the attractor of an equation of tricomi s type consider the attractor a of an equation of pendulum type with friction driven by a constant torque the results of m levi and also obtained independently by q min s xian and z jinyan show that if the friction coefficient is larger than a certain bound then a is homeomorphic to the circle we shall study the bifurcation diagram of a particular class of equations of pendulum type and show that the bounds obtained before are optimal

vibration analysis by discrete singular convolution this paper explores the utility of a discrete singular convolution algorithm for vibration analysis a number of different realizations of singular convolution kernels are selected to illustrate the present algorithm vibration analysis of strings rods beams diatomic molecules membranes waveguides and thin plates are utilized to test numerical accuracy and speed of convergence of the present approach numerical experiments indicate that the discrete singular convolution is a simple and reliable algorithm for vibration analysis

a block pseudospectral method for maxwell s equations a block pseudospectral bps method is proposed as a new way to couple pseudospectral discretizations across interfaces in computations for a linear hyperbolic system the coupling is achieved via discretized derivative matching conditions obtained from the system compared to the standard technique of imposing compatibility conditions based on characteristics of the system the bps method offers better stability and accuracy especially in the case where equation coefficients are discontinuous computational examples for maxwell s equations in nonhomogeneous media demonstrate that bps retains high accuracy over times that are orders of magnitude larger than those for not only low order methods such as yee s but also high order methods such as characteristic based spectral elements

dsc time domain solution of maxwell s equations a new computational algorithm the discrete singular convolution dsc is introduced for solving scattering and guided wave problems described by time domain maxwell s equations the dsc algorithm is utilized for the spatial discretization and the fourth order runge kutta scheme is used for the time advancing staggered meshes are used for electromagnetic fields four standard test problems a hollow air filled waveguide a dielectric slab loaded rectangular waveguide a shield microstrip line and a dielectric square are employed to illustrate the usefulness to test the accuracy and to explore the limitation of the dsc algorithm results are compared with those of finite difference scaling function multi resolution time domain and finite element based high frequency structure simulator numerical experiments indicate that the present algorithm is a promising approach for achieving high accuracy in electromagnetic wave computations

high order fdtd methods via derivative matching for maxwell s equations with material interfaces this paper introduces a series of novel hierarchical implicit derivative matching methods to restore the accuracy of high order finite difference time domain fdtd schemes of computational electromagnetics cem with material interfaces in one 1d and two spatial dimensions 2d by making use of fictitious points systematic approaches are proposed to locally enforce the physical jump conditions at material interfaces in a preprocessing stage to arbitrarily high orders of accuracy in principle while often limited by numerical instability orders up to 16 and 12 are achieved respectively in 1d and 2d detailed stability analyses are presented for the present approach to examine the upper limit in constructing embedded fdtd methods as natural generalizations of the high order fdtd schemes the proposed derivative matching methods automatically reduce to the standard fdtd schemes when the material interfaces are absent an interesting feature of the present approach is that it encompasses a variety of schemes of different orders in a single code another feature of the present approach is that it can be robustly implemented with other high accuracy time domain approaches such as the multiresolution time domain method and the local spectral time domain method to cope with material interfaces numerical experiments on both 1d and 2d problems are carried out to test the convergence examine the stability access the efficiency and explore the limitation of the proposed methods it is found that operating at their best capacity the proposed high order schemes could be over 2000 times more efficient than their fourth order versions in 2d in conclusion the present work indicates that the proposed hierarchical derivative matching methods might lead to practical high order schemes for numerical solution of time domain maxwell s equations with material interfaces

a comparison of acceleration procedures for the two dimensional periodic green s function alternate acceleration procedures are evaluated for the two dimensional free space periodic green s function two of these techniques yield exponential convergence rates and are preferable if high accuracies are required numerical aspects of these procedures are described

efficient modeling of three dimensional reverberation chambers using hybrid discrete singular convolution method of moments efficient modeling of a three dimensional reverberation chamber rc is achieved by combining the discrete singular convolution dsc method and the method of moments mom an rc usually consists of a metallic cavity and one or two conducting stirrers whose size is normally small compared to the chamber size the large cavity is efficiently modeled by the dsc method and the stirrer is simulated by the flexible mom exploiting the property of rwg basis solutions from the two methods are combined together using the equivalence principle the validity and advantages of the proposed hybrid technique are shown through comparisons with the commercial software feko employing the high efficiency of the dsc method the hybrid technique can analyze one stirrer position of a medium sized rc in a few hundred seconds on a single personal computer for which feko needs thousands of seconds cpu time the memory requirement of the proposed method is also less than that of feko furthermore our hybrid method provides efficient calculation of electric field strength at a large number of field points which is of great interest in rc analysis simulations show that our method only takes 1 7 seconds to compute electric field strength at 4026 field points

wavelets generated by using discrete singular convolution kernels this paper explores the connection between wavelet methods and an efficient computational algorithm the discrete singular convolution dsc many new dsc kernels are constructed and they are identified as wavelet scaling functions two approaches are proposed to generate wavelets from dsc kernels two well known examples the canny filter and the mexican hat wavelet are found to be special cases of the present dsc kernel generated wavelets a family of wavelet generators proposed in this paper are found to form an infinite dimensional lie group which has an invariant subgroup of translation and dilation if dsc kernels form an orthogonal system they are found to span a wavelet subspace in a multiresolution analysis

plate vibration under irregular internal supports this paper studies the problem of plate vibration under complex and irregular internal support conditions such a problem has its widely spread industrial applications and has not been addressed in the literature yet partially due to the numerical difficulties a novel computational method discrete singular convolution dsc is introduced for solving this problem the dsc algorithm exhibits controllable accuracy for approximations and shows excellent flexibility in handling complex geometries boundary conditions and internal support conditions convergence and comparison studies are carried out to check the validity and accuracy of the dsc method case studies are considered to the combination of a few different boundary conditions and irregular internal supports the latter are generated by using an image processing algorithm completely independent verifications are conducted by using the established pb 2 ritz method which is available for two relatively simpler support patterns the morphology of the first few eigenmodes is found to be localized to largest support free spatial regions

discrete singular convolution finite subdomain method for the solution of incompressible viscous flows this paper proposes a discrete singular convolution finite subdomain method dsc fsm for the analysis of incompressible viscous flows in multiply connected complex geometries the dsc algorithm has its foundation in the theory of distributions a block structured grid of fictitious overlapping interfaces is designed to decompose a complex computational geometry into a finite number of subdomains in each subdomain the governing navier stokes equations are discretized by using the dsc algorithm in space and a third order runge kutta scheme in time information exchange between fictitious overlapping zones is realized by using the dsc interpolating algorithm the taylor problem with decaying vortices could be solved to machine precision with an excellent comparison against the exact solution the reliability of the proposed method is tested by simulating the flow in a lid driven cavity the utility of the dsc fsm approach is further illustrated by two other benchmark problems viz the flow over a backward facing step and the laminar flow past a square prism the present results compare well with the numerical and experimental data available in the literature

a unified approach for the solution of the fokker planck equation this paper explores the use of a discrete singular convolution algorithm as a unified approach for numerical integration of the fokker planck equation the unified features of the discrete singular convolution algorithm are discussed it is demonstrated that different implementations of the present algorithm such as global local galerkin collocation and finite difference can be deduced from a single starting point three benchmark stochastic systems the repulsive wong process the black scholes equation and a genuine nonlinear model are employed to illustrate the robustness and to test the accuracy of the present approach for the solution of the fokker planck equation via a time dependent method an additional example the incompressible euler equation is used to further validate the present approach for more difficult problems numerical results indicate that the present unified approach is robust and accurate for solving the fokker planck equation

numerical solution of the helmholtz equation with high wavenumbers this paper investigates the pollution effect and explores the feasibility of a local spectral method the discrete singular convolution dsc algorithm for solving the helmholtz equation with high wavenumbers fourier analysis is employed to study the dispersive error of the dsc algorithm our analysis of dispersive errors indicates that the dsc algorithm yields a dispersion vanishing scheme the dispersion analysis is further confirmed by the numerical results for one and higher dimensional helmholtz equations the dsc algorithm is shown to be an essentially pollution free scheme furthermore for large scale computation the grid density of the dsc algorithm can be close to the optimal two grid points per wavelength the present study reveals that the dsc algorithm is accurate and efficient for solving the helmholtz equation with high wavenumbers copyright 2003 john wiley sons ltd

using a curvilinear grid to construct symmetry preserving discretizations for lagrangian gas dynamics the goal of this paper is to construct discretizations for the equations of lagrangian gas dynamics that preserve plane cylindrical and spherical symmetry in the solution of the original differential equations the new method uses a curvilinear grid that is reconstructed from a given logically rectangular distribution of nodes the sides of the cells of the reconstructed grid can be segments of straight lines or arcs of local circles our procedure is exact for straight lines and circles that is it reproduces rectangular and polar grids exactly we use the method of support operators to construct a conservative finite difference method that we demonstrate will preserve spatial symmetries for certain choices of the initial grid we also introduce a curvilinear version of artificial edge viscosity that also preserves symmetry we present numerical examples to demonstrate our theoretical considerations and the robustness of the new method

a spectral integral method sim for layered media a spectral integral method is presented for electromagnetic scattering from dielectric and perfectly electric conducting pec objects with a closed boundary embedded in a layered medium two dimensional layered medium green s functions are computed adaptively by using gaussian quadratures the singular terms in the green s functions and the non smooth terms in their derivatives are handled appropriately to achieve exponential convergence numerical results compared with the ones obtained by using other methods demonstrate the spectral accuracy and high efficiency of the proposed method they also confirm that the spectral integral method sim is applicable to concave objects

implementation and experiments of a hybrid algorithm of the mlfma enhanced fe bi method for open region inhomogeneous electromagnetic problems although the computational complexity of the finite element boundary integral fe bi method is significantly reduced by the multilevel fast multipole algorithm mlfma this mlfma enhanced fe bi solution experiences a very slow convergence for some complex inhomogeneous problems a hybrid algorithm combining direct methods with iterative methods is designed to speed up the rate of convergence of this mlfma enhanced fe bi solution this hybrid algorithm is efficiently implemented with the aid of a newly developed package superlu of the lu decomposition solver numerical experiments are performed for scattering by a coated northrop wing to demonstrate the efficiency of this hybrid algorithm more importantly the thorough investigation of the numerical experiments clearly shows the better accuracy stability and robustness of this hybrid algorithm over the conventional algorithms

seamless integration of global dirichlet to neumann boundary condition and spectral elements for transformation electromagnetics abstract in this paper we present an efficient spectral element method sem for solving general two dimensional helmholtz equations in anisotropic media with particular applications in accurate simulation of polygonal invisibility cloaks concentrators and circular rotators arisen from the field of transformation electromagnetics te in practice we adopt a transparent boundary condition tbc characterised by the dirichlet to neumann dtn map to reduce wave propagation in an unbounded domain to a bounded domain we then introduce a semi analytic technique to integrate the global tbc with local curvilinear elements seamlessly which is accomplished by using a novel elemental mapping and analytic formulas for evaluating global fourier coefficients on spectral element grids exactly from the perspective of te an invisibility cloak is devised by a singular coordinate transformation of maxwell s equations that leads to anisotropic materials coating the cloaked region to render any object inside invisible to observers outside an important issue resides in the imposition of appropriate conditions at the outer boundary of the cloaked region i e cloaking boundary conditions cbcs in order to achieve perfect invisibility following the spirit of yang and wang 2015 we propose new cbcs for polygonal invisibility cloaks from the essential pole conditions related to singular transformations this allows for the decoupling of the governing equations of inside and outside the cloaked regions with this efficient spectral element solver at our disposal we can study the interesting phenomena when some defects and lossy or dispersive media are placed in the cloaking layer of an ideal polygonal cloak

a 3 d enlarged cell technique ect for the conformal fdtd method in this paper we present an enlarged cell technique ect to avoid the time step reduction encountered in the conformal finite difference time domain cfdtd method due to small irregular cells truncated by metallic boundaries we focus our efforts on the discussion of the accuracy and stability of the ect and its comparison with other conformal methods especially the one called the uniformly stable conformal usc method we also provide a simplified ect which is much easier to implement in the ect a stability criterion is first constructed to identify instable irregular cells i e those having so small an area to cause instability those instable cells are then enlarged into their adjacent cells to obtain a large stable area careful treatment is performed on the communication between the intruding and intruded cells in terms of electromotive force by keeping the total electromotive force conservative this technique is verified by several 3 d numerical experiments results show that the ect is second order accurate and numerically stable at the regular courant time step limit

nodal high order methods on unstructured grids we present a convergent high order accurate scheme for the solution of linear conservation laws in geometrically complex domains as our main example we include a detailed development and analysis of a scheme for the time domain solution of maxwell s equations in a three dimensional domain the fully unstructured spatial discretization is made possible by the use of a high order nodal basis employing multivariate lagrange polynomials defined on the triangles and tetrahedra while the equations themselves are satisfied in a discontinuous galerkin form with the boundary conditions being enforced weakly through a penalty term accuracy stability and convergence of the semidiscrete approximation to maxwell s equations is established rigorously and bounds on the growth of the global divergence error are provided concerns related to efficient implementations are discussed in detail this sets the stage for the presentation of examples verifying the theoretical results and illustrating the versatility flexibility and robustness when solving two and three dimensional benchmark problems in computational electromagnetics pure scattering as well as penetration is discussed and high parallel performance of the scheme is demonstrated

lower bounds for quartic anharmonic and double well potentials rigorous and remarkably accurate lower bounds to the lower eigenvalue spectrum of the schrodinger equation with quartic anharmonic and symmetric double well potentials of the form v a b ax2 2 bx4 b 0 are presented this procedure exploits some exactly soluble model potentials and appears to be of quite general utility

discrete singular convolution and its application to the analysis of plates with internal supports part 1 theory and algorithm this paper presents a novel computational approach the discrete singular convolution dsc algorithm for analysing plate structures the basic philosophy behind the dsc algorithm for the approximation of functions and their derivatives is studied approximations to the delta distribution are constructed as either bandlimited reproducing kernels or approximate reproducing kernels unified features of the dsc algorithm for solving differential equations are explored it is demonstrated that different methods of implementation for the present algorithm such as global local galerkin collocation and finite difference can be deduced from a single starting point the use of the algorithm for the vibration analysis of plates with internal supports is discussed detailed formulation is given to the treatment of different plate boundary conditions including simply supported elastically supported and clamped edges this work paves the way for applying the dsc approach in the following paper to plates with complex support conditions which have not been fully addressed in the literature yet

solving three dimensional hexahedral finite element groundwater models by preconditioned conjugate gradient methods practical aspects of three dimensional modeling of groundwater flow in heterogeneous aquifer systems are investigated using a finite element approach particular attention is given to the properties of the conductance matrix and the efficiency of the conjugate gradient method with different preconditioners diagonal scaling incomplete cholesky decomposition incomplete factorization and modified incomplete factorization it is shown that for hexahedral trilinear finite elements the resulting matrix is except for cube shaped elements never diagonally dominant which restricts the existence of several preconditioners numerical comparison of several test problems including hypothetical and field applications with different degrees of heterogeneity show that the incomplete cholesky and the incomplete factorization preconditioners if they exist are more efficient than diagonal scaling with respect to both rate of convergence and overall computing time but diagonal scaling can be considered superior because it is always possible an m matrix transformation is proposed which guarantees the existence of all preconditioners numerical comparison of the test problems shows that this technique is very effective from the resulting preconditioners the incomplete cholesky and the incomplete factorization are shown to be the most efficient but the latter is superior from the point of view of computer storage and is recommended for all practical applications

global behavior in the dynamical equation of j j type on considere l equation de la forme 1 ecos sin asinwt on etudie le cas e 0 2 on montre que chaque trajectoire est asymptotiquement periodique ou quasi periodique

on the formulation of hybrid finite element and boundary integral methods for 3 d scattering this paper studies in detail a variety of formulations for the hybrid finite element and boundary integral fe bi method for three dimensional 3 d electromagnetic scattering by inhomogeneous objects it is shown that the efficiency and accuracy of the fe bi method depends highly on the formulation and discretization of the boundary integral equation bie used a simple analysis of the matrix condition number identifies the efficiency of the different fe bi formulations and an analysis of weighting functions shows that the traditional fe bi formulations cannot produce accurate solutions a new formulation is then proposed and numerical results show that the resulting solution has a good efficiency and accuracy and is completely immune to the problem of interior resonance finally the multilevel fast multipole algorithm mlfma is employed to significantly reduce the memory requirement and computational complexity of the proposed fe bi method

crank nicolson quasi wavelets method for solving fourth order partial integro differential equation with a weakly singular kernel in this paper we study a novel numerical scheme for the fourth order partial integro differential equation with a weakly singular kernel in the time direction a crank nicolson time stepping is used to approximate the differential term and the product trapezoidal method is employed to treat the integral term and the quasi wavelets numerical method for space discretization our interest in the present paper is a continuation of the investigation in yang et al 33 where we study discretization in time by using the forward euler scheme the comparisons of present results with the previous ones show that the present scheme is more stable and efficient for numerically solving the fourth order partial integro differential equation with a weakly singular kernel we also tested the method proposed on several one and two dimensional problems with very promising results besides in order to demonstrate the power of the quasi wavelets method in comparison with standard discretization methods we also consider the high frequency oscillation problems with the integro differential term

fast simplicial finite element algorithms using bernstein polynomials fast algorithms for applying finite element mass and stiffness operators to the b form of polynomials over d dimensional simplices are derived these rely on special properties of the bernstein basis and lead to stiffness matrix algorithms with the same asymptotic complexity as tensor product techniques in rectangular domains first special structure leading to fast application of mass matrices is developed then by factoring stiffness matrices into products of sparse derivative matrices with mass matrices fast algorithms are also obtained for stiffness matrices

a finite element boundary integral formulation for scattering by three dimensional cavity backed apertures a novel numerical technique is proposed for the electromagnetic characterization of the scattering by a three dimensional cavity backed aperture in an infinite ground plane the technique combines the finite element and boundary integral methods to formulate a system of equations for the solution of the aperture fields and those inside the cavity specifically the finite element method is used to formulate the fields in the cavity region and the boundary integral approach is used in conjunction with the equivalence principle to represent the fields above the ground plane unlike traditional approaches the proposed technique does not require a knowledge of the cavity s green s function and is therefore applicable to arbitrary shape depressions and material fillings furthermore the proposed formulation leads to a system having a partly full and partly sparse as well as symmetric and banded matrix which can be solved efficiently using special algorithms

note on nonsymmetric finite differences for maxwell s equations numerical solution methods for discretizing second derivatives are available for exam ple centered nite differences in space and nystr om or st ormer methods in time thesemethods may have advantages in accuracy and stability over their rst derivative counter parts however the imposition of boundary conditions is more dif cult in the second orderformulation

on the gibbs phenomenon and its resolution the nonuniform convergence of the fourier series for discontinuous functions and in particular the oscillatory behavior of the finite sum was already analyzed by wilbraham in 1848 this was later named the gibbs phenomenon this article is a review of the gibbs phenomenon from a different perspective the gibbs phenomenon as we view it deals with the issue of recovering point values of a function from its expansion coefficients alternatively it can be viewed as the possibility of the recovery of local information from global information the main theme here is not the structure of the gibbs oscillations but the understanding and resolution of the phenomenon in a general setting the purpose of this article is to review the gibbs phenomenon and to show that the knowledge of the expansion coefficients is sufficient for obtaining the point values of a piecewise smooth function with the same order of accuracy as in the smooth case this is done by using the finite expansion series to construct a different rapidly convergent approximation

option valuation by using discrete singular convolution this paper explores the utility of a discrete singular convolution dsc algorithm for solving the black scholes equation both european and american style options which include all nontrivial plain option pricing problems are considered to test the accuracy and to examine the efficiency of the present algorithm adaptive meshes are constructed to enhance the performance of the dsc algorithm all the present results are validated either by the analytical solution or by the standard binomial tree method extensive comparisons are carried out with two standard finite difference schemes and two binomial models of high speed convergence numerical experiments reveal that the present approach is accurate efficient and reliable for financial derivative valuations

matched interface and boundary mib for the implementation of boundary conditions in high order central finite differences high order central finite difference schemes encounter great difficulties in implementing complex boundary conditions this paper introduces the matched interface and boundary mib method as a novel boundary scheme to treat various general boundary conditions in arbitrarily high order central finite difference schemes to attain arbitrarily high order the mib method accurately extends the solution beyond the boundary by repeatedly enforcing only the original set of boundary conditions the proposed approach is extensively validated via boundary value problems initial boundary value problems eigenvalue problems and high order differential equations successful implementations are given to not only dirichlet neumann and robin boundary conditions but also more general ones such as multiple boundary conditions in high order differential equations and time dependent boundary conditions in evolution equations detailed stability analysis of the mib method is carried out the mib method is shown to be able to deliver high order accuracy while maintaining the same or similar stability conditions of the standard high order central difference approximations the application of the proposed mib method to the boundary treatment of other non standard high order methods is also considered

one way large range step methods for helmholtz waveguides a useful approach for long range computation of the helmholtz equation in a waveguide is to re formulate it as the operator differential riccati equation for the dirichlet to neumann map for waveguides with slow range dependence the piecewise range independent approximation is used to derive a second order range stepping method for this one way re formulation the range marching formulas are exact for each range independent piece and a large range step is possible if the range dependence is gradual based on a fourth order conservative exponential method for linear evolution equations a fourth order method that admits even larger range steps is developed for the one way re formulation numerical examples are used to demonstrate the improved accuracy of the fourth order method

a fast fourier transform accelerated ewald summation technique for the vector electromagnetic rectangular cavity green s function a fast fourier transform accelerated ewald method for the computation of the vector electromagnetic rectangular cavity green s function in terms of the electric field due to electric currents is presented and used in a boundary integral formulation the ewald summation technique suffers from the high frequency breakdown when it is applied to green s functions of wave problems in the case of the rectangular cavity green s function the number of necessary terms in the spectral series grows therefore cubically with frequency for a given accuracy to counteract the high frequency breakdown the evaluation of the spectral series is accelerated with an inverse fast fourier transform in this work at high frequencies a speed up of up to four orders of magnitude is achieved as an application example a reverberation chamber containing a metallic enclosure and a mode stirrer is modeled

stability and numerical dispersion analysis of a fourth order accurate fdtd method in order to obtain high order accuracy a fourth order accurate finite difference time domain fdtd method is presented by kyu pyung hwang unlike conventional fdtd methods a staggered backward differentiation scheme instead of the leapfrog scheme is used to approximate the temporal partial differential operator however the high order of its characteristic equation derived by the von neumann method makes the analysis of its numerical dispersion and stability very difficult in automatic control theory there are two effective methods for the stability analysis i e the routh hurwitz test and the jury test the combination of the von neumann method with each of the two can strictly derive the stability condition which only makes use of the coefficients of its characteristic equation without numerically solving it the method of analysis in this paper is also applicable in the stability and numerical analysis of other high order accurate fdtd methods

memory efficient modeling of reverberation chambers using hybrid recursive update discrete singular convolution method of moments a hybrid method is proposed for memory efficient analysis of reverberation chambers rcs in the hybrid method the cavity is modeled by the recursive update discrete singular convolution rudsc method and antennas and stirrers inside the cavity are simulated using the flexible method of moments mom in order to solve dsc and mom unknowns separately a layer based elimination algorithm is utilized to eliminate the dsc unknowns the mom unknowns are then solved by a direct solver once the solution for the mom model is obtained the original rc is equivalent to a cavity excited by known current sources the equivalent problem is finally solved using the rudsc method taking advantage of the layer based elimination algorithm and the recursive update technique the memory requirement of the new hybrid method is much smaller than that of using a direct solver numerical simulations are presented to show the efficacy of the proposed method it is shown that the the proposed method substantially reduces the memory cost of rc modeling which extends rc analysis to higher frequencies

a hybrid ewald spectral cavity green s function boundary element method with spectral domain acceleration for modeling of over moded cavities a hybrid ewald spectral cavity green s function cgf boundary element method bem with spectral domain acceleration is presented in contrast to all previous work on the cgf bem the presented method relieves the two major drawbacks of the cgf approach 1 the high frequency complexity is decreased using a hybrid ewald spectral representation of the cgf and 2 the costly evaluation of the interaction integrals is accelerated with a fast group spectral domain approach results demonstrate that cavities containing small medium sized and large objects are efficiently modeled over a wide frequency range

a new boundary closure scheme for the multiresolution time domain mrtd method this paper introduces a novel boundary closure treatment for the wavelet based multiresolution time domain mrtd solution of maxwell s equations accommodating non trivial boundary conditions such as the robin condition or time dependent condition has been a challenging issue in the mrtd analysis of wave scattering radiation and propagation a matched interface and boundary mib method is introduced to overcome this difficulty several numerical benchmark tests are carried out to validate the mib boundary scheme the proposed boundary treatment can also be applied to other high order finite difference time domain fdtd approaches such as the dispersion relation preserving drp method the mib boundary scheme greatly enhances the feasibility for applying the mrtd methods to more complicated electromagnetic structures

finite element algorithms for simulating three dimensional groundwater flow and solute transport in multilayer systems a finite element formulation is developed for the simulation of groundwater flow and solute transport in multilayer systems of several aquifers and aquitards this formulation is general flexibile and capable of taking full advantage of the nature of flow in such multilayer systems a fully three dimensional spatial representation can be performed for certain aquifers or for the entire flow system if needed those parts of the system that require three dimensional spatial discretization are handled effectively by combining two dimensional basis functions in the x y space and one dimensional basis functions in the z space furthermore the formulation has a desirable option to perform one dimensional representation of flow and transport in aquitards and areal representation of flow and transport in certain aquifers that do not require the fully three dimensional discretization when the one dimensional representation of an aquitard is used coupling of adjacent aquifer and aquitard layers is handled using a convolution integral approach a general solution strategy is also developed to allow systematic time stepping and cost effective matrix handling schemes for those parts of the system that require three dimensional discretization an algorithm referred to as the alternate sublayer and line sweep procedure is presented for decomposing three dimensional matrix equations this algorithm can accomodate several thousand nodal unknowns without requiring excessive core storage and cpu time four examples of transient flow and transport problems are provided these examples show verification of numerical results and demonstrate that the present finite element models are far more cost effective than earlier three dimensional models that are based on the conventional galerkin approach and direct matrix solution algorithms

comparison of taylor finite difference and window finite difference and their application in fdtd the finite difference time domain fdtd method is an important tool in numerical electromagnetic simulation there are many ways to construct a finite difference approximation such as the taylor series expansion theorem the filtering theory etc this paper aims to provide the comparison between the taylor finite difference tfd scheme based on the taylor series expansion theorem and the window finite difference wfd scheme based on the filtering theory their properties have been examined in detail separately in addition the formula of the generalized finite difference gfd scheme is presented which can include both the tfd scheme and the wfd scheme furthermore their application in the numerical solution of maxwell s equations is presented the formulas for the stability criterion and the numerical dispersion relation are derived and analyzed in order to evaluate their performance more accurately a new definition of error is presented upon it the effect of several factors including the grid resolution the courant number and the aspect ratio of the cell on the performance of the numerical dispersion is examined

a hybrid sim sem method for 3 d electromagnetic scattering problems a new method combining the spectral integral method and spectral element method sim sem is proposed to simulate 3 d electromagnetic scattering from inhomogeneous objects in this hybrid technique a special case of the finite element boundary integral fem bi combination the sem with the mixed order curl conforming vector gauss lobatto legendre gll basis functions are used to represent the interior electric field with high accuracy while the sim on a cuboid surface is used as an exact radiation boundary condition the toeplitz property of the sim matrix is utilized to reduce the memory and cpu time costs in an iterative solver by using the fast fourier transform algorithm unlike the traditional fem bi combination where the bi portion usually dominates the computational complexity the computational costs are much lower in the sim sem method numerical results verify the accuracy and capability of this method confirming that the sim sem method is a good alternative for solving scattering problems from inhomogeneous objects

discrete singular convolution for the sine gordon equation abstract this paper explores the utility of a discrete singular convolution dsc algorithm for the integration of the sine gordon equation the initial values are chosen close to a homoclinic manifold for which previous methods have encountered significant numerical difficulties such as numerically induced spatial and temporal chaos a number of new initial values are considered including a case where the initial value is exactly on the homoclinic orbit the present algorithm performs extremely well in terms of accuracy efficiency simplicity stability and reliability

dispersion in heterogeneous porous media 2 predictions for stratified and two dimensional spatially periodic systems the closure scheme which is presented in part 1 plumb and whitaker this issue is solved for a stratified system and for a two dimensional spatially periodic system by this we mean that the and regions which make up the heterogeneous media are distributed in a spatially periodic manner the solution to the closure problem yields the information necessary to determine the large scale dispersion tensor and the other coefficients that appear in the large scale dispersion equation the theoretical results for the stratified system exhibit an increase in the longitudinal dispersion coefficient of several orders of magnitude over that which would be predicted using established correlations based on laboratory experiments with seemingly homogeneous porous media for the cases studied the coefficient of the mixed space time derivative appears to be significant and this coefficient increases for those conditions which yield increasing longitudinal dispersion coefficients the results for the two dimensional spatially periodic system are in qualitative agreement with experimental observations and we found that both the size of the heterogeneity and the difference in the hydraulic conductivity between the two regions had an important influence on the large scale dispersion tensor as in the case of a stratified medium the predicted values of the coefficient of the mixed space time derivative the skewness vector indicate that this term could be a significant factor in the prediction of mass transport in heterogeneous porous media

a monotone streamline upwind finite element method for convection dominated flows a new streamline upwind approximation for treating advection dominatcd transport is presented the proposed method is quite simple to implement with existing finite element methods for this work the method is used in a two dimensional finite element computer program the new method is applied to several test cases that have been presented in previous articles to evaluate the stability and accuracy of proposed methods for treating advection dominatcd flows the results demonstrate that the new method is able to predict the test cases with a significant increase in accuracy compared to previous techniques in addition these results demonstrate conclusively that the method does not exhibit the non physical spatial oscillations that have plagued previous methods

lagrange distributed approximating functionals lagrange distributed approximating functionals ldafs are proposed as the basis for a new collocation type method for accurately approximating functions and their derivatives both on and off discrete grids example applications are presented to illustrate the use of ldafs for solving the schr umlt o dinger equation and fokker planck equation ldafs are constructed by combining the daf concept with the lagrange interpolation scheme copyright ital 1997 ital the american physical society

the effect of inversely unstable solutions on the attractor of the forced pendulum equation with friction abstract consider the attractor a of a periodically forced equation of pendulum type with linear friction in the cylinder levi and independently min xian and jinyan show that if the friction coefficient is larger than a certain bound then a is homeomorphic to the circle we shall give a topological version of the definition of inversely unstable solution of n levinson and show that the appearance of such solutions imply that a is not homeomorphic to the circle as an application we shall show that the bounds on the friction coefficient obtained before are optimal

numerical solution of the time domain maxwell equations using high accuracy finite difference methods high accuracy finite difference schemes are used to solve the two dimensional time domain maxwell equations for electromagnetic wave propagation and scattering the high accuracy schemes consist of a seven point spatial operator coupled with a six stage runge kutta time marching method two methods are studied one of which produces the maximum order of accuracy and one of which is optimized for propagation distances smaller than roughly 300 wavelengths boundary conditions are presented which preserve the accuracy of these schemes when modeling interfaces between different materials numerical experiments are performed which demonstrate the utility of the high accuracy schemes in modeling waves incident on dielectric and perfect conducting scatterers using cartesian and curvilinear grids the high accuracy schemes are shown to be substantially more efficient in both computing time and memory than a second order and a fourth order method the optimized scheme can lead to a reduction in error relative to the maximum order scheme with no additional expense especially when the number of wavelengths of travel is large

a curvilinear finite element model for simulating two well tracer tests and transport in stratified aquifers the problem of solute transport in steady nonuniform flow created by a recharging and discharging well pair is investigated numerical difficulties encountered with the standard galerkin formulation in cartesian coordinates are illustrated an improved finite element solution strategy is presented to overcome the numerical problem the strategy is based on a formulation performed in three dimensional curvilinear coordinates this formulation is developed especially for the simulation of two well injection withdrawal tracer tests conducted in homogeneous or stratified aquifers the formulation can also be applied to other types of problems involving transport in homogeneous and stratified aquifers a three dimensional finite element approximation is derived by two successive applications of upstream weighted residual and subdomain collocation procedures first the weighted residual procedure is formulated in the horizontal flow plane using curvilinear coordinates along and normal to streamlines upstream weighting is included but automatically controlled and limited to one dimension along the main flow direction second the subdomain collocation procedure is formulated in the vertical dimension spatial discretization of the three dimensional flow region is performed using a curvilinear finite element mesh that is point centered in the horizontal plane and block centered in the vertical direction general algorithms are developed for mesh generation time stepping and matrix reduction the solution algorithm is designed to accommodate several thousand nodal unknowns with minimal core storage and cpu time requirement two simulation examples are presented to verify and validate the numerical model in two dimensions using an analytical solution and experimental laboratory and field data a comparative study shows that the proposed formulation is superior to the standard galerkin formulation in cartesian coordinates both in terms of accuracy and computational efficiency the field simulation example indicates the need to conduct a more elaborate three dimensional simulation study allowing for vertical variation in hydraulic conductivity such a study has been completed and is reported in the follow up paper of this series

a new algorithm for solving some mechanical problems this paper explores the utility of a discrete singular convolution algorithm for solving certain mechanical problems benchmark mechanical systems including plate vibrations and incompressible flows are employed to illustrate the robustness and to test accuracy of the present algorithm numerical results indicate that the present approach is very accurate efficient and reliable for solving the aforementioned problems

comment on a combined laplace transform and streamline upwind approach for nonideal transport of solutes in porous media by linlin xu and mark l brusseau

a curved boundary treatment for discontinuous galerkin schemes solving time dependent problems for problems defined in a two dimensional domain with boundary conditions specified on a curve we consider discontinuous galerkin dg schemes with high order polynomial basis functions on a geometry fitting triangular mesh it is well known that directly imposing the given boundary conditions on a piecewise segment approximation boundary h will render any finite element method to be at most second order accurate unless the boundary conditions can be accurately transferred from to h in general curvilinear element method should be used to obtain high order accuracy we discuss a simple boundary treatment which can be implemented as a modified dg scheme defined on triangles adjacent to h even though integration along the curve is still necessary integrals over any curved element are avoided if the domain is convex or if is nonconvex and the true solutions can be smoothly extended to the exterior of the modified dg scheme is high order accurate in these cases numerical tests on first order and second order partial differential equations including hyperbolic systems and the scalar wave equation suggest that it is as accurate as the full curvilinear dg scheme

numerical evaluation of iterative and noniterative methods for the solution of the nonlinear richards equation several noniterative procedures for solving the nonlinear richards equation are introduced and compared to the conventional newton and picard iteration methods noniterative strategies for the numerical solution of transient nonlinear equations arise from explicit or linear time discretizations or they can be obtained by linearizing an implicit differencing scheme we present two first order accurate linearization methods a second order accurate two level implicit factored scheme and a second order accurate three level lees method the accuracy and computational efficiency of these four schemes and of the newton and picard methods are evaluated for a series of test problems simulating one dimensional flow processes in unsaturated porous media the results indicate that first order accurate schemes are inefficient compared to second order accurate methods that second order accurate noniterative schemes can be quite competitive with the iterative newton and picard methods and that the newton scheme is no less efficient than the picard method and for strongly nonlinear problems can outperform the picard scheme the two second order accurate noniterative schemes appear to be attractive alternatives to the iterative methods although there are concerns regarding the stability behavior of the three level scheme which need to be resolved we conclude that of the four noniterative strategies presented the implicit factored scheme is the most promising and we suggest improved formulations of the method

on boundary conditions and point sources in the finite element integration of the transport equation in the finite element integration of the convection dispersion equation over a groundwater domain boundary conditions of the third or cauchy type may or may not be incorporated with the variational statement while the former development formulation a requires the application of green s first identity to both the convective and the dispersive terms the latter formulation b does not although both approaches are equivalent at the theoretical level they behave differently from a computational standpoint the numerical solution proves more accurate and stable if green s lemma is not applied to the convective term since significant errors arising from the representation of the darcy velocity over the elements are avoided point sources or sinks representing injection or pumping wells can be treated as either sources sinks distributed over a small volume or external boundaries with prescribed contaminant flux it is shown that when the well bore is excluded from the flow field formulation a yields the same matrix addition for an injection abstraction well as formulation b does for an abstraction injection well reciprocity of the results the computational behaviors of these alternative modeling procedures are illustrated with a number of significant numerical examples

scattering from a large body with cracks and cavities by the fast and accurate finite element boundary integral method a large body with cracks and cavities is a typical structure widely existing in realistic targets in this paper a newly developed fast and accurate finite element boundary integral fa fe bi method is applied to compute scattering by this kind of scatterer a thorough analysis on this fa fe bi numerical technique is presented clearly demonstrating that this technique has computational complexity o n log n and memory requirement o n n is the total number of surface unknowns an inward looking approach is employed as a preconditioner to speed up the rate of convergence of iterative solvers for this structure under these techniques a powerful code is developed for this kind of scatterer whose accuracy efficiency and capability is well confirmed by various numerical results

development and application of a novel class of hierarchical tangential vector finite elements for electromagnetics tangential vector finite elements tvfes overcome most of the shortcomings of node based finite elements for electromagnetic simulations for a triangular element this paper proposes a class of hierarchical tvpes that differ from traditional tvfes the hierarchical nature of the proposed tvfes makes them ideally suited for employing an efficient selective field expansion the lowest order tvfe employed within part of the computational domain and a higher order tvfe employed within the remaining part of the computational domain this is an attractive feature not shared by nonhierarchical tvfes for which a more traditional approach the same tvfe employed throughout the computational domain must be applied for determining the scattering by composite cylinders this paper argues that the performance in terms of accuracy memory and in most cases cpu time of the proposed class of hierarchical tvfes when applying selective field expansion is superior to that of the lowest order tvfe and a traditional nonhierarchical tvfe this is the case when an artificial absorber as well as a boundary integral is used for truncating the computational domain a guideline is given as to how lowest and higher order tvfes shall be combined for optimal performance of the proposed class of hierarchical tvfes

two dimensional method of moments modelling of lossless overmoded transverse magnetic cavities we present a method of moment mom analysis technique allowing to model overmoded transverse magnetic tm cavities in two dimensions this technique uses the single series form of the green s function of the empty cavity the convergence behaviour of this series is analysed in detail

globally and locally attractive solutions for quasi periodically forced systems abstract we consider a class of differential equations x x g x f t with r d describing one dimensional dissipative systems subject to a periodic or quasi periodic diophantine forcing we study existence and properties of trajectories with the same quasi periodicity as the forcing for g x x 2 p 1 p n we show that when the dissipation coefficient is large enough there is only one such trajectory and that it describes a global attractor in the case of more general nonlinearities including g x x 2 describing the varactor equation we find that there is at least one trajectory which describes a local attractor

a novel hybridization of higher order finite element and boundary integral methods for electromagnetic scattering and radiation problems a novel hybridization of the finite element fe and boundary integral methods is presented for an efficient and accurate numerical analysis of electromagnetic scattering and radiation problems the proposed method derives an adaptive numerical absorbing boundary condition abc for the finite element solution based on boundary integral equations unlike the standard finite element boundary integral approach the proposed method is free of interior resonance and produces a purely sparse system matrix which can be solved very efficiently unlike the traditional finite element absorbing boundary condition approach the proposed method uses an arbitrarily shaped truncation boundary placed very close to the scatterer radiator to minimize the computational domain and more importantly the method produces a solution that converges to the true solution of the problem to demonstrate its great potential the proposed method is implemented using higher order curvilinear vector elements a mixed functional is designed to yield both electric and magnetic fields on an integration surface without numerical differentiation to be used in the calculation of the adaptive abc the required evaluation of boundary integrals is carried out using the multilevel fast multipole algorithm which greatly reduces both the memory requirement and cpu time the finite element equations are solved efficiently using the multifrontal algorithm a mathematical analysis is conducted to study the convergence of the method finally a number of numerical examples are given to illustrate its accuracy and efficiency

the construction of low dispersive fdtd on hexagon based on the sampling theorem with arbitrary geometry the spatial discretization can be carried out on hexagonal lattices grids instead of rectangular lattices in the two dimensional case with the concept of the directional derivative in aid a new finite difference time domain fdtd method based on hexagonal lattices h fdtd is introduced the coefficients of the h fdtd method are chosen with aim to obtain the low numerical dispersion final numerical experiment shows that the numerical anisotropy of the h fdtd method is lower than that of the conventional yee fdtd method based on rectangular lattices r fdtd in addition some numerical examples are provided to verify its good performance on the low numerical anisotropy

implicit runge kutta methods and discontinuous galerkin discretizations for linear maxwell s equations in this paper we consider implicit runge kutta methods for the time integration of linear maxwell s equations we first present error bounds for the abstract cauchy problem which respect the unboundedness of the differential operators using energy techniques the error bounds hold for algebraically stable and coercive methods such as gauss and radau collocation methods the results for the abstract evolution equation are then combined with a discontinuous galerkin discretization in space using upwind fluxes for the case that permeability and permittivity are piecewise constant functions we show error bounds for the full discretization where the constants do not deteriorate if the spatial mesh width tends to zero

numerical methods for the generalized zakharov system we present two numerical methods for the approximation of the generalized zakharov system zs the first one is the time splitting spectral tssp method which is explicit time reversible and time transverse in variant if the generalized zs is keeps the same decay rate of the wave energy as that in the generalized zs gives exact results for the plane wave solution and is of spectral order accuracy in space and second order accuracy in time the second one is to use a local spectral method the discrete singular convolution dsc for spatial derivatives and the fourth order runge kutta rk4 for time integration which is of high the same as spectral order accuracy in space and can be applied to deal with general boundary conditions in order to test accuracy and stability we compare these two methods with other existing methods fourier pseudospectral method fps and wavelet galerkin method wg for spatial derivatives combining with the rk4 for time integration as well as the standard finite difference method fd for solving the zs with a solitary wave solution furthermore extensive numerical tests are presented for plane waves solitary wave collisions in 1d as well as a 2d problem of the generalized zs numerical results show that tssp and dsc are spectralorder accuracy in space and much more accurate than fd and for stability tssp requires k o h dsc rk4 requires k o h2 for fixed acoustic speed where k is the time step and h is the spatial mesh size