
%% 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 = {568584},
Year = {2019},
Month = {June},
url = {http://dx.doi.org/10.1016/j.jcp.2019.02.031},
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 = {605626},
Year = {2019},
Month = {March},
url = {http://dx.doi.org/10.1007/s0021101810102},
Abstract = {© 2018, SpringerVerlag 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/s0021101810102},
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 = {733753},
Publisher = {Global Science Press},
Year = {2016},
Month = {September},
url = {http://dx.doi.org/10.4208/cicp.030815.240216a},
Abstract = {© 2016 GlobalScience 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 NavierStokes flow with an exact
moving boundary using the immersed interface
method},
Journal = {Siam Journal on Numerical Analysis},
Volume = {53},
Number = {4},
Pages = {20972111},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2015},
Month = {January},
ISSN = {00361429},
url = {http://www.math.duke.edu/faculty/beale/papers/nseiim.pdf},
Abstract = {© 2015 Society for Industrial and Applied Mathematics. We
prove that uniform accuracy of almost second order can be
achieved with a finite difference method applied to
NavierStokes flow at low Reynolds number with a moving
boundary, or interface, creating jumps in the velocity
gradient and pressure. Difference operators are corrected to
O(h) near the interface using the immersed interface method,
adding terms related to the jumps, on a regular grid with
spacing h and periodic boundary conditions. The force at the
interface is assumed known within an error tolerance; errors
in the interface location are not taken into account. The
error in velocity is shown to be uniformly O(h<sup>2</sup>
log h<sup>2</sup>), even at grid points near the interface,
and, up to a constant, the pressure has error
O(h<sup>2</sup> log h<sup>3</sup>). The proof uses
estimates for finite difference versions of Poisson and
diffusion equations which exhibit a gain in regularity in
maximum norm.},
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 = {12071227},
Publisher = {Global Science Press},
Year = {2013},
ISSN = {18152406},
url = {http://www.math.duke.edu/faculty/beale/papers/stokes3d2.pdf},
Abstract = {A straightforward method is presented for computing
threedimensional Stokes flow, due to forces on a surface,
with high accuracy at points near the surface. The
flowquantities arewritten as boundary integrals using the
freespaceGreen'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
GlobalScience 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 = {10731093},
Publisher = {Global Science Press},
Year = {2013},
ISSN = {18152406},
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
GlobalScience 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 = {11391153},
Publisher = {American Institute of Mathematical Sciences
(AIMS)},
Year = {2012},
Month = {June},
ISSN = {15313492},
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 smallscale
decomposition approach and does not require the iterative
solution of a system of nonlinear equations. Firstorder and
secondorder versions of the timestepping 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 meshbased
solver. The solutions are secondorder 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
NavierStokes flow},
Journal = {J. Comput. Phys.},
Volume = {231},
Number = {18},
Pages = {61596172},
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 = {24762495},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2009},
Month = {July},
ISSN = {00361429},
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 Lstable singlestep methods and
the secondorder backward difference formula, with the usual
secondorder 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
CrankNicolson method. As a consequence we show that the
numerical solution of a convectiondiffusion 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 = {33583367},
Publisher = {Elsevier BV},
Year = {2009},
Month = {May},
ISSN = {00219991},
url = {http://www.math.duke.edu/faculty/beale/papers/velcomp.pdf},
Abstract = {We present a secondorder 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 NavierStokes 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
secondorder 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 NavierStokes equations with a body force resulting
from the Stokes part. The regular velocity is obtained using
a timestepping method that combines the semiLagrangian
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 timestepping,
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 = {95100},
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 semiLagrangian methods for Stokes flow
with moving elastic interfaces},
Journal = {Journal of Computational Physics},
Volume = {227},
Number = {8},
Pages = {38963920},
Publisher = {Elsevier BV},
Year = {2008},
Month = {April},
ISSN = {00219991},
url = {http://hdl.handle.net/10161/6958 Duke open
access},
Abstract = {We present a new method for computing twodimensional Stokes
flow with moving interfaces that respond elastically to
stretching. The interface is moved by semiLagrangian
contouring: a distance function is introduced on a tree of
cells near the interface, transported by a semiLagrangian
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 secondorder
accurate},
Journal = {Journal of Computational Physics},
Volume = {227},
Number = {4},
Pages = {21952197},
Publisher = {Elsevier BV},
Year = {2008},
Month = {February},
ISSN = {00219991},
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) 7790] gives a secondorder 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) 285299]. 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 = {91119},
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. Twofluid 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 gridbased boundary integral method for elliptic problems
in three dimensions},
Journal = {Siam Journal on Numerical Analysis},
Volume = {42},
Number = {2},
Pages = {599620},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2004},
Month = {December},
ISSN = {00361429},
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 threedimensional 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 threedimensional 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 twodimensional case given by [J. T.
Beale and M.C. Lai, SIAM J. Numer. Anal., 38 (2001), pp.
19021925]. 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 = {233258},
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 firstorder kernel, we obtain a spiral rollup
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
liquidgas interface with RayleighTaylor instability. We
then apply the method to the more difficult case of
RayleighTaylor 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 = {19021925},
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. 144157]. 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 = {1826},
Publisher = {Kyoto University},
Year = {2001},
Month = {October},
ISSN = {18802818},
Key = {fds243329}
}
@article{fds243351,
Author = {Beale, JT},
Title = {A convergent boundary integral method for threedimensional
water waves},
Journal = {Mathematics of Computation},
Volume = {70},
Number = {235},
Pages = {9771029},
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 timedependent,
threedimensional, 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 wavelike
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/S0025571800012187},
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 = {241245},
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 twodimensional 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 = {17971843},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {1996},
Month = {January},
url = {http://dx.doi.org/10.1137/S0036142993245750},
Abstract = {We prove nonlinear stability and convergence of certain
boundary integral methods for timedependent water waves in
a twodimensional, 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.
12691301]. 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 = {829},
Publisher = {Elsevier BV},
Year = {1996},
Month = {January},
url = {http://dx.doi.org/10.1006/jcph.1996.0230},
Abstract = {We study the timedependent 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, timedependent Euler equations, starting with a
small disturbance in an exact ring. These solutions are
computed by a Lagrangian method, in which the
threedimensional 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 eulerstokes splitting of the navierstokes
equations},
Journal = {Communications on Pure and Applied Mathematics},
Volume = {47},
Number = {8},
Pages = {10831115},
Publisher = {WILEY},
Year = {1994},
Month = {August},
ISSN = {00103640},
url = {http://dx.doi.org/10.1002/cpa.3160470805},
Doi = {10.1002/cpa.3160470805},
Key = {fds243335}
}
@article{fds243333,
Author = {Bourgeois, AJ and Beale, JT},
Title = {Validity of the Quasigeostrophic Model for LargeScale Flow
in the Atmosphere and Ocean},
Journal = {Siam Journal on Mathematical Analysis},
Volume = {25},
Number = {4},
Pages = {10231068},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {1994},
Month = {July},
ISSN = {00361410},
url = {http://dx.doi.org/10.1137/s0036141092234980},
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 = {1011},
Pages = {127},
Publisher = {Elsevier BV},
Year = {1994},
Month = {January},
ISSN = {08957177},
url = {http://dx.doi.org/10.1016/08957177(94)901678},
Abstract = {Many physically interesting problems involve the propagation
of free surfaces in fluids with surface tension effects.
Surface tensions is an everpresent 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 LaplaceYoung 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 = {10.1016/08957177(94)901678},
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 = {12691301},
Publisher = {WILEY},
Year = {1993},
Month = {January},
ISSN = {00103640},
url = {http://dx.doi.org/10.1002/cpa.3160460903},
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 wellposedness of twofluid interfacial flows with
surface tension},
Journal = {Singularities in Fluids, Plasmas, and Optics, R. Caflisch et
al., ed., NATO ASI Series, Kluwer},
Pages = {1138},
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 = {211257},
Publisher = {WILEY},
Year = {1991},
Month = {March},
ISSN = {00103640},
url = {http://dx.doi.org/10.1002/cpa.3160440204},
Doi = {10.1002/cpa.3160440204},
Key = {fds243336}
}
@article{fds243332,
Author = {Beale, JT and Schaeffer, DG},
Title = {Nonlinear behavior of model equations which are linearly
illposed},
Journal = {Communications in Partial Differential Equations},
Volume = {13},
Number = {4},
Pages = {423467},
Publisher = {Informa UK Limited},
Year = {1988},
Month = {January},
ISSN = {03605302},
url = {http://dx.doi.org/10.1080/03605308808820548},
Doi = {10.1080/03605308808820548},
Key = {fds243332}
}
@article{fds243348,
Author = {Beale, JT},
Title = {Largetime behavior of discrete velocity boltzmann
equations},
Journal = {Communications in Mathematical Physics},
Volume = {106},
Number = {4},
Pages = {659678},
Publisher = {Springer Nature},
Year = {1986},
Month = {December},
ISSN = {00103616},
url = {http://dx.doi.org/10.1007/BF01463401},
Abstract = {We study the asymptotic behavior of equations representing
onedimensional 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 SpringerVerlag.},
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 = {56},
Pages = {10091011},
Year = {1986},
Month = {September},
ISSN = {00224715},
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 = {Convergent 3D vortex method with gridfree
stretching.},
Volume = {46},
Number = {174},
Pages = {401401},
Publisher = {JSTOR},
Year = {1986},
Month = {January},
url = {http://dx.doi.org/10.2307/2007984},
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 3D vortex method with gridfree
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 = {Largetime behavior of the Broadwell model of a discrete
velocity gas},
Journal = {Communications in Mathematical Physics},
Volume = {102},
Number = {2},
Pages = {217235},
Publisher = {Springer Nature},
Year = {1985},
Month = {June},
ISSN = {00103616},
url = {http://dx.doi.org/10.1007/BF01229378},
Abstract = {We study the behavior of solutions of the onedimensional
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 SpringerVerlag.},
Doi = {10.1007/BF01229378},
Key = {fds243345}
}
@article{fds243342,
Author = {Beale, JT and Nishida, T},
Title = {LargeTime Behavior of Viscous Surface Waves},
Journal = {NorthHolland Mathematics Studies},
Volume = {128},
Number = {C},
Pages = {114},
Publisher = {Elsevier},
Year = {1985},
Month = {January},
ISSN = {03040208},
url = {http://dx.doi.org/10.1016/S03040208(08)723557},
Abstract = {This chapter discusses the largetime 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/S03040208(08)723557},
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 = {188208},
Publisher = {Elsevier BV},
Year = {1985},
Month = {January},
ISSN = {00219991},
url = {http://dx.doi.org/10.1016/00219991(85)901767},
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 = {10.1016/00219991(85)901767},
Key = {fds243343}
}
@article{fds243341,
Author = {Beale, JT},
Title = {Largetime regularity of viscous surface
waves},
Journal = {Archive for Rational Mechanics and Analysis},
Volume = {84},
Number = {4},
Pages = {307352},
Publisher = {Springer Nature},
Year = {1984},
Month = {December},
ISSN = {00039527},
url = {http://dx.doi.org/10.1007/BF00250586},
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 3D
Euler equations},
Journal = {Communications in Mathematical Physics},
Volume = {94},
Number = {1},
Pages = {6166},
Publisher = {Springer Nature},
Year = {1984},
Month = {March},
ISSN = {00103616},
url = {http://dx.doi.org/10.1007/BF01212349},
Abstract = {The authors prove that the maximum norm of the vorticity
controls the breakdown of smooth solutions of the 3D 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 SpringerVerlag.},
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 = {2952},
Publisher = {American Mathematical Society (AMS)},
Year = {1982},
Month = {January},
url = {http://dx.doi.org/10.1090/S00255718198206582137},
Abstract = {In an earlier paper the authors introduced a new version of
the vortex method for threedimensional, incompressible
flows and proved that it converges to arbitrarily high order
accuracy, provided we assume the consistency of a discrete
approximation to the BiotSavart Law. We prove this
consistency statement here, and also derive substantially
sharper results for twodimensional flows. A complete,
simplified proof of convergence in two dimensions is
included. © 1982 American Mathematical Society.},
Doi = {10.1090/S00255718198206582137},
Key = {fds322467}
}
@article{fds243323,
Author = {Beale, JT and MAJDA, A},
Title = {Vortex Methods 2: HigherOrder Accuracy in 2 and 3
Dimensions},
Journal = {Mathematics of Computation},
Volume = {39},
Number = {159},
Pages = {2952},
Publisher = {JSTOR},
Year = {1982},
ISSN = {00255718},
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 = {127},
Publisher = {American Mathematical Society (AMS)},
Year = {1982},
ISSN = {00255718},
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/s00255718198206582125},
Key = {fds243327}
}
@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 = {359392},
Publisher = {WILEY},
Year = {1981},
Month = {January},
ISSN = {00103640},
url = {http://dx.doi.org/10.1002/cpa.3160340305},
Doi = {10.1002/cpa.3160340305},
Key = {fds243328}
}
@article{fds243326,
Author = {Beale, JT and MAJDA, A},
Title = {Rates of Convergence for Viscous Splitting of the
NavierStokes Equations},
Journal = {Mathematics of Computation},
Volume = {37},
Number = {156},
Pages = {243259},
Publisher = {JSTOR},
Year = {1981},
ISSN = {00255718},
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{fds243324,
Author = {Beale, JT},
Title = {WaterWaves Generated by a Pressure Disturbance on a Steady
Stream},
Journal = {Duke Mathematical Journal},
Volume = {47},
Number = {2},
Pages = {297323},
Publisher = {Duke University Press},
Year = {1980},
ISSN = {00127094},
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/S0012709480047195},
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 = {230263},
Publisher = {Elsevier BV},
Year = {1979},
Month = {January},
ISSN = {00220396},
url = {http://dx.doi.org/10.1016/00220396(79)901463},
Doi = {10.1016/00220396(79)901463},
Key = {fds243339}
}
@article{fds243325,
Author = {Beale, JT},
Title = {The existence of solitary water waves},
Journal = {Communications on Pure and Applied Mathematics},
Volume = {30},
Number = {4},
Pages = {373389},
Publisher = {WILEY},
Year = {1977},
Month = {July},
ISSN = {00103640},
url = {http://dx.doi.org/10.1002/cpa.3160300402},
Doi = {10.1002/cpa.3160300402},
Key = {fds243325}
}
@article{fds243322,
Author = {Beale, JT},
Title = {Eigenfunction expansions for objects floating in an open
sea},
Journal = {Communications on Pure and Applied Mathematics},
Volume = {30},
Number = {3},
Pages = {283313},
Publisher = {WILEY},
Year = {1977},
Month = {May},
ISSN = {00103640},
url = {http://dx.doi.org/10.1002/cpa.3160300303},
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 = {199222},
Year = {1977},
url = {http://dx.doi.org/10.1512/iumj.1977.26.26015},
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 socalled 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 precompact 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 = {895917},
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 = {607637},
Publisher = {Duke University Press},
Year = {1974},
Month = {September},
ISSN = {00127094},
url = {http://dx.doi.org/10.1215/s0012709474041659},
Doi = {10.1215/s0012709474041659},
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 = {12761278},
Publisher = {American Mathematical Society (AMS)},
Year = {1974},
Month = {January},
ISSN = {00029904},
url = {http://dx.doi.org/10.1090/S000299041974137146},
Doi = {10.1090/S000299041974137146},
Key = {fds243318}
}
@article{fds243319,
Author = {Beale, JT},
Title = {Scattering frequencies of resonators},
Journal = {Communications on Pure and Applied Mathematics},
Volume = {26},
Number = {4},
Pages = {549563},
Publisher = {WILEY},
Year = {1973},
Month = {July},
ISSN = {00103640},
url = {http://dx.doi.org/10.1002/cpa.3160260408},
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.
1826.},
Key = {fds10345}
}
@article{fds9732,
Author = {J. T. Beale},
Title = {Boundary Integral Methods for ThreeDimensional Water
Waves},
Journal = {Equadiff 99, Proceedings of the International Conference on
Differential Equations, Vol. 2, pp. 136978},
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, 10727.},
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 MultiFluid Flows, Y. Renardy et al., ed., pp.
24145, 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, 56575, Soc. Math. Appl. Indust., Paris,
1996.},
Key = {fds8749}
}
@article{fds9209,
Author = {J. T. Beale and E. Thomann and C. Greengard},
Title = {Operator splitting for NavierStokes and the ChorinMarsden
product formula},
Journal = {Vortex Flows and Related Numerical Methods, J. T. Beale et
al., ed., pp. 2738, 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. 2337, SpringerVerlag, New York,
1991.},
Key = {fds9203}
}
@article{fds9205,
Author = {J. T. Beale and A. Eydeland and B. Turkington},
Title = {Numerical tests of 3D vortex methods using a vortex ring
with swirl},
Journal = {Vortex Dynamics and Vortex Methods, C. Anderson and C.
Greengard, ed., pp. 19, 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.
29398, NATO ASI Series, Plenum, 1991.},
Key = {fds9206}
}
@article{fds9200,
Author = {J. T. Beale},
Title = {Largetime behavior of model gases with a discrete set of
velocities},
Journal = {Mathematics Applied to Science, J. Goldstein et al., ed. pp.
112, 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. 1932, SpringerVerlag, 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,
13748.},
Key = {fds9195}
}
@article{fds9196,
Author = {J. T. Beale},
Title = {A convergent threedimensional vortex method with gridfree
stretching},
Journal = {Math. Comp. 46 (1986), 40124 and S15S20.},
Key = {fds9196}
}
@article{fds9191,
Author = {J. T. Beale},
Title = {Largetime regularity of viscous surface
waves},
Journal = {Arch. Rational Mech. Anal. 84 (1984), 30752.},
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), 22129.},
Key = {fds9193}
}
@article{fds9190,
Author = {J. T. Beale},
Title = {Largetime regularity of viscous surface
waves},
Journal = {Contemp. Math. 17 (1983), 3133.},
Key = {fds9190}
}
@article{fds9187,
Author = {J. T. Beale and A. Majda},
Title = {Vortex methods I: Convergence in three dimensions},
Journal = {Math. Comp. 39 (1982), 127.},
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}
}
