statistical properties of shocks in burgers turbulence we consider the statistical properties of solutions of burgers equation in the limit of vanishing viscosity frac partial partial t u left x t right frac partial partial x left frac 1 2 u left x t right 2 right 0 with gaussian whitenoise initial data this system was originally proposed by burgers 1 as a crude model of hydrodynamic turbulence and more recently by zel dovichet al 12 to describe the evolution of gravitational matter at large spatio temporal scales with shocks playing the role of mass clusters we present here a rigorous proof of the scaling relationp s s 1 2 s 1 wherep s is the cumulative probability distribution of shock strengths we also show that the set of spatial locations of shocks is discrete i e has no accumulation points and establish an upper bound on the tails of the shock strength distribution namely 1 p s exp cs 3 fors 1 our method draws on a remarkable connection existing between the structure of burgers turbulence and classical probabilistic work on the convex envelope of brownian motion and related diffusion processes

a new eighth order implicit finite difference method to solve the three dimensional helmholtz equation abstract this paper introduces a novel method for deducing high order finite difference schemes for the three dimensional helmholtz equation p 2 p f the new method can reach an eighth order of accuracy it is based on an implicit formulation by computing simultaneously the unknown variable and its corresponding derivatives the scheme is obtained from taylor series expansion and plane wave theory analysis and it is constructed from a few modifications to the standard centered finite differences a classical dispersion analysis and the error between the numerical wavenumber and the exact wavenumber is given numerical experiments are presented to verify the feasibility and accuracy of the proposed methods for a broad range of values moreover the numerical method is capable to solve with great precision the monotone and oscillatory helmholtz problem with variable values thus the proposed formulation results in an attractive method easy to implement with high order of accuracy but nearly the same computational cost as those of an explicit formulation

asymptotics of karhunen loeve eigenvalues and tight constants for probability distributions of passive scalar transport in this paper we study the asymptotics of the probability distribution function for a certain model of freely decaying passive scalar transport in particular we prove rigorous large n or semiclassical asymptotics for the eigenvalues of the covariance of a fractional brownian motion using these asymptotics along with some standard large deviations results we are able to derive tight asymptotics for the rate of decay of the tails of the probability density for a generalization of the majda model of scalar intermittency originally due to vanden eijnden we are also able to derive asymptotically tight estimates for the closely related problem of small l2 ball probabilities for a fractional brownian motion

a hybrid fem mom method for 3 d electromagnetic scattering in layered medium accurate and efficient prediction of electromagnetic scattering from inhomogeneous objects in layered medium is one of the most challenging issues in engineering applications this paper presents the first 3 d higher order hybrid finite element method fem and method of moments mom for the accurate modeling of inhomogeneous dielectric objects in multilayered medium the main challenges of this paper include 1 the integration of these algorithms for layered medium and 2 the higher order computational approach involved in layered medium for high efficiency in the proposed method the mom with the layered medium dyadic green s function is used as the exact radiation boundary condition in an inhomogeneous background and the fem is applied to model the inhomogeneous objects furthermore the higher order maximally orthogonal basis functions with curl conforming and divergence conforming properties are used in the fem and mom respectively to improve the modeling capability of this algorithm for 3 d inhomogeneous objects scattering in multilayered medium this new method requires a much more tightly truncated simulation domain than the traditional fem and provides much higher flexibility than the pure surface integral equation method finally some numerical results are provided to validate the accuracy efficiency and flexibility of this method

stable electric field tdie solvers via quasi exact evaluation of mot matrix elements prior theoretical studies and experience confirm that the stability of marching on in time mot solvers pertinent to the analysis of scattering from free standing three dimensional perfect electrically conducting surfaces hinges on the accurate evaluation of mot matrix elements resulting from a galerkin discretization of the underlying time domain integral equation tdie unfortunately the accurate evaluation of the four dimensional spatial integrals involved in the expressions for these matrix elements is prohibitively expensive when performed by computational means here a method that permits the quasi exact evaluation of mot matrix elements is presented specifically the proposed method permits the analytical evaluation of three out of the four spatial integrations leaving only one integral to be evaluated numerically since the latter has finite range and a piecewise smooth integrand it can be evaluated to very high accuracy using standard quadrature rules as a result the proposed method permits the fast evaluation of mot matrix elements with arbitrary user specified accuracy extensive numerical experiments show that an mot solver for the electric field tdie that uses the proposed quasi exact method is stable for a very wide range of time step sizes and yields solutions that decay exponentially after the excitation vanishes

fast analysis of transient electromagnetic scattering phenomena using the multilevel plane wave time domain algorithm the computational complexity of classical marching on in time mot methods for solving time domain integral equations tdies pertinent to the analysis of transient scattering phenomena involving perfectly conducting targets grows as o n sub t n sub s sup 2 n sub t and n sub s denote the number of temporal and spatial degrees of freedom dof of the electric current on the target this scaling law impedes the application of these schemes to the analysis of large scale scattering phenomena the recently developed plane wave time domain pwtd algorithm permits the rapid evaluation of transient wave fields generated by temporally bandlimited sources and hence the acceleration of marching on in time based tdie solvers previously we described a two level pwtd enhanced tdie solver for analyzing electromagnetic scattering from perfectly conducting targets the computational complexity of this algorithm scales as o n sub t n sub s sup 1 5 logn sub s here a multilevel pwtd scheme for rapidly evaluating electric fields due to temporally bandlimited electric current sources is described in addition a multilevel pwtd enhanced tdie solver for analyzing electromagnetic scattering from perfectly conducting scatterers using o n sub t n sub s log sup 2 n sub s cpu resources is outlined last the accuracy and cpu memory efficiency of this solver are demonstrated by analyzing transient scattering from electrically large bodies

time domain adaptive integral method for surface integral equations an efficient marching on in time mot scheme is presented for solving electric magnetic and combined field integral equations pertinent to the analysis of transient electromagnetic scattering from perfectly conducting surfaces residing in an unbounded homogenous medium the proposed scheme is the extension of the frequency domain adaptive integral pre corrected fast fourier transform fft method to the time domain fields on the scatterer that are produced by space time sources residing on its surface are computed 1 by locally projecting for each time step all sources onto a uniform auxiliary grid that encases the scatterer 2 by computing everywhere on this grid the transient fields produced by the resulting auxiliary sources via global multilevel blocked space time ffts 3 by locally interpolating these fields back onto the scatterer surface as this procedure is inaccurate when source and observer points reside close to each other and 4 near fields are computed classically albeit pre corrected for errors introduced through the use of global ffts the proposed scheme has a computational complexity and memory requirement of o n sub t n sub s log sup 2 n sub s and o n sub s sup 3 2 when applied to quasiplanar structures and of o n sub t n sub s sup 3 2 log sup 2 n sub s and o n sub s sup 2 when used to analyze scattering from general surfaces here n sub s and n sub t denote the number of spatial and temporal degrees of freedom of the surface current density these computational cost and memory requirements are contrasted to those of classical mot solvers which scale as o n sub t n sub s sup 2 and o n sub s sup 2 respectively a parallel implementation of the scheme on a distributed memory computer cluster that uses the message passing interface is described simulation results demonstrate the accuracy efficiency and the parallel performance of the implementation

hierarchal vector basis functions of arbitrary order for triangular and tetrahedral finite elements new vector finite elements are proposed for electromagnetics the new elements are triangular or tetrahedral edge elements tangential vector elements of arbitrary polynomial order they are hierarchal so that different orders can be used together in the same mesh and p adaption is possible they provide separate representation of the gradient and rotational parts of the vector field explicit formulas are presented for generating the basis functions to arbitrary order the basis functions can be used directly or after a further stage of partial orthogonalization to improve the matrix conditioning matrix assembly for the frequency domain curl curl equation is conveniently carried out by means of universal matrices application of the new elements to the solution of a parallel plate waveguide problem demonstrates the expected convergence rate of the phase of the reflection coefficient for tetrahedral elements to order 4 in particular the full order elements have only the same asymptotic convergence rate as elements with a reduced gradient space such as the whitney element however further tests reveal that the optimum balance of the gradient and rotational components is problem dependent

solution of time domain electric field integral equation using the laguerre polynomials in this paper we propose a numerical method to obtain a solution for the time domain electric field integral equation td efie for arbitrary shaped conducting structures this method does not utilize the customary marching on in time mot solution method often used to solve a hyperbolic partial differential equation instead we solve the wave equation by expressing the transient behaviors in terms of laguerre polynomials by using these causal orthonormal basis functions for the temporal variation the time derivatives can be handled analytically in order to solve the wave equation we introduce two separate testing procedures a spatial and temporal testing by introducing first the galerkin temporal testing procedure the mot procedure is replaced by a recursive relation between the different orders of the weighted laguerre polynomials the other novelty of this approach is that through the use of the entire domain laguerre polynomials for the expansion of the temporal variation of the current the spatial and the temporal variables can be separated and the temporal variables can be integrated out for convenience we use the hertz vector as the unknown variable instead of the electric current density to verify our method we compare the results of a td efie and inverse fourier transform of a frequency domain efie

a staggered fourth order accurate explicit finite difference scheme for the time domain maxwell s equations abstract we consider a model explicit fourth order staggered finite difference method for the hyperbolic maxwell s equations appropriate fourth order accurate extrapolation and one sided difference operators are derived in order to complete the scheme near metal boundaries and dielectric interfaces an eigenvalue analysis of the overall scheme provides a necessary but not sufficient stability condition and indicates long time stability numerical results verify both the stability analysis and the scheme s fourth order convergence rate over complex domains that include dielectric interfaces and perfectly conducting surfaces for a fixed error level we find the fourth order scheme is computationally cheaper in comparison to the yee scheme by more than an order of magnitude some open problems encountered in the application of such high order schemes are also discussed

fdtd algorithm in curvilinear coordinates em scattering the finite difference time domain fdtd algorithm for the solution of electromagnetic scattering problems is formulated in generalized coordinates in two dimensions and implemented in a code with the lowest order bayliss turkel radiation boundary condition expressed in cylindrical coordinates it is shown that for a perfect conductor such a formulation leads to a stable well posed algorithm and that in regions where the curvature of the coordinate lines is not great the dispersion and anisotropy effects are negligible such effects become more pronounced in regions of high curvature leading to unphysical phase shifts the magnitude of such shifts and the amount of wavefront distortion is studied via numerical experiments using a cylindrical mesh near field results are given for two canonical shapes for each polarization the circular cylinder and cylinders of square and rectangular cross sections these results are compared with those obtained by exact eigenfunction expansion techniques with method of moments mm solutions and with solutions obtained from an alternate fdtd approach in each case agreement is excellent the propagation of a plane wave through a polar space in the absence of a scatterer is also examined and it is shown that the fdtd algorithm is capable of tracking the incident wave closely

higher order interpolatory vector bases for computational electromagnetics low order vector basis functions compatible with the nedelec 1980 representations are widely used for electromagnetic field problems higher order functions are receiving wider application but their development is hampered by the complex procedures used to generate them and lack of a consistent notation for both elements and bases in this paper fully interpolatory higher order vector basis functions of the nedelec type are defined in a unified and consistent manner for the most common element shapes it is shown that these functions can be obtained as the product of zeroth order nedelec representations and interpolatory polynomials with specially arranged arrays of interpolation points the completeness properties of the vector functions are discussed and expressions for the vector functions of arbitrary polynomial order are presented sample numerical results confirm the faster convergence of the higher order functions

a fourth order optimal finite difference scheme for the helmholtz equation with pml abstract in this paper 17 point and 25 point finite difference fd schemes for the helmholtz equation with perfectly matched layer pml in the two dimensional domain are presented it is shown that the 17 point fd scheme is inconsistent in the presence of pml however the 25 point fd scheme is pointwise consistent an error analysis for the numerical approximation of the exact wavenumber is also presented we present the global and refined 25 point finite difference schemes based on minimizing the numerical dispersion numerical experiments are given to illustrate the improvement of the accuracy and the reduction of the numerical dispersion

a higher order multilevel fast multipole algorithm for scattering from mixed conducting dielectric bodies a higher order multilevel fast multipole algorithm mlfma is presented for computing electromagnetic scattering from three dimensional bodies comprising both conducting and dielectric objects the problem is formulated using the poggio miller chang harrington wu tsai 1973 1977 approach for multiple homogeneous dielectric objects and the combined field approach for conducting objects the resultant integral equations are discretized by the method of moments mom in which the conducting and dielectric surfaces interfaces are represented by curvilinear triangular patches and the unknown equivalent electric and magnetic currents are expanded using higher order vector basis functions such a discretization yields a highly accurate representation of the unknown currents without compromising the accuracy of geometrical modeling an implicit matrix filling scheme is employed to facilitate the treatment of complex scatterers having multiple junctions the resultant numerical system is then solved by mlfma which is tailored to accommodate the material properties of dielectric scatterers and the solution is accelerated using an incomplete lu decomposition preconditioner numerical examples are presented to demonstrate the accuracy and versatility of this approach in dealing with a wide array of scattering problems

higher order hybrid fem mom technique for analysis of antennas and scatterers a novel higher order large domain hybrid computational electromagnetic technique based on the finite element method fem and method of moments mom is proposed for three dimensional analysis of antennas and scatterers in the frequency domain the geometry of the structure is modeled using generalized curved parametric hexahedral and quadrilateral elements of arbitrary geometrical orders the fields and currents on elements are modeled using curl and divergence conforming hierarchical polynomial vector basis functions of arbitrary approximation orders and the galerkin method is used for testing the elements can be as large as about two wavelengths in each dimension as multiple mom objects are possible in a global exterior region the mom part provides much greater modeling versatility and potential for applications especially in antenna problems than just as a boundary integral closure to the fem part the examples demonstrate excellent accuracy convergence efficiency and versatility of the new fem mom technique and very effective large domain meshes that consist of a very small number of large flat and curved fem and mom elements with p refined field and current distributions of high approximation orders the reduction in the number of unknowns is by two orders of magnitude when compared to available data for low order fem mom modeling

time domain integral equation analysis of scattering from composite bodies via exact evaluation of radiation fields a novel marching on in time and poggio miller chang harrington wu tsai pmchwt based time domain integral equation solver for analyzing transient scattering from composite multiregion structures comprising homogeneous penetrable volumes and perfectly conducting surfaces is presented to render the marching on in time procedure stable observer and source integrals are approximated via multipoint collocation and evaluated in closed form the proposed scheme avoids the costly evaluation of five dimensional integrals required in full space time galerkin schemes and proves to be stable for all targets investigated including coated perfect electrically conducting surfaces and multiregion penetrable volumes

higher order frequency domain computational electromagnetics a review of the higher order computational electromagnetics cem for antenna wireless and microwave engineering applications is presented higher order cem techniques use current field basis functions of higher orders defined on large e g on the order of a wavelength in each dimension curvilinear geometrical elements which greatly reduces the number of unknowns for a given problem the paper reviews all major surface volume integral and differential equation electromagnetic formulations within a higher order computational framework focusing on frequency domain solutions with a systematic and unified review of generalized curved parametric quadrilateral triangular hexahedral and tetrahedral elements and various types of higher order hierarchical and interpolatory vector basis functions in both divergence and curl conforming arrangements a large number of actual higher order techniques representing various combinations of formulations elements bases and solution procedures are identified and discussed the examples demonstrate the accuracy efficiency and versatility of higher order techniques and their advantages over low order discretizations the most important one being a much faster higher order convergence of the solution it is demonstrated that both components of the higher order modeling namely higher order geometrical modeling and higher order current field modeling are essential for accurate and efficient cem analysis of general antenna scattering and microwave structures

an application of the boundary element method to the magnetic field integral equation a boundary element method bem for the solution of electromagnetic scattering problems using the magnetic field integral equation mfie is discussed the discretized form of the mfie is written in indicial notation with no limitations placed on the order of either the geometric or functional approximation by considering several different types of boundary elements it is determined that geometric errors can be significant and degrade the accuracy of the numerical solution it is shown that a higher order approximation for the current could significantly improve the accuracy of the numerical solution the superparametric boundary element in which the geometry was given quadratic approximation and the current was given linear approximation was more efficient than elements using lower order approximations the bem results are compared to the results obtained using the dielectric bodies of revolution dbr code

higher order hierarchical legendre basis functions for electromagnetic modeling this paper presents a new hierarchical basis of arbitrary order for integral equations solved with the method of moments mom the basis is derived from orthogonal legendre polynomials which are modified to impose continuity of vector quantities between neighboring elements while maintaining most of their desirable features expressions are presented for wire surface and volume elements but emphasis is given to the surface elements in this case the new hierarchical basis leads to a near orthogonal expansion of the unknown surface current and implicitly an orthogonal expansion of the surface charge in addition all higher order terms in the expansion have two vanishing moments in contrast to existing formulations these properties allow the use of very high order basis functions without introducing ill conditioning of the resulting mom matrix numerical results confirm that the condition number of the mom matrix obtained with this new basis is much lower than existing higher order interpolatory and hierarchical basis functions as a consequence of the excellent condition numbers we demonstrate that even very high order mom systems e g tenth order can be solved efficiently with an iterative solver in relatively few iterations

an optimal compact sixth order finite difference scheme for the helmholtz equation abstract in this paper we present an optimal compact finite difference scheme for solving the 2d helmholtz equation a convergence analysis is given to show that the scheme is sixth order in accuracy based on minimizing the numerical dispersion a refined optimization rule for choosing the scheme s weight parameters is proposed numerical results are presented to demonstrate the efficiency and accuracy of the compact finite difference scheme with refined parameters

statistics of shocks in solutions of inviscid burgers equation the purpose of this paper is to analyze statistical properties of discontinuities of solutions of the inviscid burgers equation having a typical realizationb y of the brownian motion as an initial datum this case was proposed and studied numerically in the companion paper by she aurell and frisch the description of the statistics is given in terms of the behavior of the convex hull of the random process w y int limits 0 y b eta eta d eta the hausdorff dimension of the closed set of thosey where the convex hull coincides withw is also studied

a novel implementation of multilevel fast multipole algorithm for higher order galerkin s method a new approach is proposed to reduce the memory requirements of the multilevel fast multipole algorithm mlfma when applied to the higher order galerkin s method this approach represents higher order basis functions by a set of point sources such that a matrix vector multiply is equivalent to calculating the fields at a number of points from given current sources at these points the mlfma is then applied to calculate the point to point interactions this permits the use of more levels in mlfma than applying mlfma to basis to basis interactions directly and thus reduces the memory requirements significantly

an accurate scheme for the solution of the time domain integral equations of electromagnetics using higher order vector bases and bandlimited extrapolation despite the numerous advances made in increasing the computational efficiency of time domain integral equation tdie based solvers the stability and accuracy of tdie solvers remain problematic this paper introduces a new numerical method for the accurate solution of tdies for scattering from arbitrary perfectly conducting surfaces the work described in this paper uses the higher order divergence conforming basis functions of graglia et al for spatial discretization and bandlimited interpolation functions for the temporal discretization of the relevant integral equations since the basis functions used for the temporal representation are noncausal an extrapolation scheme is employed to recover the ability to solve the problem by marching on in time numerical results demonstrate that the proposed method is stable and that it exhibits superlinear convergence with regard to the spatial discretization and exponential convergence with respect to the temporal discretization

electromagnetic scattering from 3 d curved dielectric bodies using time domain integral equations a boundary integral equation bie approach is developed to calculate transient scattering from dielectric bodies the treatment is directly in terms of the e and h fields rather than magnetic and electric currents it employs curvilinear quadratic modeling which allows accurate representation of arbitrarily shaped curved bodies the treatment is isoparametric with the same quadratic representation of the spatial field variation and with the temporal variation modeled by similar quadratic elements integration employs high order gaussian quadrature with special treatment of the singular and hypersingular integrals that arise the treatment is implicit requiring the solution of a sparse matrix equation at each timestep this adds only trivially to the cost at each timestep and by freeing the timestep from the constraint that it be smaller than the smallest nodal spacing can greatly reduce the number of timesteps that must be employed additionally it produces stable results without resort to the averaging processes proposed elsewhere example calculations of scattering from a sphere a cube and an almond are presented and compared with earlier published transient results and with results from a frequency domain treatment good agreement and improved accuracy is found

global picture of self similar and non self similar decay in burgers turbulence this paper continues earlier investigations of the decay of burgers turbulence in one dimension from gaussian random initial conditions of the power law spectral type e 0 k ensuremath sim ensuremath mid k ensuremath mid n depending on the power n different characteristic regions are distinguished the main focus of this paper is to delineate the regions in wave number k and time t in which self similarity can and cannot be observed taking into account small k and large k cutoffs the evolution of the spectrum can be inferred using physical arguments describing the competition between the initial spectrum and the new frequencies generated by the dynamics for large wave numbers we always have a k ensuremath 2 region associated with the shocks when n is less than 1 the large scale part of the spectrum is preserved in time and the global evolution is self similar so that scaling arguments perfectly predict the behavior in time of the energy and integral scale if n is larger than 2 the spectrum tends for long times to a universal scaling form independent of the initial conditions with universal behavior k 2 at small wave numbers in the interval 2ln the leading behavior is self similar independent of n and with universal behavior k 2 at small wave number when 1lnl2 the spectrum has three scaling regions first a ensuremath mid k ensuremath mid n region at very small k s with a time independent constant second a k 2 region at intermediate wave numbers finally the usual k ensuremath 2 region in the remaining interval nl ensuremath 3 the small k cutoff dominates and n also plays no role we find also numerically the subleading term ensuremath sim k 2 in the evolution of the spectrum in the interval ensuremath 3lnl1 high resolution numerical simulations have been performed confirming both scaling predictions and analytical asymptotic theory

solving the volume integral equations of electromagnetic scattering time harmonic electromagnetic scattering by inhomogeneous three dimensional structures within a free space environment can be described by electric and magnetic field volume integral equations involving the free space green function a comprehensive set of galerkin projection formulations also known as moment methods for the numerical solution of these equations is presented together with comparative numerical results such formulations are widely used for particle scattering analysis optical near field calculation etc results are obtained with higher order divergence and curl conforming basis functions on iso parametric tetrahedral meshes the results demonstrate that all formulations converge with similar accuracy in the case of an analytically solvable test problem when modeling flux densities as solution variables it is argued that solenoidal function spaces should be used rather than the standard divergence conforming function spaces this assertion is supported by the results some of the formulations involve solving for curl conforming fields such fields can be discretized with fewer unknowns than divergence conforming ones implying lower computational costs additionally some formulations yield system matrices which are approximately halfway sparse meaning that computational costs will be down by a factor of 2 when iterative solvers are employed which is the case for the widely used fast methods

a novel grid robust higher order vector basis function for the method of moments a set of novel grid robust higher order vector basis functions is proposed for the method of moments mom solution of integral equations for three dimensional 3 d electromagnetic em problems these basis functions are defined over curvilinear triangular patches and represent the unknown electric current density within each patch using the lagrange interpolation polynomials the highlight of these basis functions is that the lagrange interpolation points are chosen to be the same as the nodes of the well developed gaussian quadratures as a result the evaluation of the integrals in the mom is greatly simplified additionally the surface of an object to be analyzed can be easily meshed because the new basis functions do not require the side of a triangular patch to be entirely shared by another triangular patch which is a very stringent requirement for traditional vector basis functions the proposed basis functions are implemented with point matching for the mom solution of the electric field integral equation the magnetic field integral equation and the combined field integral equation numerical examples are presented to demonstrate the higher order convergence and the grid robustness for defective meshes using the new basis functions

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

accurate solution of square scatterer as benchmark for validation of electromagnetic modeling of plate structures an infinitesimally thin square scatterer of size spl lambda spl times spl lambda excited normally by an incident plane wave which is polarized along a scatterer edge is analyzed the accurate solution of its current distribution is found in the form of a double series of basis functions which automatically satisfy the continuity equation at the plate edges and include the edge effect the coefficients that multiply basis functions are determined starting from the electric field integral equation by using the galerkin method the solution obtained for the order of approximation n 8 is adopted as a benchmark the corresponding coefficients are tabulated and graphs of such obtained current distribution are given the solution adopted as a benchmark is applied for comparison of rooftop basis functions and polynomial entire domain basis functions the relative error of the mean absolute value of current deviation is used as an error metric

preconditioning techniques for large linear systems a survey this article surveys preconditioning techniques for the iterative solution of large linear systems with a focus on algebraic methods suitable for general sparse matrices covered topics include progress in incomplete factorization methods sparse approximate inverses reorderings parallelization issues and block and multilevel extensions some of the challenges ahead are also discussed an extensive bibliography completes the paper

spatially large domain and temporally entire domain electric field integral equation method of moments for 3 d scattering analysis in time domain a novel spatially large domain and temporally entire domain method of moments mom is proposed for surface integral equation sie modeling of 3 d conducting scatterers in the time domain td the method uses higher order curved lagrange interpolation generalized quadrilateral geometrical elements higher order spatial current expansions based on hierarchical divergence conforming polynomial vector basis functions and temporal current modeling by means of orthogonal weighted associated laguerre basis functions it implements full temporal and spatial galerkin testing and marching on in degree mod scheme for an iterative solution of the final system of spatially and temporally discretized mom td equations numerical examples demonstrate excellent accuracy efficiency convergence and versatility of the new mom mod method the results also demonstrate very effective large domain mom td sie models of scatterers using flat and curved patches of electrical sizes of up to about 1 7 wavelengths at the maximum frequency in the frequency spectrum of the pulse excitation higher order current expansions of spatial orders from 2 to 8 in conjunction with entire domain laguerre temporal bases and minimal numbers of unknowns

an optimal 25 point finite difference scheme for the helmholtz equation with pml in this paper we present an optimal 25 point finite difference scheme for solving the helmholtz equation with perfectly matched layer pml in two dimensional domain based on minimizing the numerical dispersion we propose the refined choice strategy for choosing optimal parameters of the 25 point finite difference scheme numerical experiments are given to illustrate the improvement of the accuracy and the reduction of the numerical dispersion

a stable solution of time domain electric field integral equation for thin wire antennas using the laguerre polynomials in this paper a numerical method to obtain an unconditionally stable solution of the time domain electric field integral equation for arbitrary conducting thin wires is presented the time domain electric field integral equation td efie technique has been employed to analyze electromagnetic scattering and radiation problems from thin wire structures however the most popular method to solve the td efie is typically the marching on in time mot method which sometimes may suffer from its late time instability instead we solve the time domain integral equation by expressing the transient behaviors in terms of weighted laguerre polynomials by using these orthonormal basis functions for the temporal variation the time derivatives can be handled analytically and stable results can be obtained even for late time furthermore the excitation source in most scattering and radiation analysis of electromagnetic systems is typically done using a gaussian shaped pulse in this paper both a gaussian pulse and other waveshapes like a rectangular pulse or a ramp like function have been used as excitations for the scattering and radiation of thin wire antennas with and without junctions the time domain results are compared with the inverse discrete fourier transform idft of a frequency domain analysis

solving time domain electric field integral equation without the time variable an improved testing procedure using the marching on in order method to solve the time domain electric field integral equation td efie for conducting objects using the laguerre polynomials is presented exact temporal testing is performed before the spatial testing therefore the retarded terms composed of the spatial and the temporal variables can be analytically separated the uniqueness of this testing procedure is that the time variable can be analytically integrated out and the accuracy can be improved this paper is then an improvement over the earlier marching on in order method in addition this methodology is quite different from the conventional marching on in time algorithm as the present method leads to a set of final equations which need to be numerically solved containing only the spatial variables therefore there is no requirement to have a courant stability condition in this procedure how the singular integrals are treated is also discussed several examples are simulated both for radiation and scattering problem the results are compared with the inverse discrete fourier transform of the frequency domain data and they agree well

numerical evaluation of hierarchical vector finite elements on curvilinear domains in 2 d when using the finite element method computational efficiency may be improved by using higher order elements typical 2 d triangular rectilinear elements cannot exactly model curved geometries negating the advantage of higher order elements when such geometries are modeled curvilinear elements can be applied to such problems and have significantly better computational efficiency than rectilinear elements when higher order elements are used results are shown for elemental basis orders ranging from ct ln through full fifth order in rectilinear and curvilinear domains

study of mixed order basis functions for the locally corrected nystro spl uml m method a high order locally corrected nystro spl uml m lcn method employing the mixed order basis functions proposed by c 80al spl iota s 80kan and peterson is presented for the electromagnetic scattering by targets composed of both dielectric and conducting bodies an integral operator based on a combined field formulation for conducting surfaces and a mu spl uml ller formulation for dielectric surfaces is used it is found that for general scattering objects mixed order basis functions accelerate the convergence of the lcn solution can eliminate spurious charges and can significantly reduce the condition number of the impedance matrix

duffy method for evaluation of weakly singular sie potential integrals over curved quadrilaterals with higher order basis functions a duffy method for singularity cancellation is proposed for evaluation of weakly singular potential integrals defined on generalized curved parametric quadrilateral patches with polynomial basis functions such integrals arise in evaluation of galerkin matrix elements in method of moments analysis of antennas and scatterers in cases of coincident source and test elements examples demonstrate that the proposed duffy method is more accurate more rapidly converging with the increase of orders of gauss legendre integration formulas and faster to execute than four other methods for singularity treatment considered in the study

mesh generation in curvilinear domains using high order elements the ability to construct suitable computational meshes is currently a significant limiting factor in the development of compact high order algorithms in very complex geometries compact high order algorithms such as spectral element or p type finite element techniques offer the potential of high accuracy if the solution is smooth and a well behaved mapping exists between the local sub domains and a standard region in this paper we address the problem of coupling high order algorithms with unstructured mesh generation and cad representation techniques we present three complementary strategies to alleviate the problem of generation of elemental regions with singular elemental mappings the first strategy investigates the influence of an anisometric parametric surface representation on the quality of high order unstructured meshes we show that a straightforward splitting of a standard linear mesh for use in a high order method can lead to the generation of distorted possibly invalid meshes even for simple computational domains such as cubes the second strategy uses hybrid meshes applying prismatic elements near curvilinear boundaries in the third and final strategy we investigate the use of curvature driven surface discretization to incorporate higher order information about the surface into the mesh generation using the computational reconstruction of the arterial bypass graft as an example computational domain we demonstrate how the combination of all three strategies leads to a valid high order discretization of the computational domain copyright 2001 john wiley sons ltd

burgers equation with self similar gaussian initial data tail probabilities the statistical properties of solutions of the one dimensional burgers equation in the limit of vanishing viscosity are considered when the initial velocity potential is fractional brownian motion fbm we establish the asymptotic power law order for log probability of large values both velocity and shock amplitude of velocity discontinuity this confirms the conjecture of u frisch and his collaborators rigorous results for this problem were previously derived for the case of brownian motion using markov techniques our approach is based on the intrinsic properties of fbm and the theory of extreme values for gaussian processes

the decay of multiscale signals a deterministic model of burgers turbulence this work is devoted to the study of the decay of multiscale deterministic solutions of the unforced burgers equation in the limit of vanishing viscosity it is well known that burgers turbulence with a power law energy spectrum e0 k k n has a self similar regime of evolution for n 1 this regime is characterised by an integral scale l t t 2 3 n which increases with the time due to the multiple mergings of the shocks and therefore the energy of a random wave decays more slowly than the energy of a periodic signal in this paper a deterministic model of turbulence like evolution is considered we construct the initial perturbation as a piecewise linear analog of the weierstrass function the wavenumbers of this function form a weierstrass spectrum which accumulates at the origin in geometric progression reverse sawtooth functions with negative initial slope are used in this series as basic functions while their amplitudes are chosen by the condition that the distribution of energy over exponential intervals of wavenumbers is the same as for the continuous spectrum in burgers turbulence combining these two ideas allows us to obtain an exact analytical solution for the

a new family of mixed finite elements in ir 3 we introduce two families of mixed finite element on conforming inh div and one conforming inh curl these finite elements can be used to approximate the stokes system

numerical solution of the helmholtz equation in 2d and 3d using a high order nystrom discretization we show how to solve time harmonic scattering problems by means of a high order nystrom discretization of the boundary integral equations of wave scattering in 2d and 3d the novel aspect of our new method is its use of local corrections to the discretized kernel in the vicinity of the kernel singularity enhanced by local corrections the new algorithm has the simplicity and speed advantages of the traditional nystrom method but also enjoys the advantages of high order convergence for controlling solution error we explain the practical details of implementing a scattering code based on a high order nystrom discretization and demonstrate by numerical example that a scattering code based on this algorithm can achieve high order convergence to the correct answer we also demonstrate its performance advantages over a high order galerkin code

higher order hybrid method of moments physical optics modeling technique for radiation and scattering from large perfectly conducting surfaces an efficient and accurate higher order large domain hybrid computational technique based on the method of moments mom and physical optics po is proposed for analysis of large antennas and scatterers composed of perfectly conducting surfaces of arbitrary shapes the technique utilizes large generalized curvilinear quadrilaterals of arbitrary geometrical orders in both the mom and po regions it employs higher order divergence conforming hierarchical polynomial basis functions in the context of the galerkin method in the mom region and higher order divergence conforming interpolatory chebyshev type polynomial basis functions in conjunction with a point matching method in the po region the results obtained by the higher order mom po are validated against the results of the full mom analysis in three characteristic realistic examples the truly higher order and large domain nature of the technique in both mom and po regions enables a very substantial reduction in the number of unknowns and increase in accuracy and efficiency when compared to the low order small domain mom po solutions the po part of the proposed technique on the other hand allows for a dramatic reduction in the computation time and memory with respect to the pure mom higher order technique which greatly extends the practicality of the higher order mom with a smooth transition between low and high frequency applications

maximally orthogonalized higher order basis functions in large domain finite element modeling in electromagnetics curl conforming max ortho basis functions mobfs are coupled with higher order large domain curved finite elements fes the performance of the functions is compared with that of the classical and near ortho basis functions through numerical experiments it is shown that max ortho fes yield highly orthogonal mass matrices for practically arbitrarily high orders of polynomial field approximations this facilitates the usage of iterative solvers and it significantly increases their efficiency accurate and fast computation of mobfs of arbitrarily high orders is enabled by the proposed two term recurrent formula

the multifrontal method for sparse matrix solution theory and practice this paper presents an overview of the multifrontal method for the solution of large sparse symmetric positive definite linear systems the method is formulated in terms of frontal matrices update matrices and an assembly tree formal definitions of these notions are given based on the sparse matrix structure various advances to the basic method are surveyed they include the role of matrix reorderings the use of supernodes and other implementatjon techniques the use of the method in different computational environments is also described

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

finite element solution of the helmholtz equation with high wave number part i the h version of the fem abstract the paper addresses the properties of finite element solutions for the helmholtz equation the h version of the finite element method with piecewise linear approximation is applied to a one dimensional model problem new results are shown on stability and error estimation of the discrete model in all propositions assumptions are made on the magnitude of hk only where k is the wavelength and h is the stepwidth of the fe mesh previous analytical results had been shown with the assumption that k 2 h is small for medium and high wavenumber these results do not cover the meshsizes that are applied in practical applications the main estimate reveals that the error in h 1 norm of discrete solutions for the helmholtz equation is polluted when k 2 h is not small the error is then not quasioptimal i e the relation of the fe error to the error of best approximation generally depends on the wavenumber k it is noted that the pollution term in the relative error is of the same order as the phase lead of the numerical solution in the result of this analysis thorough and rigorous understanding of error behavior throughout the range of convergence is gained numerical results are presented that show sharpness of the error estimates and highlight some phenomena of the discrete solution behavior the h p version of the fem is studied in part ii

triangular prisms for edge based vector finite element analysis of conformal antennas this paper deals with the derivation of the edge based shape functions for the distorted triangular prism and their applications for the analysis of doubly curved conformal antennas in the context of the finite element method fem although the tetrahedron is often the element of choice for volume tessellation mesh generation using tetrahedra is cumbersome and the central processing unit cpu intensive on the other hand the distorted triangular prism allows for meshes which are unstructured in two dimensions and structured in the third dimension this leads to substantial simplifications in the meshing algorithm and many conformal printed antenna and microwave circuit geometries can be easily tessellated using such a mesh the new edge based shape functions are first validated by computing the eigenvalues of three different cavities rectangular cylindrical and pie shell we then proceed with their application to computing the input impedance of conformal patch antennas on planar spherical conical and other doubly curved ogival platforms where the fem mesh is terminated using an artificial absorber applied conformal to the platform use of artificial absorbers for mesh termination avoids introduction of green s functions and in contrast to absorbing boundary conditions a knowledge of the principal radii of curvature of the closure s boundary is not required

a frequency space 2 d scalar wave extrapolator using extended 25 point finite difference operator finite difference frequency domain modeling for the generation of synthetic seismograms and crosshole tomography has been an active field of research since the 1980s the generation of synthetic seismograms with the time domain finite difference technique has achieved considerable success for waveform crosshole tomography and for wider applications in seismic reverse time migration this became possible with the rapid development of high performance computers however the space frequency x finite difference modeling technique is still beyond the capability of the modern supercomputer in terms of both cost and computer memory therefore finite difference time domain modeling is much more popular among exploration geophysicists a limitation of the space frequency domain is that the recently developed nine point scheme still requires that g the number of grid points per wavelength be 5 this value is greater than for most other numerical modeling techniques for example the pseudospectral scheme

higher order interpolatory vector bases on pyramidal elements in the numerical solution of three dimensional 3 d electromagnetic field problems the regions of interest can be discretized by elements having tetrahedral brick or prismatic shape however such different shape elements cannot be linked to form a conformal mesh to this purpose pyramidal elements are required in this paper we define interpolatory higher order curl and divergence conforming vector basis functions on pyramidal elements with extension to curved pyramids and discuss their completeness properties a general procedure to obtain vector bases of arbitrary polynomial order is given and bases up to second order are explicitly reported these new elements ensure the continuity of the proper vector components across adjacent elements of equal order but different shape results to confirm the faster convergence of higher order functions on pyramids are presented

an improved marching on in degree method using a new temporal basis the marching on in degree mod method has been presented earlier for solving time domain electric field integral equations in a stable fashion this is accomplished by expanding the transient responses by a complete set of orthogonal entire domain associated laguerre functions which helps one to analytically integrate out the time variable from the final computations in a galerkin methodology so the final computations are carried out using only the spatial variables however the existing mod method suffers from low computational efficiency over a marching on in time mot method the two main causes of the computational inefficiency in the previous mod method are now addressed using a new form of the temporal basis functions and through a different computational arrangement for the green s function in this paper it is shown that incorporating these two new concepts can speed up the computational process and make it comparable to a mot algorithm sample numerical results are presented to illustrate the validity of these claims in solution of large problems using the mod method

a space time discretization criterion for a stable time marching solution of the electric field integral equation numerical techniques based on a time domain recursive solution of the electric field integral equation efie may exhibit instability phenomena induced by the joint space time discretization the above problem is addressed with specific reference to the evaluation of electromagnetic scattering from perfectly conducting bodies of arbitrary shape we analyze a particular formulation of the method of moments which relies on a triangular patch geometrical model of the exterior surface of the scattering body and operates according to a marching on in time scheme whereby the surface current distribution at a given time step is recursively evaluated as a function of the current distribution at previous steps a heuristic stability condition is devised which allows us to define a proper time step as well as a geometrical discretization criterion ensuring convergence of the numerical procedure and therefore eliminating insurgence of late time oscillations the stability condition is discussed and validated by means of a few working examples

accuracy of finite difference and finite element modeling of the scalar and elastic wave equations numerical solutions of the scalar and elastic wave equations have greatly aided geophysicists in both forward modeling and migration of seismic wave fields in complicated geologic media and they promise to be invaluable in solving the full inverse problem this paper quantitatively compares finite difference and finite element solutions of the scalar and elastic hyperbolic wave equations for the most popular implicit and explicit time domain and frequency domain techniques in addition to versatility and ease of implementation it is imperative that one choose the most cost effective solution technique for a fixed degree of accuracy to be of value a solution technique must be able to minimize 1 numerical attenuation or amplification 2 polarization errors 3 numerical anisotropy 4 errors in phase and group velocities 5 extraneous numerical parasitic modes 6 numerical diffraction and scattering and 7 errors and transmission coefficients this paper shows that in homogeneous media the

the inviscid burgers equation with initial data of brownian type the solutions to burgers equation in the limit of vanishing viscosity are investigated when the initial velocity is a brownian motion or fractional brownian motion function i e a gaussian process with scaling exponent 0 h 1 typea or the derivative thereof with scaling exponent 1 h 0 typeb largesize numerical experiments are performed helped by the fact that the solution is essentially obtained by performing a legendre transform the main result is obtained for typea and concerns the lagrangian functionx a which gives the location at timet 1 of the fluid particle which started at the locationa it is found to be a complete devil s staircase the cumulative probability of lagrangian shock intervals a also the distribution of shock amplitudes follows a a h law for small a the remaining regular lagrangian locations form a cantor set of dimensionh in eulerian coordinates the shock locations are everywhere dense the scaling properties of various statistical quantities are also found heuristic interpretations are provided for some of these results rigorous results for the case of brownian motion are established in a companion paper by ya sinai for typeb initial velocities e g white noise there are very few small shocks and shock locations appear to be isolated finally it is shown that there are universality classes of random but smooth non scaling initial velocities such that the long time large scale behavior is after rescaling the same as for typea orb

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

monomial cubature rules since stroud a compilation a bibliography of references to cubature rules which have appeared since the publication of stroud s book 1971 is presented the standard regions that are treated in this paper are the n cube the n simplex the n sphere and the entire space

a transpose free quasi minimal residual algorithm for non hermitian linear systems the biconjugate gradient method bcg for solving general non hermitian linear systems ax b and its transpose free variant the conjugate gradients squared algorithm cgs both typically exhib

transient electromagnetic scattering from dielectric objects using the electric field integral equation with laguerre polynomials as temporal basis functions in this paper we propose a time domain electric field integral equation td efie formulation for analyzing the transient electromagnetic response from three dimensional 3 d dielectric bodies the solution method in this paper is based on the galerkin s method that involves separate spatial and temporal testing procedures triangular patch basis functions are used for spatial expansion and testing functions for arbitrarily shaped 3 d dielectric structures the time domain unknown coefficients of the equivalent electric and magnetic currents are approximated using a set of orthonormal basis function that is derived from the laguerre functions these basis functions are also used as the temporal testing functions use of the laguerre polynomials as expansion functions for the transient portion of response enables one not only to handle the time derivative terms in the integral equation in an analytic fashion but also completely separates the space and the time variables thus the time variable along with the courant condition can be eliminated in a galerkin formulation using this procedure we also propose an alternative formulation using a different expansion of the magnetic current the total computational cost for this new method is similar to that of an implicit marching on in time mot efie scheme even though at each step this procedure requires more computations numerical results involving equivalent currents and far fields computed by the two proposed methods are presented and compared

statistical properties of shocks in burgers turbulence ii tail probabilities for velocities shock strengths and rarefaction intervals this paper studies the structure of the random sawtooth profile corresponding to the solution of the inviscid burgers equation with white noise initial data this function consists of a countable sequence of rarefaction waves separated by shocks we are concerned here with calculating the probabilities of rare events associated with the occurrence of very large values of the normalized velocity shock strength and rarefaction intervals we find that these quantities have tail probabilities of the form exp cx3 x 1 this cubic exponential decay of probabilities was conjectured in the companion paper 1 the calculations are done using a representation of the shock strength and length of rarefaction intervals in terms of the statistics of certain conditional diffusion processes

a modified fdtd 2 4 scheme for modeling electrically large structures with high phase accuracy a new fourth order finite difference time domain fdtd scheme has been developed that exhibits extremely low phase errors at low grid resolutions compared to the conventional fdtd scheme moreover this new scheme is capable of combining with the standard yee 1966 scheme to produce a stable hybrid algorithm the problem of wave propagation through a building is simulated using this new hybrid algorithm to demonstrate the large savings in computing resources it could afford with this new development the fdtd method can now be used to successfully model structures that are thousands of wavelengths large using the present day computer technology

fully hierarchical divergence conforming basis functions on tetrahedral cells with applications a new set of hierarchical divergence conforming vector basis functions on curvilinear tetrahedrons is presented the basis can model both mixed and full order polynomial spaces to arbitrary order as defined by raviart and thomas and nedelec solenoidal and non solenoidal components are separately represented on the element except in the case of the mixed first order space for which a decomposition procedure on the global mesh wide level is presented therefore the hierarchical aspect of the basis can be made to extend down to zero polynomial order the basis can be used to model divergence conforming quantities such as electromagnetic flux and current density fluid velocity etc within numerical methods such as the finite element method fem or integral equation based methods the basis is ideally suited to p adaptive analysis the paper concludes with two example applications the first is the fem based solution of the linearized acoustic vector wave equation where it is shown how the decomposition into solenoidal components and their complements can be used to stabilize the method at low frequencies the second is the solution of the electric field volume integral equation for electromagnetic scattering analysis where the benefits of the decomposition are again demonstrated

a dispersion minimizing compact finite difference scheme for the 2d helmholtz equation in this paper we present a dispersion minimizing compact finite difference scheme for solving the 2d helmholtz equation which is a fourth order scheme the error between the numerical wavenumber and the exact wavenumber is analyzed and a refined choice strategy based on minimizing the numerical dispersion is proposed for choosing weight parameters numerical results are presented to demonstrate the efficiency and accuracy of the compact finite difference scheme with refined parameters

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

bifractality of the devil s staircase appearing in the burgers equation with brownian initial velocity it is shown that the inverse lagrangian map for the solution of the burgers equation in the inviscid limit with brownian initial velocity presents a bifractality phase transition similar to that of the devil s staircase for the standard triadic cantor set both heuristic and rigorous derivations are given it is explained why artifacts can easily mask this phenomenon in numerical simulations

high order calderon preconditioned time domain integral equation solvers two high order accurate calderon preconditioned time domain electric field integral equation tdefie solvers are presented in contrast to existing calderon preconditioned time domain solvers the proposed preconditioner allows for high order surface representations and current expansions by using a novel set of fully localized high order div and quasi curl conforming dqcc basis functions numerical results demonstrate that the linear systems of equations obtained using the proposed basis functions converge rapidly regardless of the mesh density and of the order of the current expansion

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

optimal modeling parameters for higher order mom sie and fem mom electromagnetic simulations general guidelines and quantitative recipes for adoptions of optimal higher order parameters for computational electromagnetics cem modeling using the method of moments and the finite element method are established and validated based on an exhaustive series of numerical experiments and comprehensive case studies on higher order hierarchical cem models of metallic and dielectric scatterers the modeling parameters considered are electrical dimensions of elements h refinement polynomial orders of basis functions p refinement orders of gauss legendre integration formulas integration accuracy and geometrical curvature orders of elements in the model the goal of the study which is the first such study of higher order parameters in cem is to reduce the dilemmas and uncertainties associated with the great modeling flexibility of higher order elements basis and testing functions and integration procedures this flexibility is the principal advantage but also the greatest shortcoming of the higher order cem and to ease and facilitate the decisions to be made on how to actually use them by both cem developers and practitioners

high order finite difference methods for the helmholtz equation high order finite difference methods for solving the helmholtz equation are developed and analyzed in one and two dimensions on uniform grids the standard pointwise representation has a second order accurate local truncation error we also study two schemes which have a fourth order accurate local truncation error one of the high order schemes is based on generalizations of the pade approximation the second scheme is based on high order approximation to the derivative calculated from the helmholtz equation itself a symmetric high order representation is developed for a neumann boundary condition numerical results are presented on model problems approximated with the developed schemes

double higher order large domain volume surface integral equation method for analysis of composite wire plate dielectric antennas and scatterers a novel double higher order large domain galerkin type method of moments based on higher order geometrical modeling and higher order current modeling is proposed for analysis of composite dielectric and metallic radiation scattering structures combining the volume integral equation vie approach for dielectric parts and the surface integral equation sie approach for metallic parts of the structure the technique employs lagrange type interpolation generalized hexahedra and quadrilaterals of arbitrary geometrical mapping orders for the approximation of geometry and hierarchical divergence conforming polynomial vector basis functions of arbitrary expansion orders for the approximation of currents within the elements the double higher order vsie vie sie method is extensively validated and evaluated against the analytical solutions and the numerical results obtained by alternative higher order methods

analysis of transient scattering from composite arbitrarily shaped complex structures a time domain surface integral equation approach based on the electric field formulation is utilized to calculate the transient scattering from both conducting and dielectric bodies consisting of arbitrarily shaped complex structures the solution method is based on the method of moments mom and involves the modeling of an arbitrarily shaped structure in conjunction with the triangular patch basis functions an implicit method is described to solve the coupled integral equations derived utilizing the equivalence principle directly in the time domain the usual late time instabilities associated with the time domain integral equations are avoided by using an implicit scheme detailed mathematical steps are included along with representative numerical results

numerical evaluation of singular and near singular potential integrals a simple and efficient numerical procedure using a singularity cancellation scheme is presented for evaluating singular and near singular potential integrals with 1 r singularities the procedure not only has several advantages over singularity subtraction methods but also improves on some aspects of other singularity cancellation methods such as polar and duffy transformations a theoretical analysis is presented for triangles quadrilaterals tetrahedrons bricks and prisms and numerical results are presented for triangles and prisms

an accurate broad band method of moments using higher order basis functions and tree loop decomposition a method is presented that combines the low frequency accuracy of tree loop decomposition with higher order basis functions to achieve an electric field integral equation based method of moments that is accurate over a wide band of frequencies in general the use of higher order divergence conforming bases introduces more solenoidal degrees of freedom than the standard rao wilton glisson zeroth order bases these solenoidal degrees of freedom are counted and identified and an efficient method to construct the required higher order loops is presented numerical results demonstrate that the technique allows the use of a single mesh to acquire scattering results over a wide band of frequencies using arbitrary order basis functions

hyperbolic asymptotics in burgers turbulence and extremal processes large time asymptotics of statistical solutionu t x 1 2 of the burgers equation 1 1 is considered where x l x is a stationary zero mean gaussian process depending on a large parameterl 0 so that xi l x sim sigma l eta x l l to infty where sigma l l 2 2 log l 1 2 and x is a given standardized stationary gaussian process we prove that asl the hyperbolicly scaled random fieldsu l2t l2x converge in distribution to a random field with saw tooth trajectories defined by means of a poisson process on the plane related to high fluctuations of x which corresponds to the zero viscosity solutions at the physical level of rigor such asymptotics was considered before by gurbatov malakhov and saichev 1991

comparison of a class of subdomain and entire domain basis functions automatically satisfying kcl accuracy number of unknowns and cpu time are compared for piecewise linear subdomain basis functions and polynomial entire domain basis functions both types of functions automatically satisfy a continuity equation at wire ends and junctions according to kirchoff s current law kcl the relative root mean square rms current deviation is chosen as the error metric an electrically short scatterer a crossed wire scatterer and an electrically long scatterer are used for comparison currents are obtained by solving the electric field integral equation efie by means of the galerkin method it was shown that in most cases for the same accuracy required the entire domain approximation uses three to five times less numbers of unknowns and 10 100 times less cpu time than the subdomain approximation generally such efficiency is achieved by using entire domain expansions the order of which is up to n 5 and cannot be significantly improved by using higher order expansions

time domain bie analysis of large three dimensional electromagnetic scattering problems a time domain boundary integral equation bie solution of the magnetic field integral equation mfie for large electromagnetic scattering problems is presented it employs isoparametric curvilinear quadratic elements to model fields geometry and time dependence eliminating staircasing problems the approach is implicit which seems to provide both stability and permits arbitrary local mesh refinement to model geometrically difficult regions without the significant cost penalty explicit methods suffer error dependence on discretization is investigated accurate results are obtained with as few as five nodes per wavelength the performance both on large scatterers and on low radar cross section rcs scatterers is demonstrated including the six wavelength nasa almond two spheres a thirteen wavelength missile and a high q cavity

double higher order method of moments for surface integral equation modeling of metallic and dielectric antennas and scatterers a novel double higher order galerkin type method of moments based on higher order geometrical modeling and higher order current modeling is proposed for surface integral equation analysis of combined metallic and dielectric antennas and scatterers of arbitrary shapes the technique employs generalized curvilinear quadrilaterals of arbitrary geometrical orders for the approximation of geometry metallic and dielectric surfaces and hierarchical divergence conforming polynomial vector basis functions of arbitrary orders for the approximation of electric and magnetic surface currents within the elements the geometrical orders and current approximation orders of the elements are entirely independent from each other and can be combined independently for the best overall performance of the method in different applications the results obtained by the higher order technique are validated against the analytical solutions and the numerical results obtained by low order moment method techniques from literature the examples show excellent accuracy flexibility and efficiency of the new technique at modeling of both current variation and curvature and demonstrate advantages of large domain models using curved quadrilaterals of high geometrical orders with basis functions of high current approximation orders over commonly used small domain models and low order techniques the reduction in the number of unknowns is by an order of magnitude when compared to low order solutions