an integral equation method for epitaxial step flow growth simulations in this paper we describe an integral equation approach for simulating diffusion problems with moving interfaces the solutions are represented as moving layer potentials where the unknowns are only defined on the interfaces the resulting integro differential equation ide system is solved using spectral deferred correction sdc techniques developed for general differential algebraic equations daes and the time dependent potentials are evaluated efficiently using fast convolution algorithms the numerical solver is applied to the bcf model for the epitaxial step flow growth of crystals for which the solutions are calculated accurately instead of using quasi static approximations numerical results in 1 1 dimensions are compared with available results in the literature

a mass redistributed finite element method mr fem for acoustic problems using triangular mesh the accuracy of numerical results using standard finite element method fem in acoustic problems will deteriorate with increasing frequency due to the dispersion error such dispersion error depends on the balance between the stiffness and mass of discretization equation systems this paper reports an improved finite element method fem for solving acoustic problems by re distributing the mass in the mass matrix to tune the balance aiming to minimize the dispersion errors this is done by shifting the integration point locations when computing the entries of the mass matrix while ensuring the mass conservation the new method is verified through the detailed numerical error analysis and a strategy is also proposed for the best mass redistribution in terms of minimizing dispersion error the relative dispersion error of present mass redistributed finite element method mr fem is found to be much smaller than the fem solution in both theoretical prediction and numerical examination the present mr fem works well by using the linear triangular elements that can be generated automatically which enables automation in computation and saving computational cost in mesh generation numerical examples demonstrate the advantages of mr fem in comparison with the standard fem using the same triangular meshes and quadrilateral meshes

a novel multigrid based preconditioner for heterogeneous helmholtz problems an iterative solution method in the form of a preconditioner for a krylov subspace method is presented for the helmholtz equation the preconditioner is based on a helmholtz type differential operator with a complex term a multigrid iteration is used for approximately inverting the preconditioner the choice of multigrid components for the corresponding preconditioning matrix with a complex diagonal is validated with fourier analysis multigrid analysis results are verified by numerical experiments high wavenumber helmholtz problems in heterogeneous media are solved indicating the performance of the preconditioner

subcritical approach to sharp hardy littlewood sobolev type inequalities on the upper half space abstract in this paper we establish the reversed sharp hardy littlewood sobolev hls for short inequality on the upper half space and obtain a new hls type integral inequality on the upper half space extending an inequality found by hang wang and yan in 6 by introducing a uniform approach the extremal functions are classified via the method of moving spheres and the best constants are computed the new approach can also be applied to obtain the classical hls inequality and other similar inequalities

an optimal 9 point finite difference frequency space 2 d scalar wave extrapolator in this study a new finite difference technique is designed to reduce the number of grid points needed in frequency space domain modeling the new algorithm uses optimal nine point operators for the approximation of the laplacian and the mass acceleration terms the coefficients can be found by using the steepest descent method so that the best normalized phase curves can be obtained this method reduces the number of grid points per wavelength to 4 or less with consequent reductions of computer memory and cpu time that are factors of tens less than those involved in the conventional second order approximation formula when a band type solver is used on a scalar machine

high order accurate methods for the evaluation of layer heat potentials we discuss the numerical evaluation of single and double layer heat potentials in two dimensions on stationary and moving boundaries one of the principal difficulties in designing high order methods concerns the local behavior of the heat kernel which is both weakly singular in time and rapidly decaying in space we show that standard quadrature schemes suffer from a poorly recognized form of inaccuracy which we refer to as geometrically induced stiffness but that rules based on product integration of the full heat kernel in time are robust when combined with previously developed fast algorithms for the evolution of the history part of layer potentials diffusion processes in complex moving geometries can be computed accurately and in nearly optimal time

crystal growth and dendritic solidification we present a numerical method which computes the motion of complex solid liquid boundaries in crystal growth the model we solve includes physical effects such as crystalline anisotropy surface tension molecular kinetics and undercooling the method is based on two ideas first the equations of motion are recast as a single history dependent boundary integral equation on the solid liquid boundary a fast algorithm is used to solve the integral equation efficiently second the boundary is moved by solving a hamiltonjacobi type equation on a fixed domain formulated by osher and sethian for a function in which the boundary is a particular level set this equation is solved by finite difference schemes borrowed from the technology of hyperbolic conservation laws the two ideas are combined by constructing a smooth extension of the normal velocity off the moving boundary in a way suggested by the physics of the problem our numerical experiments show the evolution of complex crystalline shapes development of large spikes and corners dendrite formation and side branching and pieces of solid merging and breaking off freely

a second order accurate symmetric discretization of the poisson equation on irregular domains in this paper we consider the variable coefficient poisson equation with dirichlet boundary conditions on an irregular domain and show that one can obtain second order accuracy with a rather simple discretization moreover since our discretization matrix is symmetric it can be inverted rather quickly as opposed to the more complicated nonsymmetric discretization matrices found in other second order accurate discretizations of this problem multidimensional computational results are presented to demonstrate the second order accuracy of this numerical method in addition we use our approach to formulate a second order accurate symmetric implicit time discretization of the heat equation on irregular domains then we briefly consider stefan problems

compact finite difference schemes of sixth order for the helmholtz equation new compact finite difference schemes of sixth order are derived for the three dimensional helmholtz equation du k 2u f convergence characteristics and accuracy are compared and a truncation error analysis is presented for a broad range of k values

a high order solver for the heat equation in 1d domains with moving boundaries we describe a fast high order accurate method for the solution of the heat equation in domains with moving dirichlet or neumann boundaries and distributed forces we assume that the motion of the boundary is prescribed our method extends the work of greengard and strain comm pure appl math xliii 1990 pp 949 963 our scheme is based on a time space chebyshev pseudo spectral collocation discretization which is combined with a recursive product quadrature rule to accurately and efficiently approximate convolutions with green s function for the heat equation we present numerical results that exhibit up to eighth order convergence rates assuming n time steps and m spatial discretization points the evaluation of the solution of the heat equation at the same number of points in space time requires mathcal o n m log m work thus our scheme can be characterized as fast that is it is work optimal up to a logarithmic factor

blowup analysis for a nonlinear equation with negative exponent abstract in this paper we study the blowup behavior of the minimizing sequence to certain nonlinear elliptic equation with negative critical exponents we analyze the location for blowup point and the asymptotic behavior of solutions near the blowup point

new finite element finite volume level set formulation for modelling two phase incompressible flows we present a new level set method for modelling two phase incompressible flows with moving interface boundaries the method is based on a finite element fe treatment of the equations governing the motion of the interface separating two immiscible phases and a finite volume fv description of the fluid flow equations the performance of the method is demonstrated and validated using several sample problems involving two phase isothermal flows with density ratios of the order of 102 103 and viscosity ratios as high as 7 104 surface tension effects are included the present formulation displays good convergence properties and it is relatively simple to program this allows the method to be implemented into existing commercial fv based computational fluid dynamics cfd software thereby extending their application range we have successfully implemented the method in the code cfx 4

evolution implementation and application of level set and fast marching methods for advancing fronts abstract a variety of numerical techniques are available for tracking moving interfaces in this review we concentrate on techniques that result from the link between the partial differential equations that describe moving interfaces and numerical schemes designed for approximating the solutions to hyperbolic conservation laws this link gives rise to computational techniques for tracking moving interfaces in two and three space dimensions under complex speed laws we discuss the evolution of these techniques the fundamental numerical approximations involved implementation details and applications in particular we review some work on three aspects of materials sciences semiconductor process simulations seismic processing and optimal structural topology design

sharp sobolev inequalities on the sphere and the moser trudinger inequality where de denotes normalized surface measure v is the conformal gradient and q 2n n 2 a modern folklore theorem is that by taking the infinitedimensional limit of this inequality one obtains the gross logarithmic sobolev inequality for gaussian measure which determines nelson s hypercontractive estimates for the hermite semigroup see 8 one observes using conformal invariance that the above inequality is equivalent to the sharp sobolev inequality on rn for which boundedness and extremal functions can be easily calculated using dilation invariance and geometric symmetrization the roots here go back to hardy and littlewood the advantage of casting the problem on the sphere is that the role of the constants is evident and one is led immediately to the conjecture that this inequality should hold whenever possible for example 2 q 0o if n 2 this is in fact true and will be demonstrated in section 2 a clear question at this point is what is the situation in dimension 2 two important arguments 25 26 27 dealt with this issue both motivated by geometric variational problems because q goes to infinity for dimension 2 the appropriate function space is the exponential class responding in part

corrigendum to subcritical approach to sharp hardy littlewood sobolev type inequalities on the upper half space adv math 312 2017 1 45

wavenumber explicit convergence analysis for galerkin discretizations of the helmholtz equation we develop a stability and convergence theory for a class of highly indefinite elliptic boundary value problems bvps by considering the helmholtz equation at high wavenumber k as our model problem the key element in this theory is a novel k explicit regularity theory for helmholtz bvps that is based on decomposing the solution into two parts the first part has the sobolev regularity properties expected of second order elliptic pdes but features k independent regularity constants the second part is an analytic function for which k explicit bounds for all derivatives are given this decomposition is worked out in detail for several types of bvps namely the helmholtz equation in bounded smooth domains or convex polygonal domains with robin boundary conditions and in exterior domains with dirichlet boundary conditions we present an error analysis for the classical hp version of the finite element method hp fem where the dependence on the mesh width h the approximation order p and the wavenumber k is given explicitly in particular under the assumption that the solution operator for helmholtz problems is polynomially bounded in k it is shown that quasi optimality is obtained under the conditions that kh p is sufficiently small and the polynomial degree p is at least o log k

a fast algorithm for the evaluation of heat potentials a fast algorithm for the evaluation of heat potentials to appear in communications and pure and applied mathematics l greengard j strain may 1989 abstract numerical methods for solving the heat equation via potential theory have been hampered by the high cost of evaluating heat potentials when m points are used in the discretization of the boundary and n time steps are computed an amount of work of the order o n 2 m 2 has traditionally been required in this paper we present an algorithm which requires an amount of work of the order o n m and we observe speedups of five orders of magnitude for large scale problems thus the method makes it possible to solve the heat equation by potential theory in practical situations keywords heat equation potential theory fast algorithms integral equations introduction a classical approach to the solution of the heat equation u t u q t in a space time domain t t 0 t see fig 1 is through the use of heat potentials 3 7 given zero initial conditions one seeks a representation of u as a single layer heat potential z t z s x t k x x 0 t t 0 x 0 t 0 dx 0 dt 0 t 0 or a double layer heat potential d x t z t z t 0 k x x 0 t t 0 x 0 t 0 dx 0 dt 0 n 0 where k is a fundamental solution of the heat equation in some region containing q n 0 denotes the unit t outward normal to t t at x and is a surface density defined on t t 0 t yale university and courant institute of mathematical sciences new york university the work of this author was supported in part by the office of naval research under grant n00014 89 j 1527 in part by ibm under grant p00038437 and in part by a nsf mathematical sciences postdoctoral fellowship courant institute of mathematical sciences new york university the work of this author was supported by darpa afosr contract no f 49620 87 c 0065

a simple level set method for solving stefan problems a simple level set method for solving stefan problems is presented this method can be applied to problems involving dendritic solidification our method consists of an implicit finite difference scheme for solving the heat equation and a level set approach for capturing the front between solid and liquid phases of a pure substance our method is accurate with respect to some exact solutions of the stefan problem results indicate that this method can handle topology changes and complicated interfacial shapes and that it can numerically simulate many of the physical features of dendritic solidification

efficient thermal field computation in phase field models we solve the phase field equations in two dimensions to simulate crystal growth in the low undercooling regime the novelty is the use of a fast solver for the free space heat equation to compute the thermal field this solver is based on the efficient direct evaluation of the integral representation of the solution to the constant coefficient free space heat equation with a smooth source term the computational cost and memory requirements of the new solver are reasonable and no artificial boundary conditions are needed this allows one to solve for the thermal field in a computational domain whose size depends only on the size of the growing crystal and not on the extent of the thermal field which can result in significant computational savings in the low undercooling regime

a lagrangian particle level set method we present a novel particle level set method for capturing interfaces the level set equation is solved in a lagrangian frame using particles that carry the level set information a key aspect of the method involves a consistent remeshing procedure for the regularization of the particle locations when the particle map gets distorted by the advection field the lagrangian description of the level set method is inherently adaptive and exact in the case of solid body motions the efficiency and accuracy of the method is demonstrated in several benchmark problems in two and three dimensions involving pure advection and curvature induced motion of the interface the simplicity of the particle description is shown to be well suited for real time simulations of surfaces involving cutting and reconnection as in virtual surgery environments

tree methods for moving interfaces fast adaptive numerical methods for solving moving interface problems are presented the methods combine a level set approach with frequent redistancing and semi lagrangian time stepping schemes which are explicit yet unconditionally stable a quadtree mesh is used to concentrate computational effort on the interface so the methods move an interface withndegrees of freedom ino nlogn work per time step efficiency is increased by taking large time steps even for parabolic curvature flows the methods compute accurate viscosity solutions to a wide variety of difficult moving interface problems involving merging anisotropy faceting and curvature

compact 2d and 3d sixth order schemes for the helmholtz equation with variable wave number several studies have presented compact fourth order accurate finite difference approximation for the helmholtz equation in two or three dimensions several of these formulae allow for the wave number k to be variable other papers have extended this further to include variable coefficients within the laplacian which models non homogeneous materials in electromagnetism later papers considered more accurate compact sixth order methods but these were restricted to constant k in this paper we extend these compact sixth order schemes to variable k in both two and three dimensions results on 2d and 3d problems with known analytic solutions verify the sixth order accuracy we demonstrate that for large wave numbers the second order scheme cannot produce comparable results with reasonable grid sizes

the role of the green s function in a non linear elliptic equation involving the critical sobolev exponent this paper is concerned with non linear elliptic problems of the type pe u u n 2 n 2 eu u 0 on u 0 on where is a smooth and bounded domain in rn n 4 and e 0 we show that if the ue are solutions of pe which concentrate around a point as e 0 then this point cannot be on the boundary of and is a critical point of the regular part of the green s function conversely we show that for n 5 and any non degenerate critical point x0 of the regular part of the green s function there exist solutions of pe concentrating around x0 as e 0

a rapidly converging domain decomposition method for the helmholtz equation a new domain decomposition method is introduced for the heterogeneous 2 d and 3 d helmholtz equations transmission conditions based on the perfectly matched layer pml are derived that avoid artificial reflections and match incoming and outgoing waves at the subdomain interfaces we focus on a subdivision of the rectangular domain into many thin subdomains along one of the axes in combination with a certain ordering for solving the subdomain problems and a gmres outer iteration when combined with multifrontal methods the solver has near linear cost in examples due to very small iteration numbers that are essentially independent of problem size and number of subdomains it is to our knowledge only the second method with this property next to the moving pml sweeping method

implementing continuous scale morphology via curve evolution abstract a new approach to digital implementation of continuous scale mathematical morphology is presented the approach is based on discretization of evolution equations associated with continuous multiscale morphological operations those equations and their corresponding numerical implementation can be derived either directly from mathematical morphology definitions or from curve evolution theory the advantages of the proposed approach over the classical discrete morphology are demonstrated

numerical approximation of a thermally driven interface using finite elements a two dimensional finite element model for dendritic solidification has been developed that is based on the direct solution of the energy equation over a fixed mesh the model tracks the position of the sharp solid liquid interface using a set of marker points placed on the interface the simulations require calculation of the temperature gradients on both sides of the interface in the direction normal to it at the interface the heat flux is discontinuous due to the release of latent heat during the solidification melting process two ways to calculate the temperature gradients at the interface evaluating their interpolants at gauss points were proposed using known one and two dimensional solutions to stable solidification problems the stefan problem it was shown that the method converges with second order accuracy when applied to the unstable solidification of a crystal into an undercooled liquid it was found that the numerical solution is extremely sensitive to the mesh size and the type of approximation used to calculate the temperature gradients at the interface i e different approximations and different meshes can yield different solutions the cause of these difficulties is examined the effect of different types of interpolation on the simulations is investigated and the necessary criteria to ensure converged solutions are established copyright 2003 john wiley sons ltd

shape instabilities and pattern formation in solidification a new method for numerical solution of the moving boundary problem abstract when a solidification front is advancing into a region of supercooled liquid its shape is subject to instabilities which can lead to complex modes of growth the evaluation of the growth pattern is determined by the interaction of the driving force of the instability due to heat diffusion with the restabilizing force due to surface tension and its full development is a highly nonlinear process a new method for solving the two phase heat diffusion problem under these conditions is described the method is based on a single domain treatment weak formulation of the moving boundary problem with the temperature field near the interface approximated by a piecewise linear function in the spirit of the finite element technique calculated results for a model problem are in excellent agreement with the predictions of an analytical approximation

a fast semi lagrangian contouring method for moving interfaces abstract general moving interface problems are solved by a new approach evaluating an explicit semi lagrangian advection formula with efficient geometric algorithms and extracting the moving interface with a fast new contouring technique the new approach decouples spatial and temporal resolutions and grid free adaptive refinement of the interface increases accuracy dramatically a modular implementation with a fast new intrinsic geometry module computes highly accurate solutions to geometric moving interface problems involving merging anisotropy and faceting with a high order embedded geometry module it solves second order problems involving curvature dynamic topology and nonlocal interactions

multiscale finite difference diffusion monte carlo method for simulating dendritic solidification abstract we present a novel hybrid computational method to simulate accurately dendritic solidification in the low undercooling limit where the dendrite tip radius is one or more orders of magnitude smaller than the characteristic spatial scale of variation of the surrounding thermal or solutal diffusion field the first key feature of this method is an efficient multiscale diffusion monte carlo dmc algorithm which allows off lattice random walkers to take longer and concomitantly rarer steps with increasing distance away from the solid liquid interface as a result the computational cost of evolving the large scale diffusion field becomes insignificant when compared to that of calculating the interface evolution the second key feature is that random walks are only permitted outside of a thin liquid layer surrounding the interface inside this layer and in the solid the diffusion equation is solved using a standard finite difference algorithm that is interfaced with the dmc algorithm using the local conservation law for the diffusing quantity here we combine this algorithm with a previously developed phase field formulation of the interface dynamics and demonstrate that it can accurately simulate three dimensional dendritic growth in a previously unreachable range of low undercoolings that is of direct experimental relevance

a boundary integral approach to unstable solidification we consider the supercooled stefan problem with a general anisotropic curvature and velocity dependent boundary condition on the moving interface we present numerical methods based on an integral equation formulation and including a new algorithm for moving curves with curvature dependent velocity these methods compute a periodic interface with o t accuracy where t is the time step previous work has been limited to short time spans and achieved slightly less than o t12 accuracy accurate numerical results are seen to agree with the predictions of linear stability theory this agreement has eluded previous authors because their numerical methods suffered from grid effects and their linear stability theory was incorrect we study the long time evolution of an unstable interface our computations exhibit the beginnings of a sidebranching instability when the boundary condition includes anisotropy and tip splitting in the isotropic case

a dispersion minimizing finite difference scheme and preconditioned solver for the 3d helmholtz equation in this paper a new 27 point finite difference method is presented for solving the 3d helmholtz equation with perfectly matched layer pml which is a second order scheme and pointwise consistent with the equation an error analysis is made between the numerical wavenumber and the exact wavenumber and a refined choice strategy based on minimizing the numerical dispersion is proposed for choosing weight parameters a full coarsening multigrid based preconditioned bi cgstab method is developed for solving the linear system stemming from the helmholtz equation with pml by the finite difference scheme the shifted laplacian is extended to precondition the 3d helmholtz equation and a spectral analysis is given the discrete preconditioned system is solved by the bi cgstab method with a multigrid method used to invert the preconditioner approximately full coarsening multigrid is employed and a new matrix based prolongation operator is constructed accordingly numerical results are presented to demonstrate the efficiency of both the new 27 point finite difference scheme with refined parameters and the preconditioned bi cgstab method with the 3d full coarsening multigrid

a generalized finite element method for solving the helmholtz equation in two dimensions with minimal pollution when using the galerkin fem for solving the helmholtz equation in two dimensions the error of the corresponding solution differs substantially from the error of the best approximation and this effect increases with higher wave number k in this paper we will design a generalized finite element method gfem for the helmholtz equation such that the pollution effect is minimal

a parallel sweeping preconditioner for heterogeneous 3d helmholtz equations a parallelization of a sweeping preconditioner for three dimensional helmholtz equations without large cavities is introduced and benchmarked for several challenging velocity models the setup and application costs of the sequential preconditioner are shown to be o gamma 2 n 4 3 and o gamma n log n where gamma omega denotes the modestly frequency dependent number of grid points per perfectly matched layer several computational and memory improvements are introduced relative to using black box sparse direct solvers for the auxiliary problems and competitive runtimes and iteration counts are reported for high frequency problems distributed over thousands of cores two open source packages are released along with this paper parallel sweeping preconditioner psp and the underlying distributed multifrontal solver clique

shortening three dimensional curves via two dimensional flows in this paper a curve evolution approach for the computation of geodesic curves on 3d surfaces is presented the algorithm is based on deforming via the curve shortening flow an arbitrary initial curve ending at two given surface points the 3d curve shortening flow is first transformed into an equivalent 2d one this 2d flow is implemented using an efficient numerical algorithm for curve evolution with fixed end points

efficient symmetric discretization for the poisson heat and stefan type problems with robin boundary conditions we present a novel and efficient method for solving the poisson equation the heat equation and stefan type problems with robin boundary conditions over potentially moving arbitrarily shaped domains the method utilizes a level set framework thus it has all of the benefits of a sharp implicitly represented interface such as the ease of handling complex topological changes this method is straightforward to implement and leads to a linear system that is symmetric and positive definite which can be inverted efficiently with standard iterative methods this approach is second order accurate for both the poisson and heat equations and first order accurate for the stefan problem we demonstrate the accuracy in the l 1 and l norms

boundary integral formulation of the two dimensional symmetric model of dendritic growth abstract selection of steady needle crystals in the symmetric model of dendritic growth is considered the diffusion equation and associated kinematic and thermodynamic boundary condition are recast into a nonlinear equation through the use of boundary integral methods the equation is solved numerically and for the range of peclet numbers considered it is found that a smooth solution exists only if anisotropy is included in the capillarity term of the gibbs thompson condition the behavior of the selected velocity and tip radius as a function of undercooling is also examined a linear integral equation is also derived to examine the stability of the steady state

blowup analysis for integral equations on bounded domains abstract consider the integral equation f q 1 x f y x y n d y f x 0 x where r n is a smooth bounded domain for 1 n the existence of energy maximizing positive solution in the subcritical case 2 q 2 n n and nonexistence of energy maximizing positive solution in the critical case q 2 n n are proved in 6 for n the existence of energy minimizing positive solution in the subcritical case 0 q 2 n n and nonexistence of energy minimizing positive solution in the critical case q 2 n n are also proved in 4 based on these in this paper the blowup behaviour of energy maximizing positive solution as q 2 n n in the case of 1 n and the blowup behaviour of energy minimizing positive solution as q 2 n n in the case of n are analyzed we see that for 1 n the blowup behaviour obtained is quite similar to that of the elliptic equation involving the subcritical sobolev exponent but for n different phenomena appear

a parallel multigrid based preconditioner for the 3d heterogeneous high frequency helmholtz equation we investigate the parallel performance of an iterative solver for 3d heterogeneous helmholtz problems related to applications in seismic wave propagation for large 3d problems the computation is no longer feasible on a single processor and the memory requirements increase rapidly therefore parallelization of the solver is needed we employ a complex shifted laplace preconditioner combined with the bi cgstab iterative method and use a multigrid method to approximate the inverse of the resulting preconditioning operator a 3d multigrid method with 2d semi coarsening is employed we show numerical results for large problems arising in geophysical applications

a dispersion minimizing scheme for the 3 d helmholtz equation based on ray theory we develop a new dispersion minimizing compact finite difference scheme for the helmholtz equation in 2 and 3 dimensions the scheme is based on a newly developed ray theory for difference equations a discrete helmholtz operator and a discrete operator to be applied to the source and the wavefields are constructed their coefficients are piecewise polynomial functions of hk chosen such that phase and amplitude errors are minimal the phase errors of the scheme are very small approximately as small as those of the 2 d quasi stabilized fem method and substantially smaller than those of alternatives in 3 d assuming the same number of gridpoints per wavelength is used in numerical experiments accurate solutions are obtained in constant and smoothly varying media using meshes with only five to six points per wavelength and wave propagation over hundreds of wavelengths when used as a coarse level discretization in a multigrid method the scheme can even be used with down to three points per wavelength tests on 3 d examples with up to 108 degrees of freedom show that with a recently developed hybrid solver the use of coarser meshes can lead to corresponding savings in computation time resulting in good simulation times compared to the literature

accurate finite difference methods for time harmonic wave propagation finite difference methods for solving problems of time harmonic acoustics are developed and analyzed multi dimensional inhomogeneous problems with variable possibly discontinuous coefficients are considered accounting for the effects of employing non uniform grids a weighted average representation is less sensitive to transition in wave resolution due to variable wave numbers or non uniform grids than the standard pointwise representation further enhancement in method performance is obtained by basing the stencils on generalizations of pade approximation or generalized definitions of the derivative reducing spurious dispersion anisotropy and reflection and by improving the representation of source terms the resulting schemes have fourth order accurate local truncation error on uniform grids and third order in the non uniform case guidelines for discretization pertaining to grid orientation and resolution are presented

dynamics of layered interfaces arising from phase boundaries the dynamics of a material in two phases is studied in the context of phase field models based on a landau ginzburg free energy functional they consist of a system of two nonlinear diffusion equat

fast potential theory ii layer potentials and discrete sums we present three new families of fast algorithms for classical potential theory based on ewald summation and fast transforms of gaussians and fourier series ewald summation separates the green function for a cube into a high frequency localized part and a rapidly converging fourier series each part can then be evaluated efficiently with appropriate fast transform algorithms our algorithms are naturally suited to the use of green functions with boundary conditions imposed on the boundary of a cube rather than free space green functions our first algorithm evaluates classical layer potentials on the boundary of a d dimensional domain with d equal to two or three the quadrature error is o hm e where h is the mesh size on the boundary and m is the order of quadrature used the algorithm evaluates the discretized potential using n elements at o n points in o n log n arithmetic operations the constant in o n log n depends logarithmically on the desired error tolerance our second scheme evaluates a layer potential on the domain itself with the same accuracy it produces md values using n boundary elements in o n md log m arithmetic operations our third method evaluates a discrete sum of values of the green function of the type which occur in particle methods it attains error at a cost o na log n where a 2 1 dd and d is the hausdorff dimension of the set where the sources concentrate in the limit n thus it is o n log n when the sources do not cluster too much and close to o n log n in the important practical case when the points are uniformly distributed over a hypersurface we also sketch an o n log n algorithm based on special functions two dimensional numerical results are presented for all three algorithms layer potentials are evaluated to second order accuracy in times which exhibit considerable speedups even over a reasonably sophisticated direct calculation discrete sum calculations are speeded up astronomically our algorithm reduces the cpu time required for a calculation with 40 000 points from six months to one hour

symmetry breaking for integral equation of henon type on unit ball abstract in this paper our purpose is to study the nonlinear integral equation of henon type 0 1 u x q 1 x y u y x y n d y where is the unit ball in r n centred at the origin 0 0 and 1 n we prove that there exists a positive radial solution of problem 0 1 provided that q 2 n n 2 and analyse the symmetry breaking of the maximizer of supremum q on l q for large enough

adaptive mesh computation of polycrystalline pattern formation using a renormalization group reduction of the phase field crystal model we implement an adaptive mesh algorithm for calculating the space and time dependence of the atomic density field in microscopic material processes our numerical approach uses the systematic renormalization group formulation of a phase field crystal model of a pure material to provide the underlying equations for the complex amplitude of the atomic density field a quantity that is spatially uniform except near topological defects grain boundaries and other lattice imperfections our algorithm employs a hybrid formulation of the amplitude equations combining cartesian and polar decompositions of the complex amplitude we show that this approach leads to an acceleration by three orders of magnitude in model calculations of polycrystalline grain growth in two dimensions

front tracking finite element method for dendritic solidification 765 abstract a two dimensional finite element method capable of tracking sharp interfaces is developed the method is based on a fixed mesh of bilinear isoparametric elements and is extremely simple and easy to use the interfaces are tracked with a set of marker points that define their position at all times several different approaches to finding the interface position and velocity are discussed and their effect on the convergence rate is examined it is shown through numerical examples that the approximation to the interface position converges with second order accuracy in two dimensional problems as opposed to previously developed finite difference algorithms that are only first order accurate the approximation to the interface velocity is shown to be of first order a number of examples are examined including several that provide a careful comparison with previously published results

a multigrid method for the helmholtz equation with optimized coarse grid corrections we study the convergence of multigrid schemes for the helmholtz equation focusing in particular on the choice of the coarse scale operators let g rm c denote the number of points per wavelength at the coarse level if the coarse scale solutions are to approximate the true solutions then the oscillatory nature of the solutions implies the requirement g rm c 2 however in examples the requirement is more like g rm c gtrsim 10 in a trade off involving also the amount of damping present and the number of multigrid iterations we conjecture that this is caused by the difference in phase speeds between the coarse and fine scale operators standard 5 point finite differences in two dimensions are our first example a new coarse scale 9 point operator is constructed to match the fine scale phase speeds we then compare phase speeds and multigrid performance of standard schemes with a scheme using the new operator the required g rm c is reduced from about 10 to about 3 5 with less dam

on the two phase stefan problem with interfacial energy and entropy abstract the classical stefan theory for the melting of a solid or the freezing of a liquid is too simplistic to describe phenomena such as supercooling in which a liquid supports temperatures below its freezing point or superheating the analog for solids or dendritic growth in which simple shapes evolve to complicated tree like structures this paper develops a general theory for two phase phenomena of this type it develops partial differential equations satisfied in the phase regions and free boundary conditions satisfied on the interface between phases and gives arguments which indicate that the resulting boundary value problems predict the formation of dendrites keywords equilibrium liapunoo functions mathematical physics

motion of level sets by mean curvature i we continue our investigation of the level set technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature the principal assertion of this paper is a kind of reconciliation with the geometric measure theoretic approach pioneered by k brakke we prove that almost every level set of the solution to the mean curvature evolution pde is in fact aunit density varifold moving according to its mean curvature in particular a e level set is endowed with a kind of geometric structure the proof utilizes compensated compactness methods to pass to limits in various geometric expressions

level set methods an overview and some recent results the level set method was devised by s osher and j a sethian 1988 j comput phys 79 12 49 as a simple and versatile method for computing and analyzing the motion of an interface in two or three dimensions bounds a possibly multiply connected region the goal is to compute and analyze the subsequent motion of under a velocity field v this velocity can depend on position time the geometry of the interface and the external physics the interface is captured for later time as the zero level set of a smooth at least lipschitz continuous function x t i e t x x t 0 is positive inside negative outside and is zero on t topological merging and breaking are well defined and easily performed in this review article we discuss recent variants and extensions including the motion of curves in three dimensions the dynamic surface extension method fast methods for steady state problems diffusion generated motion and the variational level set approach we also give a user s guide to the level set dictionary and technology and couple the method to a wide variety of problems involving external physics such as compressible and incompressible possibly reacting flow stefan problems kinetic crystal growth epitaxial growth of thin films vortex dominated flows and extensions to multiphase motion we conclude with a discussion of applications to computer vision and image processing

numerical simulation of dendritic solidification with convection two dimensional geometry a front tracking method is presented for simulations of dendritic growth of pure substances in the presence of flow the liquid solid interface is explicitly tracked and the latent heat released during solidification is calculated using the normal temperature gradient near the interface a projection method is used to solve the navier stokes equations the no slip condition on the interface is enforced by setting the velocities in the solid phase to zero the method is validated through a comparison with an exact solution for a stefan problem a grid refinement test and a comparison with a solution obtained by a boundary integral method three sets of two dimensional simulations are presented a comparison with the simulations of beckermann et al j comput phys 154 468 1999 a study of the effect of different flow velocities and a study of the effect of the prandtl number on the growth of a group of dendrites growing together the simulations show that on the upstream side the dendrite tip velocity is increased due to the increase in the temperature gradient and the formation of side branches is promoted the flow has the opposite effect on the downstream side the results are in good qualitative agreement with published experimental results even though only the two dimensional aspects are examined here

linearly first and second order unconditionally energy stable schemes for the phase field crystal model in this paper we develop a series of linear unconditionally energy stable numerical schemes for solving the classical phase field crystal model the temporal discretizations are based on the first order euler method the second order backward differentiation formulas bdf2 and the second order crank nicolson method respectively the schemes lead to linear elliptic equations to be solved at each time step and the induced linear systems are symmetric positive definite we prove that all three schemes are unconditionally energy stable rigorously various classical numerical experiments in 2d and 3d are performed to validate the accuracy and efficiency of the proposed schemes

linear stability of planar solidification fronts abstract we review extend and modify the classical linear stability theory of planar solidification fronts in langer s moving symmetric model using a new integral equation for the front position we compute an exact linear stability equation and solve it exactly for an important special case the classical theory is seen to be the long time limit of the exact solution finally we show how to treat a general planar front using a short time approximation our conclusions generalize those of previous analyses by treating initial temperature field perturbations and the resulting transient growth phenomena

an efficient boundary integral method for the mullins sekerka problem we use a boundary integral technique to study the two space dimensional mullins sekerka free boundary problem which originates from a study of solidification and liquidation of materials of negligible specific heat this is an area preserving and curve shortening motion evolution equations for the free boundaries are derived in terms of the tangent angle and total arclength which makes a small scale decomposition possible and the fourier transform a powerful tool in numerical calculations with this formulation implicit schemes can be implemented to avoid the difficult numerical stiffness associated with explicit schemes we can compute solutions up to the time when there is a topological change i e when particles touch or break up our numerical results for systems of a single particle or multi particles provide some valuable information in the particle dynamics such as the circularization of each individual particle and the mass transfer between different particles during particle interactions

velocity effects in unstable solidification the supercooled stefan problem with surface tension is considered as a model of unstable solidification with a general anisotropic curvature and velocity dependent boundary condition on the moving interface the problem is reformulated in terms of a history dependent singular integral equation for the velocity of the boundary using this equation a new linear stability analysis of a flat interface is carried out and the smoothing role of velocity dependence and the destabilizing effect of anisotropy are demonstrated the results disagree with previous analyses because transient effects due to the initial temperature field are included and numerical results are presented that confirm the analysis presented here it is found that velocity effects cannot increase the range of unstable modes beyond that permitted by surface tension

curvature and the evolution of fronts the evolution of a front propagating along its normal vector field with speedf dependent on curvaturek is considered the change in total variation of the propagating front is shown to depend only ondf dk only wherek changes sign analysis of the casef k 1 ek where e is a constant shows that curvature plays a role similar to that of viscosity in burgers equation for e 0 and non convex initial data the curvature blows up corners develop and an entropy condition can be formulated to provide an explicit construction for a weak solution beyond the singularity we then numerically show that the solution as e goes to zero converges to the constructed weak solution numerical methods based on finite difference schemes for marker particles along the front are shown to be unstable in regions where the curvature builds as a remedy we show that front tracking based on volume of fluid techniques can be used together with the entropy condition to provide transition from the classical to weak solution

level sets of viscosity solutions some applications to fronts and rendez vous problems the authors treat some applications of hamilton jacobi equations to the study of a flame front propagation model and the rendez vows problem the solution of both problems requires the determination of the level sets of the viscosity solution for the corresponding equation in the flame front propagation model described here it is assumed that the evolution is driven by a vector field satisfying a transversality condition at time t 0 the evolution in the normal direction with variable velocity c x geq 0 is considered as a special case this approach is constructive permitting the numerical solution of such problems

the chebyshev fast gauss and nonuniform fast fourier transforms and their application to the evaluation of distributed heat potentials we present a method for the fast and accurate computation of distributed heat potentials in two dimensions the distributed source is assumed to be given in terms of piecewise space time chebyshev polynomials we discretize uniformly in time whereas in space the polynomials are defined on the leaf nodes of a quadtree the quadtree can vary at each time step we combine a product integration rule with fast algorithms fast heat potentials nonuniform fft fast gauss transform to obtain a high order accurate method with optimal complexity if n is the number of time steps m is the maximum number of leaf nodes over all the time steps and the input contains a qth order polynomial representation of f then our method requires o q 3mnlogm work to evaluate the heat potential at arbitrary mn space time target locations the overall convergence rate of the method is of order q we present numerical experiments for q 4 8 and 16 and we verify the theoretical convergence rate of the method when the solution is sufficiently smooth the 16th order variant results in significant computational savings even in the case in which we require only a few digits of accuracy

nonlinear integral equations on bounded domains abstract in this paper we introduce and study some nonlinear integral equations on bounded domains that are related to the sharp hardy littlewood sobolev inequality existence results as well as nonexistence results are obtained

negative power nonlinear integral equations on bounded domains abstract this is the continuation of our previous work 10 where we introduced and studied some nonlinear integral equations on bounded domains that are related to the sharp hardy littlewood sobolev inequality in this paper we introduce some nonlinear integral equations on bounded domains that are related to the sharp reversed hardy littlewood sobolev inequality these are integral equations with nonlinear term involving negative exponents existence results as well as nonexistence results are obtained

an efficient direct bem numerical scheme for phase change problems using fourier series abstract due to the moving boundary nature of the solidification and melting problems the application of the standard time domain boundary element formulation of the problem presents the difficulty of the numerical evaluation of the history dependence representing more and more work as time proceeds in that regard we present an algorithm based on a green s function which allows a fourier representation of the domain integral in the reinitialization scheme of the direct boundary element method formulation of the problem from which it is possible to carry out the history integration in a recursive manner to obtain the reinitialization approach in the integral representation formula the green s function corresponding to zero temperature in a box containing the original domain will be used instead of using the classical free space fundamental solution this green s function is given in terms of the original fundamental solution plus a regular solution of the heat equation inside the domain in consideration it can therefore be used in the integral representation formula of the heat equation direct formulation to obtain the solution of a heat problem in such domain the mentioned green s function can be obtained by the images method and the resulting source series can also be rewritten in terms of a double fourier series that will be used in the domain integral of the integral representation formula to transform such integral into equivalent surface integrals in this work a new reinitialization scheme has been applied to solve moving boundary problems to detect the location of the phase change interface in solidification and melting problems the numerical results were compared with analytical solutions in the case where they were available showing a good agreement between the results giving special emphasis to the interface location and the temperature distribution

a dispersion minimizing finite difference scheme for the helmholtz equation based on point weighting abstract in this paper we develop a new dispersion minimizing finite difference scheme for the helmholtz equation with perfectly matched layer pml in two dimensional domain which is a second order 9 point scheme to discretize the second derivative operator we employ a linear combination of a point and its neighboring grid points to replace each of the five points in the traditional central difference scheme based on minimizing the numerical dispersion the combination weights are determined by minimizing the numerical dispersion with a flexible selection strategy the new scheme is simple rotation free and pointwise consistent with the equation which is different from the classical rotated 9 point difference scheme obtained by combining the cartesian coordinate system and the rotated system moreover it is a robust scheme even if the step sizes of different directions are not equal convergence analysis and dispersion analysis are given several numerical examples are presented to illustrate the numerical convergence and effectiveness of the new scheme

3d finite difference frequency domain modeling of visco acoustic wave propagation using a massively parallel direct solver a feasibility study we present a finite difference frequency domain method for 3d visco acoustic wave propagation modeling in the frequency domain the underlying numerical problem is the resolution of a large sparse system of linear equations whose right hand side term is the source this system is solved with a massively parallel direct solver we first present an optimal 3d finite difference stencil for frequency domain modeling the method is based on a parsimonious staggered grid method differential operators are discretized with second order accurate staggered grid stencils on different rotated coordinate systems to mitigate numerical anisotropy an antilumped mass strategy is implemented to minimize numerical dispersion the stencil incorporates 27 grid points and spans two grid intervals dispersion analysis shows that four grid points per wavelength provide accurate simulations in the 3d domain to assess the feasibility of the method for frequency domain full waveform inversion we computed simulations in the 3d seg eage overthrust model for frequencies 5 7 and 10 hz results confirm the huge memory requirement of the factorization several hundred figabytes but also the cpu efficiency of the resolution phase few seconds per shot heuristic scalability analysis suggests that the memory complexity of the factorization is o 35n 4 for a n 3 grid our method may provide a suitable tool to perform frequency domain full waveform inversion using a large distributed memory platform further investigation is still necessary to assess more quantitatively the respective merits and drawbacks of time and frequency domain modeling of wave propagation to perform 3d full waveform inversion