Publications of J. Thomas Beale

%% Papers Published   
@article{fds342192,
   Author = {Tlupova, S and Beale, JT},
   Title = {Regularized single and double layer integrals in 3D Stokes
             flow},
   Journal = {Journal of Computational Physics},
   Volume = {386},
   Pages = {568-584},
   Year = {2019},
   Month = {June},
   Abstract = {© 2019 Elsevier Inc. 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 singular
             Stokeslet and stresslet kernels are regularized and, for the
             nearly singular case, corrections are added to reduce the
             regularization error. These corrections are derived
             analytically for both the Stokeslet and the stresslet using
             local asymptotic analysis. For the case of evaluating the
             integrals on the surface, as needed when solving integral
             equations, we design high order regularizations for both
             kernels that do not require corrections. This approach is
             direct in that it does not require grid refinement or
             special quadrature near the singularity, and therefore does
             not increase the computational complexity of the overall
             algorithm. Numerical tests demonstrate the uniform
             convergence rates for several surfaces in both the singular
             and near singular cases, as well as the importance of
             corrections when two surfaces are close to each
             other.},
   Doi = {10.1016/j.jcp.2019.02.031},
   Key = {fds342192}
}

@article{fds340892,
   Author = {Beale, JT and Ying, W},
   Title = {Solution of the Dirichlet problem by a finite difference
             analog of the boundary integral equation},
   Journal = {Numerische Mathematik},
   Volume = {141},
   Number = {3},
   Pages = {605-626},
   Year = {2019},
   Month = {March},
   Abstract = {© 2018, Springer-Verlag GmbH Germany, part of Springer
             Nature. 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 can be solved by replacing
             the integral operator with a finite difference calculation
             on a regular grid. A practical method of this type has been
             developed by the second author. In this paper we prove the
             validity of a simplified version of this method for the
             Dirichlet problem in a general domain in R 2 or R 3 . Given
             a boundary value, we solve for a discrete version of the
             density of the double layer potential using a low order
             interface method. It produces the Shortley–Weller solution
             for the unknown harmonic function with accuracy O(h 2 ). We
             prove the unique solvability for the density, with bounds in
             norms based on the energy or Dirichlet norm, using
             techniques which mimic those of exact potentials. The
             analysis reveals that this crude method maintains much of
             the mathematical structure of the classical integral
             equation. Examples are included.},
   Doi = {10.1007/s00211-018-1010-2},
   Key = {fds340892}
}

@article{fds322466,
   Author = {Beale, JT and Ying, W and Wilson, JR},
   Title = {A Simple Method for Computing Singular or Nearly Singular
             Integrals on Closed Surfaces},
   Journal = {Communications in Computational Physics},
   Volume = {20},
   Number = {3},
   Pages = {733-753},
   Publisher = {Global Science Press},
   Year = {2016},
   Month = {September},
   Abstract = {© 2016 Global-Science Press. 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 kernel
             and corrections are added for regularization and
             discretization, which are found from analysis near the
             singular point. The surface integrals are computed from a
             new quadrature rule using surface points which project onto
             grid points in coordinate planes. The method does not
             require coordinate charts on the surface or special
             treatment of the singularity other than the corrections. The
             accuracy is about O(h 3), where h is the spacing in the
             background grid, uniformly with respect to the point of
             evaluation, on or near the surface. Improved accuracy is
             obtained for points on the surface. The treecode of Duan and
             Krasny for Ewald summation is used to perform sums.
             Numerical examples are presented with a variety of
             surfaces.},
   Doi = {10.4208/cicp.030815.240216a},
   Key = {fds322466}
}

@article{fds243316,
   Author = {Beale, JT},
   Title = {Uniform Error Estimates for Navier--Stokes Flow with an
             Exact Moving Boundary Using the Immersed Interface
             Method},
   Journal = {Siam Journal on Numerical Analysis},
   Volume = {53},
   Number = {4},
   Pages = {2097-2111},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2015},
   Month = {January},
   ISSN = {0036-1429},
   url = {http://www.math.duke.edu/faculty/beale/papers/nseiim.pdf},
   Doi = {10.1137/151003441},
   Key = {fds243316}
}

@article{fds243354,
   Author = {Tlupova, S and Beale, JT},
   Title = {Nearly singular integrals in 3D stokes flow},
   Journal = {Communications in Computational Physics},
   Volume = {14},
   Number = {5},
   Pages = {1207-1227},
   Publisher = {Global Science Press},
   Year = {2013},
   ISSN = {1815-2406},
   url = {http://www.math.duke.edu/faculty/beale/papers/stokes3d2.pdf},
   Abstract = {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 evaluate the integrals near
             the boundary, the singular kernels are regularized and a
             simple quadrature is applied in coordinate charts. High
             order accuracy is obtained by adding special corrections for
             the regularization and discretization errors, derived here
             using local asymptotic analysis. Numerical tests demonstrate
             the uniform convergence rates of the method. © 2013
             Global-Science Press.},
   Doi = {10.4208/cicp.020812.080213a},
   Key = {fds243354}
}

@article{fds243355,
   Author = {Ying, W and Beale, JT},
   Title = {A fast accurate boundary integral method for potentials on
             closely packed cells},
   Journal = {Communications in Computational Physics},
   Volume = {14},
   Number = {4},
   Pages = {1073-1093},
   Publisher = {Global Science Press},
   Year = {2013},
   ISSN = {1815-2406},
   url = {http://www.math.duke.edu/faculty/beale/papers/cpcells2.pdf},
   Abstract = {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 a boundary integralmethod in
             two dimensions which is specially designed tomaintain second
             order accuracy even if boundaries are arbitrarily close.
             Themethod uses a regularization of the integral kernel which
             admits analytically determined corrections to maintain
             accuracy. For boundaries with many components we use the
             fast multipolemethod for efficient summation. We compute
             electric potentials on a domain with cells whose
             conductivity differs from that of the surrounding medium. We
             first solve an integral equation for a source term on the
             cell interfaces and then find values of the potential near
             the interfaces via integrals. Finally we use a Poisson
             solver to extend the potential to a regular grid covering
             the entire region. A number of examples are presented. We
             demonstrate that increased refinement is not needed to
             maintain accuracy as interfaces become very close. © 2013
             Global-Science Press.},
   Doi = {10.4208/cicp.210612.240113a},
   Key = {fds243355}
}

@article{fds243357,
   Author = {Layton, AT and Beale, JT},
   Title = {A partially implicit hybrid method for computing interface
             motion in stokes flow},
   Journal = {Discrete and Continuous Dynamical Systems Series
             B},
   Volume = {17},
   Number = {4},
   Pages = {1139-1153},
   Publisher = {American Institute of Mathematical Sciences
             (AIMS)},
   Year = {2012},
   Month = {June},
   ISSN = {1531-3492},
   url = {http://www.math.duke.edu/faculty/beale/papers/dcdsb.pdf},
   Abstract = {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 of the computational region. The
             implicit time integration is based on the small-scale
             decomposition approach and does not require the iterative
             solution of a system of nonlinear equations. First-order and
             second-order versions of the time-stepping method are
             derived systematically, and numerical results indicate that
             both methods are substantially more stable than explicit
             methods. At each time level, the Stokes equations are solved
             using a hybrid approach combining nearly singular integrals
             on a band of mesh points near the interface and a mesh-based
             solver. The solutions are second-order accurate in space and
             preserve the jump discontinuities across the interface.
             Finally, the hybrid method can be used as an alternative to
             adaptive mesh refinement to resolve boundary layers that are
             frequently present around a stiff immersed
             interface.},
   Doi = {10.3934/dcdsb.2012.17.1139},
   Key = {fds243357}
}

@article{fds243356,
   Author = {Beale, JT},
   Title = {Partially implicit motion of a sharp interface in
             Navier-Stokes flow},
   Journal = {J. Comput. Phys.},
   Volume = {231},
   Number = {18},
   Pages = {6159-6172},
   Publisher = {Elsevier BV},
   Year = {2012},
   url = {http://www.math.duke.edu/faculty/beale/papers/pimpl2.pdf},
   Doi = {10.1016/j.jcp.2012.05.018},
   Key = {fds243356}
}

@article{fds243358,
   Author = {Beale, JT},
   Title = {Smoothing properties of implicit finite difference methods
             for a diffusion equation in maximum norm},
   Journal = {Siam Journal on Numerical Analysis},
   Volume = {47},
   Number = {4},
   Pages = {2476-2495},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2009},
   Month = {July},
   ISSN = {0036-1429},
   url = {http://www.math.duke.edu/faculty/beale/papers/parab.pdf},
   Abstract = {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 difference formula, with the usual
             second-order Laplacian, we show that solutions of the scheme
             gai n first spatial differences boundedly, and also second
             differences except for logarithmic factors, with respect to
             nonhomogeneous terms. A weaker property is shown for the
             Crank-Nicolson method. As a consequence we show that the
             numerical solution of a convection-diffusion equation with
             an interface can allow O(h) truncation error near the
             interface and still have a solution with uniform O(h 2)
             accuracy and first differences of uniform accuracy almost
             O(h 2). © 2009 Society for Industrial and Applied
             Mathematics.},
   Doi = {10.1137/080731645},
   Key = {fds243358}
}

@article{fds243353,
   Author = {Beale, JT and Layton, AT},
   Title = {A velocity decomposition approach for moving interfaces in
             viscous fluids},
   Journal = {Journal of Computational Physics},
   Volume = {228},
   Number = {9},
   Pages = {3358-3367},
   Publisher = {Elsevier BV},
   Year = {2009},
   Month = {May},
   ISSN = {0021-9991},
   url = {http://www.math.duke.edu/faculty/beale/papers/velcomp.pdf},
   Abstract = {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 the moving interface. We decompose the
             velocity into a "Stokes" part and a "regular" part. The
             first part is determined by the Stokes equations and the
             singular interfacial force. The Stokes solution is obtained
             using the immersed interface method, which gives
             second-order accurate values by incorporating known jumps
             for the solution and its derivatives into a finite
             difference method. The regular part of the velocity is given
             by the Navier-Stokes equations with a body force resulting
             from the Stokes part. The regular velocity is obtained using
             a time-stepping method that combines the semi-Lagrangian
             method with the backward difference formula. Because the
             body force is continuous, jump conditions are not necessary.
             For problems with stiff boundary forces, the decomposition
             approach can be combined with fractional time-stepping,
             using a smaller time step to advance the interface quickly
             by Stokes flow, with the velocity computed using boundary
             integrals. The small time steps maintain numerical
             stability, while the overall solution is updated on a larger
             time step to reduce computational cost. © 2009 Elsevier
             Inc. All rights reserved.},
   Doi = {10.1016/j.jcp.2009.01.023},
   Key = {fds243353}
}

@article{fds243359,
   Author = {Beale, JT and Chopp, D and LeVeque, R and Li, Z},
   Title = {Correction to the article A comparison of the extended
             finite element method with the immersed interface method for
             elliptic equations with discontinuous coefficients and
             singular sources by Vaughan et al.},
   Journal = {Commun. Appl. Math. Comput. Sci.},
   Volume = {3},
   Number = {1},
   Pages = {95-100},
   Publisher = {Mathematical Sciences Publishers},
   Year = {2008},
   Month = {August},
   url = {http://www.math.duke.edu/faculty/beale/papers/camcoscorr.pdf},
   Doi = {10.2140/camcos.2008.3.95},
   Key = {fds243359}
}

@article{fds243360,
   Author = {Beale, JT and Strain, J},
   Title = {Locally corrected semi-Lagrangian methods for Stokes flow
             with moving elastic interfaces},
   Journal = {Journal of Computational Physics},
   Volume = {227},
   Number = {8},
   Pages = {3896-3920},
   Publisher = {Elsevier BV},
   Year = {2008},
   Month = {April},
   ISSN = {0021-9991},
   url = {http://hdl.handle.net/10161/6958 Duke open access
             repository},
   Abstract = {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, transported by a semi-Lagrangian
             time step and then used to contour the new interface. The
             velocity field in a periodic box is calculated as a
             potential integral resulting from interfacial and body
             forces, using a technique based on Ewald summation with
             analytically derived local corrections. The interfacial
             stretching is found from a surprisingly natural formula. A
             test problem with an exact solution is constructed and used
             to verify the speed, accuracy and robustness of the
             approach. © 2007 Elsevier Inc. All rights
             reserved.},
   Doi = {10.1016/j.jcp.2007.11.047},
   Key = {fds243360}
}

@article{fds243361,
   Author = {Beale, JT},
   Title = {A proof that a discrete delta function is second-order
             accurate},
   Journal = {Journal of Computational Physics},
   Volume = {227},
   Number = {4},
   Pages = {2195-2197},
   Publisher = {Elsevier BV},
   Year = {2008},
   Month = {February},
   ISSN = {0021-9991},
   url = {http://www.math.duke.edu/faculty/beale/papers/ddel.pdf},
   Abstract = {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 surface integrals using values on a
             regular background grid. The delta function is found using a
             technique of Mayo [A. Mayo, The fast solution of Poisson's
             and the biharmonic equations on irregular regions, SIAM J.
             Numer. Anal. 21 (1984) 285-299]. It can be expressed
             naturally using a level set function. © 2007 Elsevier Inc.
             All rights reserved.},
   Doi = {10.1016/j.jcp.2007.11.004},
   Key = {fds243361}
}

@article{fds243362,
   Author = {Thomas Beale and J and Layton, AT},
   Title = {On the accuracy of finite difference methods for elliptic
             problems with interfaces},
   Journal = {Communications in Applied Mathematics and Computational
             Science},
   Volume = {1},
   Number = {1},
   Pages = {91-119},
   Publisher = {Mathematical Sciences Publishers},
   Year = {2006},
   Month = {January},
   url = {http://www.math.duke.edu/faculty/beale/papers/alayton.pdf},
   Abstract = {In problems with interfaces, the unknown or its derivatives
             may have jump discontinuities. Finite difference methods,
             including the method of A. Mayo and the immersed interface
             method of R. LeVeque and Z. Li, maintain accuracy by adding
             corrections, found from the jumps, to the difference
             operator at grid points near the interface and by modifying
             the operator if necessary. It has long been observed that
             the solution can be computed with uniform O(h2) accuracy
             even if the truncation error is O.h/ at the interface, while
             O(h2) in the interior. We prove this fact for a class of
             static interface problems of elliptic type using discrete
             analogues of estimates for elliptic equations. Moreover, we
             show that the gradient is uniformly accurate to O.h2 log
             .1=h//. Various implications are discussed, including the
             accuracy of these methods for steady fluid flow governed by
             the Stokes equations. Two-fluid problems can be handled by
             first solving an integral equation for an unknown jump.
             Numerical examples are presented which confirm the
             analytical conclusions, although the observed error in the
             gradient is O(h2).},
   Doi = {10.2140/camcos.2006.1.91},
   Key = {fds243362}
}

@article{fds243364,
   Author = {Beale, JT},
   Title = {A grid-based boundary integral method for elliptic problems
             in three dimensions},
   Journal = {Siam Journal on Numerical Analysis},
   Volume = {42},
   Number = {2},
   Pages = {599-620},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2004},
   Month = {December},
   ISSN = {0036-1429},
   url = {http://www.math.duke.edu/faculty/beale/papers/gbbim.pdf},
   Abstract = {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 boundary using special corrections to a
             standard quadrature. We treat the Dirichlet problem for a
             harmonic function with a prescribed boundary value in a
             bounded three-dimensional region with a smooth boundary. The
             solution is a double layer potential, whose strength is
             found by solving an integral equation of the second kind.
             The boundary surface is represented by rectangular grids in
             overlapping coordinate systems, with the boundary value
             known at the grid points. A discrete form of the integral
             equation is solved using a regularized form of the kernel.
             It is proved that the discrete solution converges to the
             exact solution with accuracy O(h p), p < 5, depending on the
             smoothing parameter. Once the dipole strength is found, the
             harmonic function can be computed from the double layer
             potential. For points close to the boundary, the integral is
             nearly singular, and accurate computation is not routine. We
             calculate the integral by summing over the boundary grid
             points and then adding corrections for the smoothing and
             discretization errors using formulas derived here; they are
             similar to those in the two-dimensional case given by [J. T.
             Beale and M.-C. Lai, SIAM J. Numer. Anal., 38 (2001), pp.
             1902-1925]. The resulting values of the solution are
             uniformly of O(h p) accuracy, p < 3. With a total of N
             points, the calculation could be done in essentially O(N)
             operations if a rapid summation method is used. © 2004
             Society for Industrial and Applied Mathematics.},
   Doi = {10.1137/S0036142903420959},
   Key = {fds243364}
}

@article{fds243363,
   Author = {Baker, GR and Beale, JT},
   Title = {Vortex blob methods applied to interfacial
             motion},
   Journal = {Journal of Computational Physics},
   Volume = {196},
   Number = {1},
   Pages = {233-258},
   Publisher = {Elsevier BV},
   Year = {2004},
   Month = {May},
   url = {http://www.math.duke.edu/faculty/beale/papers/baker.pdf},
   Abstract = {We develop a boundary integral method for computing the
             motion of an interface separating two incompressible,
             inviscid fluids. The velocity integral is regularized, so
             that the vortex sheet on the interface is replaced by a sum
             of "blobs" of vorticity. The regularization allows control
             of physical instabilities. We design a class of high order
             blob methods and analyze the errors. Numerical tests suggest
             that the blob size should be scaled with the local spacing
             of the interfacial markers. For a vortex sheet in one fluid,
             with a first-order kernel, we obtain a spiral roll-up
             similar to Krasny [J. Comput. Phys. 65 (1986) 292], but the
             higher order kernels lead to more detailed structure. We
             verify the accuracy of the new method by computing a
             liquid-gas interface with Rayleigh-Taylor instability. We
             then apply the method to the more difficult case of
             Rayleigh-Taylor flow separating two fluids of positive
             density, a case for which the regularization appears to be
             essential, as found by Kerr and Tryggvason [both J. Comput.
             Phys. 76 (1988) 48; 75 (1988) 253]. We use a "blob"
             regularization in certain local terms in the evolution
             equations as well as in the velocity integral. We find
             strong evidence that improved spatial resolution with fixed
             blob size leads to a converged, regularized solution without
             numerical instabilities. However, it is not clear that there
             is a weak limit as the regularization is decreased. © 2003
             Elsevier Inc. All rights reserved.},
   Doi = {10.1016/j.jcp.2003.10.023},
   Key = {fds243363}
}

@article{fds243352,
   Author = {Beale, JT and Lai, MC},
   Title = {A method for computing nearly singular integrals},
   Journal = {Siam Journal on Numerical Analysis},
   Volume = {38},
   Number = {6},
   Pages = {1902-1925},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2001},
   Month = {December},
   url = {http://www.math.duke.edu/faculty/beale/papers/nearsing.ps},
   Abstract = {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 standard quadrature rule. Then we
             add corrections for the errors due to smoothing and
             discretization, which are found by asymptotic analysis. We
             prove an error estimate for the corrected value, uniform
             with respect to the point of evaluation. One application is
             a simple method for solving the Dirichlet problem for
             Laplace's equation on a grid covering an irregular region in
             the plane, similar to an earlier method of A. Mayo [SIAM J.
             Sci. Statist. Comput., 6 (1985), pp. 144-157]. This approach
             could also be used to compute the pressure gradient due to a
             force on a moving boundary in an incompressible fluid.
             Computational examples are given for the double layer
             potential and for the Dirichlet problem.},
   Doi = {10.1137/S0036142999362845},
   Key = {fds243352}
}

@article{fds243329,
   Author = {Beale, JT},
   Title = {Discretization of Layer Potentials and Numerical Methods for
             Water Waves (Tosio Kato's Method and Principle for Evolution
             Equations in Mathematical Physics)},
   Journal = {Rims Kokyuroku},
   Volume = {1234},
   Pages = {18-26},
   Publisher = {Kyoto University},
   Year = {2001},
   Month = {October},
   ISSN = {1880-2818},
   Key = {fds243329}
}

@article{fds243351,
   Author = {Beale, JT},
   Title = {A convergent boundary integral method for three-dimensional
             water waves},
   Journal = {Mathematics of Computation},
   Volume = {70},
   Number = {235},
   Pages = {977-1029},
   Publisher = {American Mathematical Society (AMS)},
   Year = {2001},
   Month = {July},
   url = {http://www.math.duke.edu/faculty/beale/papers/mathcomp.ps},
   Abstract = {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 transported according to a computed
             velocity. An integral equation arising from potential theory
             is solved for the normal velocity. A new method is developed
             for the integration of singular integrals, in which the
             Green's function is regularized and an efficient local
             correction to the trapezoidal rule is computed. The sums
             replacing the singular integrals are treated as discrete
             versions of pseudodifferential operators and are shown to
             have mapping properties like the exact operators. The scheme
             is designed so that the error is governed by evolution
             equations which mimic the structure of the original problem,
             and in this way stability can be assured. The wave-like
             character of the exact equations of motion depends on the
             positivity of the operator which assigns to a function on
             the surface the normal derivative of its harmonic extension;
             similarly, the stability of the scheme depends on
             maintaining this property for the discrete operator. With n
             grid points, the scheme can be implemented with essentially
             O(n) operations per time step.},
   Doi = {10.1090/S0025-5718-00-01218-7},
   Key = {fds243351}
}

@article{fds243347,
   Author = {Beale, JT and Hou, TY and Lowengrub, J},
   Title = {Stability of boundary integral methods for water
             waves},
   Journal = {Ams Ims Siam Joint Summer Research Conference},
   Pages = {241-245},
   Year = {1996},
   Month = {January},
   Abstract = {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 free surface can be avoided. The
             case considered is rather special, but analysis has provided
             a definitive answer in this case, and the treatment of this
             case might partially clarify the numerical issues in the
             more realistic problems. The results are presented in
             detail. A calculation of an overturning wave illustrates the
             resolution made possible by a fully stable numerical method.
             A careful study of an overturning wave by method of this
             class was presented previously.},
   Key = {fds243347}
}

@article{fds243349,
   Author = {Beale, JT and Hou, TY and Lowengrub, J},
   Title = {Convergence of a boundary integral method for water
             waves},
   Journal = {Siam Journal on Numerical Analysis},
   Volume = {33},
   Number = {5},
   Pages = {1797-1843},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {1996},
   Month = {January},
   Abstract = {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 underlying solution remains fairly
             regular (and a sign condition holds in the case without
             surface tension). Thus, numerical instabilities are ruled
             out even in a fully nonlinear regime. The analysis is based
             on delicate energy estimates, following a framework
             previously developed in the continuous case [Beale, Hou, and
             Lowengrub, Comm. Pure Appl. Math., 46 (1993), pp.
             1269-1301]. No analyticity assumption is made for the
             physical solution. Our study indicates that the numerical
             methods must satisfy certain compatibility conditions in
             order to be stable. Violation of these conditions will lead
             to numerical instabilities. A breaking wave is calculated as
             an illustration.},
   Doi = {10.1137/S0036142993245750},
   Key = {fds243349}
}

@article{fds243350,
   Author = {Lifschitz, A and Suters, WH and Beale, JT},
   Title = {The onset of instability in exact vortex rings with
             swirl},
   Journal = {Journal of Computational Physics},
   Volume = {129},
   Number = {1},
   Pages = {8-29},
   Publisher = {Elsevier BV},
   Year = {1996},
   Month = {January},
   Abstract = {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. We begin with vortex rings
             which are exact solutions of the Euler equations of
             inviscid, incompressible fluid flow, axisymmetric, and
             traveling along the axis; swirl refers to the component of
             velocity around the axis. Exact vortex rings with swirl can
             be computed reliably using a variational method.
             Quantitative predictions can then be made for the maximum
             growth rates of localized instabilities of small amplitude,
             using asymptotic analysis as in geometric optics. The
             predicted growth rates are compared with numerical solutions
             of the full, time-dependent Euler equations, starting with a
             small disturbance in an exact ring. These solutions are
             computed by a Lagrangian method, in which the
             three-dimensional flow is represented by a collection of
             vortex elements, moving according to their induced velocity.
             The computed growth rates are typically found to be about
             half of the predicted maximum, and the dependence on
             location and ring parameters qualitatively matches the
             predictions. The comparison of these two very different
             methods for estimating the growth of instabilities serves to
             check the realm of validity of each approach. © 1996
             Academic Press, Inc.},
   Doi = {10.1006/jcph.1996.0230},
   Key = {fds243350}
}

@article{fds243335,
   Author = {Beale, JT and Greengard, C},
   Title = {Convergence of euler-stokes splitting of the navier-stokes
             equations},
   Journal = {Communications on Pure and Applied Mathematics},
   Volume = {47},
   Number = {8},
   Pages = {1083-1115},
   Publisher = {WILEY},
   Year = {1994},
   Month = {August},
   ISSN = {0010-3640},
   Doi = {10.1002/cpa.3160470805},
   Key = {fds243335}
}

@article{fds243333,
   Author = {Bourgeois, AJ and Beale, JT},
   Title = {Validity of the Quasigeostrophic Model for Large-Scale Flow
             in the Atmosphere and Ocean},
   Journal = {Siam Journal on Mathematical Analysis},
   Volume = {25},
   Number = {4},
   Pages = {1023-1068},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {1994},
   Month = {July},
   ISSN = {0036-1410},
   Doi = {10.1137/s0036141092234980},
   Key = {fds243333}
}

@article{fds243346,
   Author = {Beale, JT and Hou, TY and Lowengrub, JS and Shelley,
             MJ},
   Title = {Spatial and temporal stability issues for interfacial flows
             with surface tension},
   Journal = {Mathematical and Computer Modelling},
   Volume = {20},
   Number = {10-11},
   Pages = {1-27},
   Publisher = {Elsevier BV},
   Year = {1994},
   Month = {January},
   ISSN = {0895-7177},
   Abstract = {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 equations of motion. Accurate simulation of
             these interfaces presents a problem of considerable
             difficulty on several levels. First, even for stably
             stratified flows like water waves, it turns out that
             straightforward spatial discretizations (of the boundary
             integral formulation) generate numerical instability.
             Second, surface tension introduces a large number of
             derivatives through the Laplace-Young boundary condition.
             This induces severe time step restrictions for explicit time
             integration methods. In this paper, we present a class of
             stable spatial discretizations and we present a
             reformulation of the equations of motion that make apparent
             how to remove the high order time step restrictions
             introduced by the surface tension. This paper is a review of
             the results given in [1,2]. © 1994.},
   Doi = {10.1016/0895-7177(94)90167-8},
   Key = {fds243346}
}

@article{fds243320,
   Author = {Beale, JT and Hou, TY and Lowengrub, JS},
   Title = {Growth rates for the linearized motion of fluid interfaces
             away from equilibrium},
   Journal = {Communications on Pure and Applied Mathematics},
   Volume = {46},
   Number = {9},
   Pages = {1269-1301},
   Publisher = {WILEY},
   Year = {1993},
   Month = {January},
   ISSN = {0010-3640},
   Abstract = {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 presumed time‐dependent solution, are
             wellposed; that is, linear disturbances have a bounded rate
             of growth. If surface tension is neglected, the linear
             equations are well‐posed provided the underlying exact
             motion satisfies a condition on the acceleration of the
             interface relative to gravity, similar to the criterion
             formulated by G. I. Taylor. If surface tension is included,
             the linear equations are well‐posed without
             qualifications, whether the fluid is above or below the
             interface. An interesting qualitative structure is found for
             the linear equations. A Lagrangian approach is used, like
             that of numerical work such as [3], except that the
             interface is assumed horizontal at infinity. Certain
             integral equations which occur, involving double layer
             potentials, are shown to be solvable in the present case. ©
             1993 John Wiley & Sons, Inc. Copyright © 1993 Wiley
             Periodicals, Inc., A Wiley Company},
   Doi = {10.1002/cpa.3160460903},
   Key = {fds243320}
}

@article{fds9208,
   Author = {J. T. Beale and T. Y. Hou and J. S. Lowengrub},
   Title = {On the well-posedness of two-fluid interfacial flows with
             surface tension},
   Journal = {Singularities in Fluids, Plasmas, and Optics, R. Caflisch et
             al., ed., NATO ASI Series, Kluwer},
   Pages = {11-38},
   Year = {1993},
   Key = {fds9208}
}

@article{fds243336,
   Author = {Beale, JT},
   Title = {Exact solitary water waves with capillary ripples at
             infinity},
   Journal = {Communications on Pure and Applied Mathematics},
   Volume = {44},
   Number = {2},
   Pages = {211-257},
   Publisher = {WILEY},
   Year = {1991},
   Month = {March},
   ISSN = {0010-3640},
   Doi = {10.1002/cpa.3160440204},
   Key = {fds243336}
}

@article{fds243332,
   Author = {Beale, JT and Schaeffer, DG},
   Title = {Nonlinear behavior of model equations which are linearly
             ill-posed},
   Journal = {Communications in Partial Differential Equations},
   Volume = {13},
   Number = {4},
   Pages = {423-467},
   Publisher = {Informa UK Limited},
   Year = {1988},
   Month = {January},
   ISSN = {0360-5302},
   Doi = {10.1080/03605308808820548},
   Key = {fds243332}
}

@article{fds243348,
   Author = {Beale, JT},
   Title = {Large-time behavior of discrete velocity boltzmann
             equations},
   Journal = {Communications in Mathematical Physics},
   Volume = {106},
   Number = {4},
   Pages = {659-678},
   Publisher = {Springer Nature},
   Year = {1986},
   Month = {December},
   ISSN = {0010-3616},
   Abstract = {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 a state in which each component is a
             traveling wave without interaction. The techniques are
             similar to those in an earlier treatment of the Broadwell
             model [1]. © 1986 Springer-Verlag.},
   Doi = {10.1007/BF01463401},
   Key = {fds243348}
}

@article{fds243330,
   Author = {BEALE, JT},
   Title = {ANALYSIS OF VORTEX METHODS FOR INCOMPRESSIBLE-FLOW},
   Journal = {Journal of Statistical Physics},
   Volume = {44},
   Number = {5-6},
   Pages = {1009-1011},
   Year = {1986},
   Month = {September},
   ISSN = {0022-4715},
   url = {http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:A1986E184600018&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=47d3190e77e5a3a53558812f597b0b92},
   Key = {fds243330}
}

@article{fds243344,
   Author = {Beale, JT},
   Title = {A Convergent 3-D Vortex Method With Grid-Free
             Stretching},
   Journal = {Mathematics of Computation},
   Volume = {46},
   Number = {174},
   Pages = {401-401},
   Publisher = {JSTOR},
   Year = {1986},
   Month = {April},
   Abstract = {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 explicitly on the arrangement of
             the vorticity elements in a Lagrangian frame. Thus, it could
             be used naturally in a more general context in which
             boundaries and viscosity are present. It is also shown that
             previous estimates for the velocity approximation can be
             improved by taking into account the fact that the integral
             kernel has an average value of zero. Implications for the
             design of the method are discussed. (A)},
   Doi = {10.2307/2007984},
   Key = {fds243344}
}

@article{fds340682,
   Author = {Beale, JT},
   Title = {Convergent 3-D vortex method with grid-free
             stretching.},
   Year = {1986},
   Month = {January},
   Abstract = {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 explicitly on the arrangement of
             the vorticity elements in a Lagrangian frame. Thus, it could
             be used naturally in a more general context in which
             boundaries and viscosity are present. It is also shown that
             previous estimates for the velocity approximation can be
             improved by taking into account the fact that the integral
             kernel has an average value of zero. Implications for the
             design of the method are discussed. (A)},
   Key = {fds340682}
}

@article{fds243345,
   Author = {Beale, JT},
   Title = {Large-time behavior of the Broadwell model of a discrete
             velocity gas},
   Journal = {Communications in Mathematical Physics},
   Volume = {102},
   Number = {2},
   Pages = {217-235},
   Publisher = {Springer Nature},
   Year = {1985},
   Month = {June},
   ISSN = {0010-3616},
   Abstract = {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 a free state in which all the mass
             travels outward at speed 1. The limiting behavior is stable
             with respect to the initial state. No smallness assumptions
             are made. © 1985 Springer-Verlag.},
   Doi = {10.1007/BF01229378},
   Key = {fds243345}
}

@article{fds243342,
   Author = {Beale, JT and Nishida, T},
   Title = {Large-Time Behavior of Viscous Surface Waves},
   Journal = {North-Holland Mathematics Studies},
   Volume = {128},
   Number = {C},
   Pages = {1-14},
   Publisher = {Elsevier},
   Year = {1985},
   Month = {January},
   ISSN = {0304-0208},
   Abstract = {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 equation. © 1985, Elsevier Inc. All
             rights reserved.},
   Doi = {10.1016/S0304-0208(08)72355-7},
   Key = {fds243342}
}

@article{fds243343,
   Author = {Beale, JT and Majda, A},
   Title = {High order accurate vortex methods with explicit velocity
             kernels},
   Journal = {Journal of Computational Physics},
   Volume = {58},
   Number = {2},
   Pages = {188-208},
   Publisher = {Elsevier BV},
   Year = {1985},
   Month = {January},
   ISSN = {0021-9991},
   Abstract = {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 exact solutions in two dimensions. It
             is found that the higher order methods yield a considerably
             more accurate representation of the velocity field than
             those of lower order for moderate integration times. On the
             other hand, the velocity field computed by the point vortex
             method has very poor accuracy at locations other than the
             particle trajectories. © 1985.},
   Doi = {10.1016/0021-9991(85)90176-7},
   Key = {fds243343}
}

@article{fds243341,
   Author = {Beale, JT},
   Title = {Large-time regularity of viscous surface
             waves},
   Journal = {Archive for Rational Mechanics and Analysis},
   Volume = {84},
   Number = {4},
   Pages = {307-352},
   Publisher = {Springer Nature},
   Year = {1984},
   Month = {December},
   ISSN = {0003-9527},
   Doi = {10.1007/BF00250586},
   Key = {fds243341}
}

@article{fds243340,
   Author = {Beale, JT and Kato, T and Majda, A},
   Title = {Remarks on the breakdown of smooth solutions for the 3-D
             Euler equations},
   Journal = {Communications in Mathematical Physics},
   Volume = {94},
   Number = {1},
   Pages = {61-66},
   Publisher = {Springer Nature},
   Year = {1984},
   Month = {March},
   ISSN = {0010-3616},
   Abstract = {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 maximum vorticity necessarily
             grows without bound as the critical time approaches;
             equivalently, if the vorticity remains bounded, a smooth
             solution persists. © 1984 Springer-Verlag.},
   Doi = {10.1007/BF01212349},
   Key = {fds243340}
}

@article{fds332857,
   Author = {Beale, JT and Majda, AJ},
   Title = {Explicit smooth velocity kernels for vortex
             methods.},
   Year = {1983},
   Month = {January},
   Abstract = {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 accuracy of the method depends on
             replacing the integral kernel with a smooth approximation.
             The purpose of this note is to construct smooth kernels of
             arbitrary order of accuracy which are given by simple,
             explicit formulae.},
   Key = {fds332857}
}

@article{fds322467,
   Author = {Beale, JT and Majda, A},
   Title = {Vortex methods. II. Higher order accuracy in two and three
             dimensions},
   Journal = {Mathematics of Computation},
   Volume = {39},
   Number = {159},
   Pages = {29-29},
   Publisher = {American Mathematical Society (AMS)},
   Year = {1982},
   Month = {September},
   Doi = {10.1090/s0025-5718-1982-0658213-7},
   Key = {fds322467}
}

@article{fds243323,
   Author = {Beale, JT and Majda, A},
   Title = {Vortex Methods. II: Higher Order Accuracy in Two and Three
             Dimensions},
   Journal = {Mathematics of Computation},
   Volume = {39},
   Number = {159},
   Pages = {29-29},
   Publisher = {JSTOR},
   Year = {1982},
   Month = {July},
   ISSN = {0025-5718},
   url = {http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:A1982NY40800002&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=47d3190e77e5a3a53558812f597b0b92},
   Doi = {10.2307/2007618},
   Key = {fds243323}
}

@article{fds243327,
   Author = {Beale, JT and MAJDA, A},
   Title = {Vortex Methods 1: Convergence in 3 Dimensions},
   Journal = {Mathematics of Computation},
   Volume = {39},
   Number = {159},
   Pages = {1-27},
   Publisher = {American Mathematical Society (AMS)},
   Year = {1982},
   ISSN = {0025-5718},
   url = {http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:A1982NY40800001&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=47d3190e77e5a3a53558812f597b0b92},
   Doi = {10.1090/s0025-5718-1982-0658212-5},
   Key = {fds243327}
}

@article{fds243326,
   Author = {Beale, JT and Majda, A},
   Title = {Rates of Convergence for Viscous Splitting of the
             Navier-Stokes Equations},
   Journal = {Mathematics of Computation},
   Volume = {37},
   Number = {156},
   Pages = {243-243},
   Publisher = {JSTOR},
   Year = {1981},
   Month = {October},
   ISSN = {0025-5718},
   url = {http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:A1981MP67700001&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=47d3190e77e5a3a53558812f597b0b92},
   Doi = {10.2307/2007424},
   Key = {fds243326}
}

@article{fds243328,
   Author = {Beale, JT},
   Title = {The initial value problem for the navier‐stokes equations
             with a free surface},
   Journal = {Communications on Pure and Applied Mathematics},
   Volume = {34},
   Number = {3},
   Pages = {359-392},
   Publisher = {WILEY},
   Year = {1981},
   Month = {January},
   ISSN = {0010-3640},
   Doi = {10.1002/cpa.3160340305},
   Key = {fds243328}
}

@article{fds243324,
   Author = {Beale, JT},
   Title = {Water-Waves Generated by a Pressure Disturbance on a Steady
             Stream},
   Journal = {Duke Mathematical Journal},
   Volume = {47},
   Number = {2},
   Pages = {297-323},
   Publisher = {Duke University Press},
   Year = {1980},
   ISSN = {0012-7094},
   url = {http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:A1980KA08800002&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=47d3190e77e5a3a53558812f597b0b92},
   Doi = {10.1215/S0012-7094-80-04719-5},
   Key = {fds243324}
}

@article{fds243339,
   Author = {Beale, JT},
   Title = {The existence of cnoidal water waves with surface
             tension},
   Journal = {Journal of Differential Equations},
   Volume = {31},
   Number = {2},
   Pages = {230-263},
   Publisher = {Elsevier BV},
   Year = {1979},
   Month = {January},
   ISSN = {0022-0396},
   Doi = {10.1016/0022-0396(79)90146-3},
   Key = {fds243339}
}

@article{fds243325,
   Author = {Thomas Beale and J},
   Title = {The existence of solitary water waves},
   Journal = {Communications on Pure and Applied Mathematics},
   Volume = {30},
   Number = {4},
   Pages = {373-389},
   Publisher = {WILEY},
   Year = {1977},
   Month = {July},
   ISSN = {0010-3640},
   Doi = {10.1002/cpa.3160300402},
   Key = {fds243325}
}

@article{fds243322,
   Author = {Thomas Beale and J},
   Title = {Eigenfunction expansions for objects floating in an open
             sea},
   Journal = {Communications on Pure and Applied Mathematics},
   Volume = {30},
   Number = {3},
   Pages = {283-313},
   Publisher = {WILEY},
   Year = {1977},
   Month = {May},
   ISSN = {0010-3640},
   Doi = {10.1002/cpa.3160300303},
   Key = {fds243322}
}

@article{fds243338,
   Author = {BEALE, JT},
   Title = {ACOUSTIC SCATTERING FROM LOCALLY REACTING
             SURFACES},
   Journal = {Indiana University Mathematics Journal},
   Volume = {26},
   Number = {2},
   Pages = {199-222},
   Year = {1977},
   Abstract = {A theory is developed for scattering from surfaces that are
             nonporous and locally reacting in the sense that wave motion
             along the surface is negligible. It is assumed that a small
             part of the surface reacts to the excess pressure due to the
             wave like a resistive harmonic oscillator. This boundary
             condition differs from others for the acoustic equation in
             that it does not have the so-called coercive property.
             However, with certain assumptions on the parameters
             occurring in the boundary behavior, it is possible to find a
             special class of initial data, dense in the energy norm,
             whose solutions form pre-compact sets with respect to local
             energy. As a consequence, the local decay of arbitrary
             solutions of finite energy is established.},
   Doi = {10.1512/iumj.1977.26.26015},
   Key = {fds243338}
}

@article{fds243337,
   Author = {Beale, JT},
   Title = {Spectral Properties of an Acoustic Boundary
             Condition},
   Journal = {Indiana University Mathematics Journal},
   Volume = {25},
   Number = {9},
   Pages = {895-917},
   Year = {1976},
   Abstract = {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 semigroup generator A is then
             obtained. Unlike the generators associated with the usual
             boundary conditions, which have compact resolvent and
             spectrum consisting of discrete eigenvalues, A always has
             essential spectrum. Moreover, if the parameters occurring in
             the boundary condition are constant, there are sequences of
             eigenvalues converging to the essential spectrum.},
   Key = {fds243337}
}

@article{fds243317,
   Author = {Beale, JT},
   Title = {Purely imaginary scattering frequencies for exterior
             domains},
   Journal = {Duke Mathematical Journal},
   Volume = {41},
   Number = {3},
   Pages = {607-637},
   Publisher = {Duke University Press},
   Year = {1974},
   Month = {September},
   ISSN = {0012-7094},
   Doi = {10.1215/s0012-7094-74-04165-9},
   Key = {fds243317}
}

@article{fds243318,
   Author = {Beale, JT and Rosencrans, SI},
   Title = {Acoustic boundary conditions},
   Journal = {Bulletin of the American Mathematical Society},
   Volume = {80},
   Number = {6},
   Pages = {1276-1278},
   Publisher = {American Mathematical Society (AMS)},
   Year = {1974},
   Month = {January},
   ISSN = {0002-9904},
   Doi = {10.1090/S0002-9904-1974-13714-6},
   Key = {fds243318}
}

@article{fds243319,
   Author = {Beale, JT},
   Title = {Scattering frequencies of resonators},
   Journal = {Communications on Pure and Applied Mathematics},
   Volume = {26},
   Number = {4},
   Pages = {549-563},
   Publisher = {WILEY},
   Year = {1973},
   Month = {July},
   ISSN = {0010-3640},
   Doi = {10.1002/cpa.3160260408},
   Key = {fds243319}
}

@article{fds10348,
   Author = {J. T. Beale},
   Title = {Methods for computing singular and nearly singular
             integrals},
   Journal = {J. Turbulence, vol. 3, (2002), article 041 (4
             pp.)},
   url = {http://www.math.duke.edu/faculty/beale/papers/jot.pdf},
   Key = {fds10348}
}

@article{fds10345,
   Author = {J. T. Beale},
   Title = {Discretization of Layer Potentials and Numerical Methods for
             Water Waves},
   Journal = {Proc. of Workshop on Kato's Method and Principle for
             Evolution Equations in Mathematical Physics, H. Fujita, S.
             T. Kuroda, H.Okamoto, eds., Univ. of Tokyo Press, pp.
             18-26.},
   Key = {fds10345}
}

@article{fds9732,
   Author = {J. T. Beale},
   Title = {Boundary Integral Methods for Three-Dimensional Water
             Waves},
   Journal = {Equadiff 99, Proceedings of the International Conference on
             Differential Equations, Vol. 2, pp. 1369-78},
   url = {http://www.math.duke.edu/faculty/beale/papers/equadiff.ps},
   Key = {fds9732}
}

@article{fds8750,
   Author = {J. T. Beale and T.Y. Hou and J.S. Lowengrub},
   Title = {Stability of Boundary Integral Methods for Water
             Waves},
   Journal = {Nonlinear Evolutionary Partial Differential Equations, X. X.
             Ding and T.P. Liu eds., A.M.S., 1997, 107-27.},
   Key = {fds8750}
}

@article{fds8748,
   Author = {J. T. Beale and T.Y. Hou and J.S. Lowengrub},
   Title = {Stability of Boundary Integral Methods for Water
             Waves},
   Journal = {Advances in Multi-Fluid Flows, Y. Renardy et al., ed., pp.
             241-45, SIAM, Philadelphia, 1996.},
   Key = {fds8748}
}

@article{fds8749,
   Author = {J. T. Beale and A. Lifschitz and W.H. Suters},
   Title = {A Numerical and Analytical Study of Vortex Rings with
             Swirl},
   Journal = {Vortex Flows and Related Numerical Methods, II, ESAIM Proc.
             1, 565-75, Soc. Math. Appl. Indust., Paris,
             1996.},
   Key = {fds8749}
}

@article{fds9209,
   Author = {J. T. Beale and E. Thomann and C. Greengard},
   Title = {Operator splitting for Navier-Stokes and the Chorin-Marsden
             product formula},
   Journal = {Vortex Flows and Related Numerical Methods, J. T. Beale et
             al., ed., pp. 27-38, NATO ASI Series, Kluwer,
             1993.},
   Key = {fds9209}
}

@article{fds9203,
   Author = {J. T. Beale},
   Title = {The approximation of weak solutions to the Euler equations
             by vortex elements},
   Journal = {Multidimensional Hyperbolic Problems and Computations, J.
             Glimm et al., ed., pp. 23-37, Springer-Verlag, New York,
             1991.},
   Key = {fds9203}
}

@article{fds9205,
   Author = {J. T. Beale and A. Eydeland and B. Turkington},
   Title = {Numerical tests of 3-D vortex methods using a vortex ring
             with swirl},
   Journal = {Vortex Dynamics and Vortex Methods, C. Anderson and C.
             Greengard, ed., pp. 1-9, A.M.S., 1991.},
   Key = {fds9205}
}

@article{fds9206,
   Author = {J. T. Beale},
   Title = {Solitary water waves with ripples beyond all
             orders},
   Journal = {Asymptotics beyond All Orders, H. Segur et al., ed., pp.
             293-98, NATO ASI Series, Plenum, 1991.},
   Key = {fds9206}
}

@article{fds9200,
   Author = {J. T. Beale},
   Title = {Large-time behavior of model gases with a discrete set of
             velocities},
   Journal = {Mathematics Applied to Science, J. Goldstein et al., ed. pp.
             1-12, Academic Press, Orlando, 1988.},
   Key = {fds9200}
}

@article{fds9201,
   Author = {J. T. Beale},
   Title = {On the accuracy of vortex methods at large
             times},
   Journal = {Computational Fluid Dynamics and Reacting Gas Flows, B.
             Engquist et al., ed., pp. 19-32, Springer-Verlag, New York,
             1988.},
   Key = {fds9201}
}

@article{fds9195,
   Author = {J. T. Beale},
   Title = {Existence, regularity, and decay of viscous surface
             waves},
   Journal = {Nonlinear Systems of Partial Differential Equations in
             Applied Mathematics, Part 2, Lectures in Applied
             Mathematics, Vol. 23, A.M.S., Providence, 1986,
             137-48.},
   Key = {fds9195}
}

@article{fds9196,
   Author = {J. T. Beale},
   Title = {A convergent three-dimensional vortex method with grid-free
             stretching},
   Journal = {Math. Comp. 46 (1986), 401-24 and S15-S20.},
   Key = {fds9196}
}

@article{fds9191,
   Author = {J. T. Beale},
   Title = {Large-time regularity of viscous surface
             waves},
   Journal = {Arch. Rational Mech. Anal. 84 (1984), 307-52.},
   Key = {fds9191}
}

@article{fds9193,
   Author = {J. T. Beale and A. Majda},
   Title = {Vortex methods for fluid flow in two or three
             dimensions},
   Journal = {Contemp. Math. 28 (1984), 221-29.},
   Key = {fds9193}
}

@article{fds9190,
   Author = {J. T. Beale},
   Title = {Large-time regularity of viscous surface
             waves},
   Journal = {Contemp. Math. 17 (1983), 31-33.},
   Key = {fds9190}
}

@article{fds9187,
   Author = {J. T. Beale and A. Majda},
   Title = {Vortex methods I: Convergence in three dimensions},
   Journal = {Math. Comp. 39 (1982), 1-27.},
   Key = {fds9187}
}

@article{fds9189,
   Author = {J. T. Beale and A. Majda},
   Title = {The design and numerical analysis of vortex
             methods},
   Journal = {Transonic, Shock, and Multidimensional Flows, R. E. Meyer,
             ed., Academic Press, New York, 1982.},
   Key = {fds9189}
}


%% Papers Submitted   
@article{fds226858,
   Author = {J. t. Beale and W. YIng and J. R. Wilson},
   Title = {A simple method for computing singular or nearly singular
             integrals on closed surfaces},
   Journal = {Commun. Comput. Phys.},
   Year = {2015},
   Month = {August},
   url = {http://www.math.duke.edu/faculty/beale/papers/bywcicp.pdf},
   Key = {fds226858}
}