a convergence rate theorem for finite difference approximations to delta functions we prove a rate of convergence theorem for approximations to certain integrals over codimension one manifolds in r n the type of manifold involved here is defined by the zero level set of a smooth mapping u r email protected r our approximations are based on the two finite difference methods for discretizing delta functions presented in 16 we included a convergence proof in that paper but only proved rates of convergence in some greatly simplified situations numerical experiments indicated that our two methods were at least first and second order accurate respectively in this note we prove those empirical convergence rates for the two algorithms under fairly general hypotheses

variance estimation using auxiliary information abstract variance estimators under several sample designs are proposed and compared when auxiliary information is available improvement of the bias and mean squared error over some estimators commonly used in practice is illustrated a monte carlo comparison of the estimators is also presented

improvement in estimating the population mean in simple random sampling using information on auxiliary attribute this paper proposes some estimators for the population mean by adapting the estimator in singh et al 2008 5 to the ratio estimators presented in kadilar and cingi 2006 2 we obtain mean square error mse equation for all proposed estimators and show that all proposed estimators are always more efficient than ratio estimator in naik and gupta 1996 3 and singh et al 2008 5 the results have been illustrated numerically by taking some empirical population considered in the literature

on the periodic solutions of periodic multi species kolmogorov type competitive system with delays and feedback controls abstract in this paper we consider a periodic multi species kolmogorov type competitive system with delays and feedback controls applying schauder s fixed point theorem a criterion on the existence of the positive periodic solution of the system are obtained as application the existence of the positive periodic solution of a kind of nonlinear competitive model with delays and feedback controls is discussed

improved estimation of population variance using information on auxiliary attribute in simple random sampling singh and kumar 2011 19 suggested estimators for calculating population variance using auxiliary attributes this paper proposes a family of estimators based on an adaptation of the estimators presented by kadilar and cingi 2004 5 and singh et al 2007 16 and introduces a new family of estimators using auxiliary attributes the expressions of the mean square errors mses of the adapted and proposed families are derived it is shown that adapted estimators and suggested estimators are more efficient than singh and kumar 2011 19 estimators the theoretical findings are supported by a numerical example

an improved estimator using two auxiliary attributes in the present study we propose two new estimators for population mean using kadilar and cingi 2005 2 and lu et al 2010 9 estimators in the case when the information is available in form of attributes expressions for the mse s of the proposed estimator are derived up to the first degree of approximation the theoretical conditions have also been verified by a numerical example it has been shown that the proposed estimators are more efficient than usual regression estimators

highly accurate surface and volume integration on implicit domains by means of moment fitting summary we introduce a new method for the numerical integration over curved surfaces and volumes defined by a level set function the method is based on the solution of a small linear system based on a simplified variant of the moment fitting equations numerical experiments suggest that the accuracy of the resulting quadrature rules exceeds the accuracy of traditional methods by orders of magnitude using moments up to an order of p the measured experimental orders of convergence exceed hp consequently their construction is very efficient because only coarse computational grids are required the conceptual simplicity allows for the application on very general grid types which is demonstrated by numerical experiments on quadrilateral triangular and hexahedral grids copyright 2013 john wiley sons ltd

a level set approach for computing solutions to incompressible two phase flow a level set approach for computing solutions to incompressible two phase flow is presented the interface between the two fluids is considered to be sharp and is described as the zero level set of a smooth function a new treatment of the level set method allows us to efficiently maintain the level set function as the signed distance from the interface we never have to explicitly reconstruct or find the zero level set consequently we are able to handle arbitrarily complex topologies large density and viscosity ratios and surface tension on relatively coarse grids we use a second order projection method along with a second order upwinded procedure for advecting the momentum and level set equations we consider the motion of air bubbles and water drops we also compute flows with multiple fluids such as air oil and water

short note a proof that a discrete delta function is second order accurate 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

immersed interface methods for stokes flow with elastic boundaries or surface tension a second order accurate interface tracking method for the solution of incompressible stokes flow problems with moving interfaces on a uniform cartesian grid is presented the interface may consist of an elastic boundary immersed in the fluid or an interface between two different fluids the interface is represented by a cubic spline along which the singularly supported elastic or surface tension force can be computed the stokes equations are then discretized using the second order accurate finite difference methods for elliptic equations with singular sources developed in our previous paper siam j numer anal 31 1994 pp 1019 1044 the resulting velocities are interpolated to the interface to determine the motion of the interface an implicit quasi newton method is developed that allows reasonable time steps to be used

diffusion generated motion for grain growth in two and three dimensions an efficient algorithm for accurately simulating curvature flow for large networks of curves in two dimensions and surfaces in three dimensions on uniform grids is proposed this motion arises in the technologically important problem of simulating grain boundary motion in polycrystalline materials in this formulation grain boundaries are zero level sets of signed distance functions curvature motion is achieved by first diffusing locally maintained signed distance functions followed by a reinitialization step a technique is devised to allow a single signed distance function to represent a large subset of spatially separated grains hundreds of thousands of grains can be simulated using a small number of signed distance functions in this work 32 in two dimensions and 64 in three dimensions are more than sufficient using modest computational hardware

discretization of dirac delta functions in level set methods discretization of singular functions is an important component in many problems to which level set methods have been applied we present two methods for constructing consistent approximations to dirac delta measures concentrated on piecewise smooth curves or surfaces both methods are designed to be convenient for level set simulations and are introduced to replace the commonly used but inconsistent regularization technique that is solely based on a regularization parameter proportional to the mesh size the first algorithm is based on a tensor product of regularized one dimensional delta functions it is independent of the irregularity relative to the grid in the second method the regularization is constructed from a one dimensional regularization that is extended to multi dimensions with a variable support depending on the orientation of the singularity relative to the computational grid convergence analysis and numerical results are given

robust second order accurate discretizations of the multi dimensional heaviside and dirac delta functions we present a robust second order accurate method for discretizing the multi dimensional heaviside and the dirac delta functions on irregular domains the method is robust in the following ways 1 integrations of source terms on a co dimension one surface are independent of the underlying grid and therefore stable under perturbations of the domain s boundary 2 the method depends only on the function value of a level function not on its derivatives we present the discretizations in tabulated form to make their implementations straightforward we present numerical results in two and three spatial dimensions to demonstrate the second order accuracy in the l 1 norm in the case of the solution of pdes with singular source terms in the case of evaluating the contribution of singular source terms on interfaces the method is also second order accurate in the l norm

regular article a pde based fast local level set method we develop a fast method to localize the level set method of osher and sethian 1988 j comput phys 79 12 and address two important issues that are intrinsic to the level set method a how to extend a quantity that is given only on the interface to a neighborhood of the interface b how to reset the level set function to be a signed distance function to the interface efficiently without appreciably moving the interface this fast local level set method reduces the computational effort by one order of magnitude works in as much generality as the original one and is conceptually simple and easy to implement our approach differs from previous related works in that we extract all the information needed from the level set function or functions in multiphase flow and do not need to find explicitly the location of the interface in the space domain the complexity of our method to do tasks such as extension and distance reinitialization is o n where n is the number of points in space not o n log n as in works by sethian 1996 proc nat acad sci 93 1591 and helmsen and co workers 1996 spie microlithography ix p 253 this complexity estimation is also valid for quite general geometrically based front motion for our localized method

the fast solution of poisson s and the biharmonic equations on irregular regions we present fast methods for solving laplace s and the biharmonic equations on irregular regions with smooth boundaries the methods used for solving both equations make use of fast poisson solvers on a rectangular region in which the irregular region is embedded they also both use an integral equation formulation of the problem where the integral equations are fredholm integral equations of the second kind the main idea is to use the integral equation formulation to define a discontinuous extension of the solution to the rest of the rectangular region fast solvers are then used to compute the extended solution aside from solving the equations we have also been able to compute derivatives of the solutions with little loss of accuracy when the data was sufficiently smooth

ratio estimators in simple random sampling this study proposes ratio estimators by adapting the estimators type of ray and singh j ind stat assoc 19 1981 147 to traditional and the other ratio type estimators in simple random sampling in literature theoretically mean square error mse equations of all proposed ratio estimators are obtained and compared with each other by these comparisons the conditions which make each proposed estimator more efficient than the others are found the theoretical results are supported by a numerical illustration

discretizing delta functions via finite differences and gradient normalization in j d towers two methods for discretizing a delta function supported on a level set j comput phys 220 2007 915 931 the author presented two closely related finite difference methods referred to here as fdm1 and fdm2 for discretizing a delta function supported on a manifold of codimension one defined by the zero level set of a smooth mapping u r n r these methods were shown to be consistent meaning that they converge to the true solution as the mesh size h 0 in the codimension one setting in this paper we concentrate on n 3 but generalize our methods to codimensions other than one now the level set function is generally a vector valued mapping u r n r m 1 m n 3 seemingly reasonable algorithms based on simple products of approximate delta functions are not generally consistent when applied to these problems motivated by this we instead use the wedge product formalism to generalize our fdm algorithms and this approach results in accurate often consistent approximations with the goal of ensuring consistency in general we propose a new gradient normalization process that is applied before our fdm algorithms these combined algorithms seem to be consistent in all reasonable situations with numerical experiments indicating o h 2 convergence for our new gradient normalized fdm2 algorithm in the full codimension setting m n our gradient normalization processing also improves accuracy when using more standard approximate delta functions this combination also yields approximations that appear to be consistent

the numerical approximation of a delta function with application to level set methods it is shown that a discrete delta function can be constructed using a technique developed by anita mayo the fast solution of poisson s and the biharmonic equations on irregular regions siam j sci comput 21 1984 285 299 for the numerical solution of elliptic equations with discontinuous source terms this delta function is concentrated on the zero level set of a continuous function in two space dimensions this corresponds to a line and a surface in three space dimensions delta functions that are first and second order accurate are formulated in both two and three dimensions in terms of a level set function the numerical implementation of these delta functions achieves the expected order of accuracy

a boundary condition capturing method for poisson s equation on irregular domains interfaces have a variety of boundary conditions or jump conditions that need to be enforced the ghost fluid method gfm was developed to capture the boundary conditions at a contact discontinuity in the inviscid euler equations and has been extended to treat more general discontinuities such as shocks detonations and deflagrations and compressible viscous flows in this paper a similar boundary condition capturing approach is used to develop a new numerical method for the variable coefficient poisson equation in the presence of interfaces where both the variable coefficients and the solution itself may be discontinuous this new method is robust and easy to implement even in three spatial dimensions furthermore the coefficient matrix of the associated linear system is the standard symmetric matrix for the variable coefficient poisson equation in the absence of interfaces allowing for straightforward application of standard black box solvers

on the accuracy of the complex step finite difference method abstract the complex step finite difference method csfdm is a very simple methodology that can be implemented in well known numerical techniques helping to improve for instance in the wave propagation problem time and or space derivative based wavefields we clarify differences between the csfdm and previous implementations of the complex step cs derivative approximation in well known numerical techniques we study dispersion properties for the one way and two way wave equations using the finite difference method fdm the pseudospectral method psm the finite element method fem and the csfdm under the influence of a plane wave and ricker source time functions we show the gain in numerical accuracy offered by the methodology of the csfdm over the fdm psm and fem we finally discuss consequences of the csfdm in future scenarios and propose directions of study in this area

analysis of a one dimensional model for the immersed boundary method numerical methods are studied for the one dimensional heat equation with a singular forcing term u t u xx c t delta x alpha t the delta function delta x is replaced by a discrete approximation d h x and the resulting equation is solved by a crank nicolson method on a uniform grid the accuracy of this method is analyzed for various choices of d h the case where c t is specified and also the case where c is determined implicitly by a constraint on the solution at the point a are studied these problems serve as a model for the immersed boundary method of peskin for incompressible flow problems in irregular regions some insight is gained into the accuracy that can be achieved and the importance of choosing appropriate discrete delta functions

level set minimization of potential controlled hadwiger valuations for molecular solvation a level set method is developed for the numerical minimization of a class of hadwiger valuations with a potential on a set of three dimensional bodies such valuations are linear combinations of the volume surface area and surface integral of mean curvature the potential increases rapidly as the body shrinks beyond a critical size the combination of the hadwiger valuation and the potential is the mean field free energy functional of the solvation of non polar molecules in the recently developed variational implicit solvent model this functional of surfaces is minimized by the level set evolution in the steepest decent of the free energy the normal velocity of this surface evolution consists of both the mean and gaussian curvatures and a lower order forcing term arising from the potential the forward euler method is used to discretize the time derivative with a dynamic time stepping that satisfies a cfl condition the normal velocity is decomposed into two parts the first part consists of both the mean and gaussian curvature terms it is of parabolic type with parameter correction and is discretized by central differencing the second part has all the lower order terms it is of hyperbolic type and is discretized by an upwinding scheme new techniques of local level set method and numerical integration are developed numerical tests demonstrate a second order convergence of the method examples of application to the modeling of molecular solvation are presented

an implicit interface boundary integral method for poisson s equation on arbitrary domains abstract we propose a simple formulation for constructing boundary integral methods to solve poisson s equation on domains with smooth boundaries defined through their signed distance function our formulation is based on averaging a family of parameterizations of an integral equation defined on the boundary of the domain where the integrations are carried out in the level set framework using an appropriate jacobian by the coarea formula the algorithm operates in the euclidean space and does not require any explicit parameterization of the boundaries we present numerical results in two and three dimensions

a front tracking method for viscous incompressible multi fluid flows abstract a method to simulate unsteady multi fluid flows in which a sharp interface or a front separates incompressible fluids of different density and viscosity is described the flow field is discretized by a conservative finite difference approximation on a stationary grid and the interface is explicitly represented by a separate unstructured grid that moves through the stationary grid since the interface deforms continuously it is necessary to restructure its grid as the calculations proceed in addition to keeping the density and viscosity stratification sharp the tracked interface provides a natural way to include surface tension effects both two and three dimensional full numerical simulations of bubble motion are presented

a front tracking method for the computations of multiphase flow direct numerical simulations of multiphase flows using a front tracking method are presented the method is based on writing one set of governing equations for the whole computational domain and treating the different phases as one fluid with variable material properties interfacial terms are accounted for by adding the appropriate sources as functions at the boundary separating the phases the unsteady navier stokes equations are solved by a conventional finite volume method on a fixed structured grid and the interface or front is tracked explicitly by connected marker points interfacial source terms such as surface tension are computed on the front and transferred to the fixed grid advection of fluid properties such as density is done by following the motion of the front the method has been implemented for fully three dimensional flows as well as for two dimensional and axisymmetric ones first the method is described for the flow of two or more isothermal phases the representation of the moving interface and its dynamic restructuring as well as the transfer of information between the moving front and the fixed grid are discussed applications and extensions of the method to homogeneous bubbly flows atomization flows with variable surface tension solidification and boiling are then presented

a high order discontinuous galerkin method for compressible flows with immersed boundaries summary we present a higher order discretization scheme for the compressible euler and navier stokes equations with immersed boundaries our approach makes use of a discontinuous galerkin dg discretization in a domain that is implicitly defined by means of a level set function the zero iso contour of this level set function is considered as an additional domain boundary where we weakly enforce boundary conditions in the same manner as in boundary fitted cells in order to retain the full order of convergence of the scheme it is crucial to perform volume and surface integrals in boundary cells with high accuracy this is achieved using a linear moment fitting strategy moreover we apply a non intrusive cell agglomeration technique that averts problems with very small and ill shaped cuts the robustness accuracy and convergence properties of the scheme are assessed in several two dimensional test cases for the steady compressible euler and navier stokes equations approximation orders range from zero to four even though the approach directly generalizes to even higher orders in all test cases with a sufficiently smooth solution the experimental order of convergence matches the expected rate for dg schemes this article is protected by copyright all rights reserved

the rapid evaluation of volume integrals of potential theory on general regions abstract we present a new method for the rapid high order accurate evaluation of certain volume integrals in potential theory on general irregular regions the kernels of the integrals are either a fundamental solution or a linear combination of the derivatives of a fundamental solution of a second order linear elliptic differential equation instead of using a standard quadrature formula or the exact evaluation of any integral the methods rely on rapid methods of solving the differential equation which the kernel is the solution of therefore the number of operations needed to evaluate the volume integral is essentially equal to the number of operations needed to solve the differential equation on a rectangular region with a regular grid and the method requires no evaluation of the kernel