%% 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} }