusion equation in maximum norm

Journal Article SIAM Journal on Numerical Analysis · July 20, 2009 We prove a regularity property of finite difference schemes for the heat or diffusion equation μ t = δμ in maximum norm with large time steps. For a class of time discretizations including L-stable single-step methods and the second-order backward differen ... Full text Cite

A velocity decomposition approach for moving interfaces in viscous fluids

Journal Article Journal of Computational Physics · May 20, 2009 We present a second-order accurate method for computing the coupled motion of a viscous fluid and an elastic material interface with zero thickness. The fluid flow is described by the Navier-Stokes equations, with a singular force due to the stretching of ... Full text Cite

Locally corrected semi-Lagrangian methods for Stokes flow with moving elastic interfaces

Journal Article Journal of Computational Physics · April 1, 2008 We present a new method for computing two-dimensional Stokes flow with moving interfaces that respond elastically to stretching. The interface is moved by semi-Lagrangian contouring: a distance function is introduced on a tree of cells near the interface, ... Full text Open Access Cite

A proof that a discrete delta function is second-order accurate

Journal Article Journal of Computational Physics · February 1, 2008 It is proved that a discrete delta function introduced by Smereka [P. Smereka, The numerical approximation of a delta function with application to level set methods, J. Comput. Phys. 211 (2006) 77-90] gives a second-order accurate quadrature rule for surfa ... Full text Cite

A grid-based boundary integral method for elliptic problems in three dimensions

Journal Article SIAM Journal on Numerical Analysis · December 1, 2004 We develop a simple, efficient numerical method of boundary integral type for solving an elliptic partial differential equation in a three-dimensional region using the classical formulation of potential theory. Accurate values can be found near the boundar ... Full text Cite

Vortex blob methods applied to interfacial motion

Journal Article J. Comput. Phys. · 2004 Link to item Cite

A method for computing nearly singular integrals

Journal Article SIAM Journal on Numerical Analysis · December 1, 2001 We develop a method for computing a nearly singular integral, such as a double layer potential due to sources on a curve in the plane, evaluated at a point near the curve. The approach is to regularize the singularity and obtain a preliminary value from a ... Full text Cite

Extrapolated regularization of nearly singular integrals on surfaces

Journal Article Advances in Computational Mathematics · August 1, 2024 We present a method for computing nearly singular integrals that occur when single or double layer surface integrals, for harmonic potentials or Stokes flow, are evaluated at points nearby. Such values could be needed in solving an integral equation when o ... Full text Cite

Solving partial differential equations on closed surfaces with planar cartesian grids

Journal Article SIAM Journal on Scientific Computing · January 1, 2020 We present a general purpose method for solving partial differential equations on a closed surface, based on a technique for discretizing the surface introduced by Wenjun Ying and Wei-Cheng Wang [J. Comput. Phys., 252 (2013), pp. 606{624] which uses projec ... Full text Cite

Regularized single and double layer integrals in 3D Stokes flow

Journal Article Journal of Computational Physics · June 1, 2019 We present a numerical method for computing the single layer (Stokeslet) and double layer (stresslet) integrals in Stokes flow. The method applies to smooth, closed surfaces in three dimensions, and achieves high accuracy both on and near the surface. The ... Full text Cite

Solution of the Dirichlet problem by a finite difference analog of the boundary integral equation

Journal Article Numerische Mathematik · March 13, 2019 Several important problems in partial differential equations can be formulated as integral equations. Often the integral operator defines the solution of an elliptic problem with specified jump conditions at an interface. In principle the integral equation ... Full text Cite

A Simple Method for Computing Singular or Nearly Singular Integrals on Closed Surfaces

Journal Article Communications in Computational Physics · September 1, 2016 We present a simple, accurate method for computing singular or nearly singular integrals on a smooth, closed surface, such as layer potentials for harmonic functions evaluated at points on or near the surface. The integral is computed with a regularized ke ... Full text Cite

Uniform error estimates for Navier-Stokes flow with an exact moving boundary using the immersed interface method

Journal Article SIAM Journal on Numerical Analysis · January 1, 2015 We prove that uniform accuracy of almost second order can be achieved with a finite difference method applied to Navier-Stokes flow at low Reynolds number with a moving boundary, or interface, creating jumps in the velocity gradient and pressure. Differenc ... Full text Cite

A fast accurate boundary integral method for potentials on closely packed cells

Journal Article Communications in Computational Physics · 2013 Boundary integral methods are naturally suited for the computation of harmonic functions on a region having inclusions or cells with different material properties. However, accuracy deteriorates when the cell boundaries are close to each other. We present ... Full text Cite

Nearly singular integrals in 3D stokes flow

Journal Article Communications in Computational Physics · 2013 A straightforward method is presented for computing three-dimensional Stokes flow, due to forces on a surface, with high accuracy at points near the surface. The flowquantities arewritten as boundary integrals using the free-spaceGreen's function. To evalu ... Full text Cite

A partially implicit hybrid method for computing interface motion in stokes flow

Journal Article Discrete and Continuous Dynamical Systems - Series B · June 1, 2012 We present a partially implicit hybrid method for simulating the motion of a stiff interface immersed in Stokes flow, in free space or in a rectangular domain with boundary conditions. We assume the interface is a closed curve which remains in the interior ... Full text Cite

Smoothing properties of implicit finite difference methods for a diffusion equation in maximum norm

Journal Article SIAM Journal on Numerical Analysis · July 20, 2009 We prove a regularity property of finite difference schemes for the heat or diffusion equation μ t = δμ in maximum norm with large time steps. For a class of time discretizations including L-stable single-step methods and the second-order backward differen ... Full text Cite

A velocity decomposition approach for moving interfaces in viscous fluids

Journal Article Journal of Computational Physics · May 20, 2009 We present a second-order accurate method for computing the coupled motion of a viscous fluid and an elastic material interface with zero thickness. The fluid flow is described by the Navier-Stokes equations, with a singular force due to the stretching of ... Full text Cite

Locally corrected semi-Lagrangian methods for Stokes flow with moving elastic interfaces

Journal Article Journal of Computational Physics · April 1, 2008 We present a new method for computing two-dimensional Stokes flow with moving interfaces that respond elastically to stretching. The interface is moved by semi-Lagrangian contouring: a distance function is introduced on a tree of cells near the interface, ... Full text Open Access Cite

A proof that a discrete delta function is second-order accurate

Journal Article Journal of Computational Physics · February 1, 2008 It is proved that a discrete delta function introduced by Smereka [P. Smereka, The numerical approximation of a delta function with application to level set methods, J. Comput. Phys. 211 (2006) 77-90] gives a second-order accurate quadrature rule for surfa ... Full text Cite

A grid-based boundary integral method for elliptic problems in three dimensions

Journal Article SIAM Journal on Numerical Analysis · December 1, 2004 We develop a simple, efficient numerical method of boundary integral type for solving an elliptic partial differential equation in a three-dimensional region using the classical formulation of potential theory. Accurate values can be found near the boundar ... Full text Cite

Vortex blob methods applied to interfacial motion

Journal Article J. Comput. Phys. · 2004 Link to item Cite

A method for computing nearly singular integrals

Journal Article SIAM Journal on Numerical Analysis · December 1, 2001 We develop a method for computing a nearly singular integral, such as a double layer potential due to sources on a curve in the plane, evaluated at a point near the curve. The approach is to regularize the singularity and obtain a preliminary value from a ... Full text Cite

A convergent boundary integral method for three-dimensional water waves

Journal Article Mathematics of Computation · July 1, 2001 We design a boundary integral method for time-dependent, three-dimensional, doubly periodic water waves and prove that it converges with O(h3) accuracy, without restriction on amplitude. The moving surface is represented by grid points which are transporte ... Full text Cite

The onset of instability in exact vortex rings with swirl

Journal Article Journal of Computational Physics · January 1, 1996 We study the time-dependent behavior of disturbances to inviscid vortex rings with swirl, using two different approaches. One is a linearized stability analysis for short wavelengths, and the other is direct flow simulation by a computational vortex method ... Full text Cite

Convergence of a boundary integral method for water waves

Journal Article SIAM Journal on Numerical Analysis · January 1, 1996 We prove nonlinear stability and convergence of certain boundary integral methods for time-dependent water waves in a two-dimensional, inviscid, irrotational, incompressible fluid, with or without surface tension. The methods are convergent as long as the ... Full text Cite

Stability of boundary integral methods for water waves

Journal Article AMS-IMS-SIAM Joint Summer Research Conference · January 1, 1996 This paper studies the numerical stability of method of boundary integral type, in which the free surface is tracked explicitly. The focus is on two-dimensional motions, periodic in the horizontal direction, so that issues of boundary conditions for the fr ... Cite

Spatial and temporal stability issues for interfacial flows with surface tension

Journal Article Mathematical and Computer Modelling · November 1, 1994 Many physically interesting problems involve the propagation of free surfaces in fluids with surface tension effects. Surface tensions is an ever-present physical effect that is often neglected due to the difficulties associated with its inclusion in the e ... Full text Cite

Convergence of euler‐stokes splitting of the navier‐stokes equations

Journal Article Communications on Pure and Applied Mathematics · August 1994 AbstractWe consider approximation by partial time steps of a smooth solution of the Navier‐Stokes equations in a smooth domain in two or three space dimensions with no‐slip boundary condition. For small k Full text Cite

Validity of the Quasigeostrophic Model for Large-Scale Flow in the Atmosphere and Ocean

Journal Article SIAM Journal on Mathematical Analysis · July 1994 Full text Cite

Growth rates for the linearized motion of fluid interfaces away from equilibrium

Journal Article Communications on Pure and Applied Mathematics · January 1, 1993 We consider the motion of a two‐dimensional interface separating an inviscid, incompressible, irrotational fluid, influenced by gravity, from a region of zero density. We show that under certain conditions the equations of motion, linearized about a presum ... Full text Cite

Exact solitary water waves with capillary ripples at infinity

Journal Article Communications on Pure and Applied Mathematics · March 1991 AbstractWe prove the existence of solitary water waves of elevation, as exact solutions of the equations of steady inviscid flow, taking into account the effect of surface tension on the free surface. In contrast to the cas ... Full text Cite

Nonlinear behavior of model equations which are linearly ill-posed

Journal Article Communications in Partial Differential Equations · January 1988 Full text Cite

Large-time behavior of discrete velocity boltzmann equations

Journal Article Communications In Mathematical Physics · December 1, 1986 We study the asymptotic behavior of equations representing one-dimensional motions in a fictitious gas with a discrete set of velocities. The solutions considered have finite mass but arbitrary amplitude. With certain assumptions, every solution approaches ... Full text Cite

Analysis of Vortex Methods for Incompressible Flow

Journal Article JOURNAL OF STATISTICAL PHYSICS · September 1986 Link to item Cite

Convergent 3-D vortex method with grid-free stretching.

Journal Article · January 1, 1986 This document proves the convergence of a vortex method for three dimensional, incompressible, inviscid flow without boundaries. This version differs from an earlier one whose convergence was shown in another work in that the calculation does not depend ex ... Cite

Convergent 3-D vortex method with grid-free stretching.

Journal Article · January 1, 1986 This document proves the convergence of a vortex method for three dimensional, incompressible, inviscid flow without boundaries. This version differs from an earlier one whose convergence was shown in another work in that the calculation does not depend ex ... Cite

Large-time behavior of the Broadwell model of a discrete velocity gas

Journal Article Communications in Mathematical Physics · June 1, 1985 We study the behavior of solutions of the one-dimensional Broadwell model of a discrete velocity gas. The particles have velocity ±1 or 0; the total mass is assumed finite. We show that at large time the interaction is negligible and the solution tends to ... Full text Cite

High order accurate vortex methods with explicit velocity kernels

Journal Article Journal of Computational Physics · January 1, 1985 Vortex methods of high order accuracy are developed for inviscid, incompressible fluid flow in two or three space dimensions. The velocity kernels are smooth functions given by simple, explicit formulas. Numerical results are given for test problems with e ... Full text Cite

Large-Time Behavior of Viscous Surface Waves

Journal Article North-Holland Mathematics Studies · January 1, 1985 This chapter discusses the large-time behavior of viscous surface waves. It presents global in time solutions to a free surface problem of the viscous incompressible fluid, which is formulated as the motion of the fluid, governed by the Navier–Stokes equat ... Full text Cite

Large-time regularity of viscous surface waves

Journal Article Archive for Rational Mechanics and Analysis · December 1, 1984 Full text Cite

Remarks on the breakdown of smooth solutions for the 3-D Euler equations

Journal Article Communications in Mathematical Physics · March 1, 1984 The authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3-D Euler equations. In other words, if a solution of the Euler equations is initially smooth and loses its regularity at some later time, then the m ... Full text Cite

Explicit smooth velocity kernels for vortex methods.

Journal Article · January 1, 1983 The authors showed the convergence of a class of vortex methods for incompressible, inviscid flow in two or three space dimensions. These methods are based on the fact that the velocity can be determined from the vorticity by a singular integral. The accur ... Cite

Vortex Methods 1: Convergence in 3 Dimensions

Journal Article MATHEMATICS OF COMPUTATION · 1982 Link to item Cite

Vortex methods. ii: Higher order accuracy in two and three dimensions

Journal Article Mathematics of Computation · January 1, 1982 In an earlier paper the authors introduced a new version of the vortex method for three-dimensional, incompressible flows and proved that it converges to arbitrarily high order accuracy, provided we assume the consistency of a discrete approximation to the ... Full text Cite

The initial value problem for the navier‐stokes equations with a free surface

Journal Article Communications on Pure and Applied Mathematics · January 1, 1981 Full text Cite

The existence of cnoidal water waves with surface tension

Journal Article Journal of Differential Equations · January 1, 1979 Full text Cite

The existence of solitary water waves

Journal Article Communications on Pure and Applied Mathematics · July 1977 Full text Cite

Eigenfunction expansions for objects floating in an open sea

Journal Article Communications on Pure and Applied Mathematics · May 1977 Full text Cite

ACOUSTIC SCATTERING FROM LOCALLY REACTING SURFACES

Journal Article INDIANA UNIVERSITY MATHEMATICS JOURNAL · 1977 Full text Cite

Spectral Properties of an Acoustic Boundary Condition

Journal Article Indiana University Mathematics Journal · 1976 A boundary condition is studied for the wave equation occurring in theoretical acoustics. The initial value problem in a bounded domain is solved by semigroup methods in a Hilbert space of data with finite energy. A description of the spectrum of the semig ... Cite

Purely imaginary scattering frequencies for exterior domains

Journal Article Duke Mathematical Journal · September 1, 1974 Full text Cite

Acoustic boundary conditions

Journal Article Bulletin of the American Mathematical Society · January 1, 1974 Full text Cite

Scattering frequencies of resonators

Journal Article Communications on Pure and Applied Mathematics · January 1, 1973 Full text Cite