interpolation of electromagnetic radiated fields over a plane by nonuniform samples an iterative algorithm is developed which allows the interpolation of two dimensional band limited functions from nonuniformly distributed samples such a scheme is based on optimal sampling expansions of central type and is applied to the interpolation of electromagnetic fields over a plane the algorithm can be applied whenever the average sample s density is higher than the nyquist one and there is a one to one correspondence between the nonuniform samples and a lattice of the uniform ones which associates at each uniform sampling point the nearest nonuniform sampling point as compared with the optimal linear estimation algorithm the procedure is computationally much more efficient many numerical simulations have assessed the rapid convergence of the method as well as its stability with respect to both absolute and relative errors in the data

a sparse data fast fourier transform sdfft a multilevel algorithm that efficiently fourier transforms sparse spatial data to sparse spectral data with controllable error is presented the algorithm termed sparse data fast fourier transform sdfft is particularly useful for signal processing applications where only part of the k space is to be computed regardless of whether it is a regular region like an angular section of the ewald sphere or it consists of completely arbitrary points scattered in the k space in addition like the various nonuniform fast fourier transforms the o nlogn algorithm can deal with a sparse nonuniform spatial domain in this paper the parabolic reflector antenna problem is studied as an example to demonstrate its use in the computation of far field patterns due to arbitrary aperture antennas and antenna arrays the algorithm is also promising for various applications such as backprojection tomography diffraction tomography and synthetic aperture radar imaging

a conservative fourier pseudo spectral method for the nonlinear schrodinger equation a fourier pseudo spectral method that conserves mass and energy is developed for a two dimensional nonlinear schrodinger equation by establishing the equivalence between the semi norm in the fourier pseudo spectral method and that in the finite difference method we are able to extend the result in ref 56 to prove that the optimal rate of convergence of the new method is in the order of o n r 2 in the discrete l 2 norm without any restrictions on the grid ratio where n is the number of modes used in the spectral method and is the time step size a fast solver is then applied to the discrete nonlinear equation system to speed up the numerical computation for the high order method numerical examples are presented to show the efficiency and accuracy of the new method

a canonical transformation relating the lagrangian and eulerian description of ideal hydrodynamics abstract we derive by means of a canonical transformation the variational principle of seliger and whitham for the eulerian description of ideal hydrodynamics from the more familiar variational principle that yields the equations of motion of an ideal fluid in the lagrangian description

linear pdes and numerical methods that preserve a multisymplectic conservation law multisymplectic methods have recently been proposed as a generalization of symplectic ode methods to the case of hamiltonian pdes their excellent long time behavior for a variety of hamiltonian wave equations has been demonstrated in a number of numerical studies a theoretical investigation and justification of multisymplectic methods is still largely missing in this paper we study linear multisymplectic pdes and their discretization by means of numerical dispersion relations it is found that multisymplectic methods in the sense of bridges and reich phys lett a 284 2001 pp 184 193 and reich j comput phys 157 2000 pp 473 499 such as gauss legendre runge kutta methods possess a number of desirable properties such as nonexistence of spurious roots and conservation of the sign of the group velocity a certain cfl type restriction on delta t delta x might be required for methods higher than second order in time it is also demonstrated by means of the explicit midpoint method that multistep methods may exhibit spurious roots in the numerical dispersion relation for any value of delta t delta x despite being multisymplectic in the sense of discrete variational mechanics j e marsden g p patrick and s shkoller commun math phys 199 1999 pp 351 395

stability of nonlinear elliptic systems with distributed parameters and variable boundary data in this paper nonlinear partial differential equations of the elliptic type with the dirichlet boundary data are investigated some sufficient conditions under which the solutions of considered equations depend continuously on parameters and boundary conditions are proved the proofs of main results are based on variational methods in the final part of the paper we give a short survey of the results and methods related to the question of stability of the boundary value problems

on the spatial bandwidth of scattered fields it is shown that the scattered fields are almost space bandlimited functions the effective bandwidth w is introduced and evaluated for a very general scattering system as well as the error made using functions bandlimited to w w for representing the scattered field the effective bandwidth is very simply related to the maximum dimension of the scattering system the error drops to negligible values for modest increases of w compared to w in the case of large scatterers important consequences of the above general results are finally stressed

an energy preserving crank nicolson galerkin spectral element method for the two dimensional nonlinear schrodinger equation abstract a crank nicolson galerkin spectral element method for solving the nonlinear schrodinger nls equation in two dimensions is proposed in this paper our key idea is twofolds first the 2d nls equation is rewritten as an infinite dimensional hamiltonian pde and the hamiltonian pde is discreted by using the galerkin spectral element gse method in space second we cast the resulted odes into a finite dimensional canonical hamiltonian system and discrete the system by using the crank nicolson cn method the relay leads to a fully discretized and energy preserved scheme without grid ratio restrictions the order of convergence of our new method is o 2 h 2 if the discrete l 2 norm is employed the fast fourier transform and the matrix diagonalization method are applied to the new method to increase computing efficiency numerical examples are given to further illustrate the conservation properties and convergence of the energy preserving scheme constructed

two classes of linearly implicit local energy preserving approach for general multi symplectic hamiltonian pdes abstract two classes of efficient and robust schemes are proposed for the general multi symplectic hamiltonian systems using the invariant energy quadratization ieq approach the schemes are linear second order accurate local energy preserving and preserve the global energy they are not restricted to specific forms of the nonlinear part of the state function and only require solving linear equations at each time step we applied the new schemes to various multi symplectic hamiltonian pdes to demonstrate their effectiveness computational efficiency and accuracy

magnitude of the truncation error in sampling expansions of band limited signals bounds are established for the truncation error in the sampling theorem for signals with radian bandwidth w w 0 the assumptions are either differenfiability conditions upon the fourier transform of the signal or on the rate of decay of the first derivative of the signal itself finally our theorems are applied to two new examples worked out in detail

multi symplectic runge kutta collocation methods for hamiltonian wave equations a number of conservative pdes like various wave equations allow for a multi symplectic formulation which can be viewed as a generalization of the symplectic structure of hamiltonian odes we show that gauss legendre collocation in space and time leads to multi symplectic integrators i e to numerical methods that preserve a symplectic conservation law similar to the conservation of symplecticity under a symplectic method for hamiltonian odes we also discuss the issue of conservation of energy and momentum since time discretization by a gauss legendre method is computational rather expensive we suggest several semi explicit multi symplectic methods based on gauss legendre collocation in space and explicit or linearly implicit symplectic discretizations in time

algorithmic framework for group analysis of differential equations and its application to generalized zakharov kuznetsov equations abstract in this paper we explain in more details the modern treatment of the problem of group classification of systems of partial differential equations pdes from the algorithmic point of view more precisely we revise the classical lie algorithm of construction of symmetries of differential equations describe the group classification algorithm and discuss the process of reduction of systems of pdes to systems of equations with smaller number of independent variables in order to construct invariant solutions the group classification algorithm and reduction process are illustrated by the example of the generalized zakharov kuznetsov gzk equations of form u t f u x x x g u x y y h u x 0 as a result a complete group classification of the gzk equations is performed and a number of new interesting nonlinear invariant models which have non trivial invariance algebras are obtained lie symmetry reductions and exact solutions for two important invariant models i e the classical and modified zakharov kuznetsov equations are constructed the algorithmic framework for group analysis of differential equations presented in this paper can also be applied to other nonlinear pdes

the sampling window corresp the concept of the sampling window is introduced for the central interpolation of finite energy band limited functions the sampling window does not increase the rate of convergence of the truncation error series as do various convergence factors but does significantly reduce the truncation error bound

multisymplectic geometry local conservation laws and fourier pseudospectral discretization for the good boussinesq equation the multisymplectic geometry for the good boussinesq equation is presented in this paper the multisymplectic form and the local energy and momentum conservation laws are derived directly from the variational principle the multisymplectic hamiltonian formulation is also presented based on the multisymplectic formulation we investigate the corresponding multisymplectic fourier pseudospectral discretization

phaseless antenna measurement on non redundant sample points via leith upatnieks holography a scheme for reducing the number of sample points in phaseless acquisition based on leith upatnieks holography is presented an appropriate choice of the phase of the reference field enables to drastically decrease the sampling rate by acquiring the so called reduced field instead of the real radiated field the reference phase changes are enabled by applying the recent developments on the synthesis of the reference wave moreover optimal sampling interpolation can be applied to the retrieved field so that the real field can be computed at any point on the acquisition surface the method is illustrated by means of multiple simulations for observation domains such as straight lines azimuthal circles as well as for a plane polar acquisition scheme

multisymplectic geometry variational integrators and nonlinear pdes this paper presents a geometric variational approach to continuous and discrete mechanics and field theories using multisymplectic geometry we show that the existence of the fundamental geometric structures as well as their preservation along solutions can be obtained directly from the variational principle in particular we prove that a unique multisymplectic structure is obtained by taking the derivative of an action function and use this structure to prove covariant generalizations of conservation of symplecticity and noether s theorem natural discretization schemes for pdes which have these important preservation properties then follow by choosing a discrete action functional in the case of mechanics we recover the variational symplectic integrators of veselov type while for pdes we obtain covariant spacetime integrators which conserve the corresponding discrete multisymplectic form as well as the discrete momentum mappings corresponding to symmetries we show that the usual notion of symplecticity along an infinite dimensional space of fields can be naturally obtained by making a spacetime split all of the aspects of our method are demonstrated with a nonlinear sine gordon equation including computational results and a comparison with other discretization schemes

exact sampling approach for reflector antennas analysis a new method for computing the far field of a possibly shaped or deformed reflector antenna based on a sampling representation of the radiation integral is presented using a projection technique the required samples can be efficiently computed using a two dimensional fast fourier transform fft whereas the radiated field is reconstructed via standard sampling expansion numerical examples and computation time analysis are reported showing the effectiveness of the approach particularly for deformed reflectors and or plane cut field evaluation

the truncation error in the application of sampling series to electromagnetic problems truncation error interpolation methods used in the evaluation and representation of scattered fields are discussed three main points are considered the choice of the sampling point lattice the choice of the type of sampling expansion and the determination of the number of terms of the sampling series that must be retained to insure a negligible truncation error the aim is to provide a clear reference frame for the applications which allow the optimal choice for the reconstruction algorithm the cases of bounded and square integrable band limited functions are separately dealt with in order to exploit the properties of each method a general analysis of the multidimensional truncation error is carried out no particular assumption on the sampling expansion is made except for that concerning the use of a rectangular sampling lattice simple and effective expressions relating the multidimensional error to the corresponding one dimensional bounds are derived the one dimensional error is then analyzed in detail with reference to the most relevant case of central interpolation by very simple methods error bounds for all the sampling expansions explicitly considered in the literature are derived in a systematic way expressions providing the values of the sampling rate and number of retained samples which minimize the overall computer time and memory requirement are derived some numerical examples are presented and briefly discussed

a multisymplectic pseudospectral method for seismic modeling seismic modeling is a foundation of geophysical date processing a multisymplectic pseudospectral method for seismic modeling is presented in this paper this method takes the multisymplectic structure of the scalar wave equation into account and possesses discrete multisymplectic conservation laws numerical experiments demonstrate the remarkable capability of this new method for long time simulations

functionally weighted lagrange interpolation of band limited signals from nonuniform samples a modification of the conventional lagrange interpolator is proposed in this paper that allows one to approximate a band limited signal from its own nonuniform samples with high accuracy the modification consists in applying the lagrange method to the signal but pre multiplied by a fixed function and then solving for the desired signal value its efficiency lies in the fact that the fixed function is independent of the sampling instants it is shown in this paper that the function can be selected so that the interpolation error decreases exponentially with the number of samples for the case in which the sampling instants have a maximum deviation from a uniform grid this paper includes a low complexity recursive implementation of the method its accuracy is validated in the numerical examples by comparison with several interpolators in the literature and by deriving upper and lower bounds for its maximum error

cylindrical quasi solitons of the zakharov kuznetsov equation abstract evolutions and interactions of two dimensional solitary waves of the zakharov kuznetsov equation are investigated numerically formations of cylindrical bell shaped pulses are observed in the initial value problems a single bell shaped pulse propagates stably without any deformation like a soliton two similar pulses exchange their amplitudes without merging and two dissimilar ones undergo overtaking collision after a collision of two pulses the strong pulse becomes somewhat stronger and the weak one becomes weaker with radiation of ripples so that the collision process is slightly inelastic generated ripples are very small for center to center collision of similar pulses the cylindrically symmetric pulses of the zakharov kuznetov equation are thus found to behave approximately soliton like some properties of collision process are interpreted in terms of the conservation laws

on the preservation of phase space structure under multisymplectic discretization in this paper we explore the local and global properties of multisymplectic discretizations based on finite differences and fourier spectral approximations multisymplectic ms schemes are developed for two benchmark nonlinear wave equations the sine gordon and nonlinear schrodinger equations we examine the implications of preserving the ms structure under discretization on the numerical scheme s ability to preserve phase space structure as measured by the nonlinear spectrum of the governing equation we find that the benefits of multisymplectic integrators include improved resolution of the local conservation laws dynamical invariants and complicated phase space structures

modeling the scalar wave equation with nystrom methods high accuracy numerical schemes for modeling of the scalar wave equation based on nystrom methods are developed in this paper space is discretized by using the pseudospectral algorithm for the time discretization nystrom methods are used a fourth order symplectic nystrom method with pseudospectral spatial discretization is presented this scheme is compared with a commonly used second order scheme and a fourth order nonsymplectic nystrom method for a typical time step size the second order scheme exhibits spatial dispersion errors for long time simulations while both fourth order schemes do not suffer from these errors numerical comparisons show that the fourth order symplectic algorithm is more accurate than the fourth order nonsymplectic one the capability of the symplectic nystrom method in approximately preserving the discrete energy for long time simulations is also demonstrated

sampling method using prefiltered band limited green s functions for the solution of electromagnetic integral equations in the context of far field scattering the spatial spectrum of fields or currents on a scatterer may be approximated as band limited on this premise one can greatly simplify the computation of radiation integrals by eliminating in advance those spectral components of the green s function not expected to be present in the fields this approach enables one to dispense with the notion of basis functions and instead represent fields and currents as well as the impedance matrix representing the green s function entirely in terms of a sparse set of samples thereby obviating the need for integration moreover by taking full advantage of the band limited character of the solution this sampling theoretical approach yields an efficient representation with a minimal number of degrees of freedom the method was explored for scattering from two dimensional perfect electric conductors including ellipsoidal cylinders and flat strips in both polarizations using uniform as well as nonuniform sampling and yielded surprisingly good results

optimal error estimate of a linear fourier pseudo spectral scheme for two dimensional klein gordon schrodinger equations abstract the focus of this paper is on the optimal error bounds of a fourier pseudo spectral conservative scheme for solving the 2 dimensional nonlinear klein gordon schrodinger equations the proposed fourier pseudo spectral scheme not only conserves the mass and energy in the discrete level but also is efficient in practical computation because only two linear systems need to be solved at each time step based on the equivalence between the semi norm derived by the fourier pseudo spectral method and that by the finite difference method the pseudo spectral solution of the proposed scheme is proved strictly to be bounded and convergent with the order of o n r 2 in the discrete l 2 norm where n is the number of nodes and is the time step size some numerical experiments are carried out to validate the theoretical analysis

on nonparaxial nonlinear schrodinger type equations abstract in this paper the one dimensional nonparaxial nonlinear schrodinger equation is considered this was proposed as an alternative to the classical nonlinear schrodinger equation in those situations where the assumption of paraxiality may fail the paper contributes to the mathematical properties of the equation in a two fold way first some theoretical results on linear well posedness hamiltonian and multi symplectic formulations are derived then we propose to take into account these properties in order to deal with the numerical approximation in this sense different numerical procedures that preserve the hamiltonian and multi symplectic structures are discussed and illustrated with numerical experiments

multi symplectic methods for the ito type coupled kdv equation abstract in this paper we find that the ito type coupled kdv equation can be written as a multi symplectic hamiltonian partial differential equation pde then multi symplectic fourier pseudospectral method and multi symlpectic wavelet collocation method are constructed for this equation in the numerical experiments we show the effectiveness of the proposed methods some comparisons between the proposed methods are also made with respect to global conservation properties

numerical methods for hamiltonian pdes the paper provides an introduction and survey of conservative discretization methods for hamiltonian partial differential equations the emphasis is on variational symplectic and multi symplectic methods the derivation of methods as well as some of their fundamental geometric properties are discussed basic principles are illustrated by means of examples from wave and fluid dynamics

on multisymplecticity of sinc gauss legendre collocation discretizations for some hamiltonian pdes abstract in this paper a new multisymplectic scheme based on sinc collocation discretization in space and gauss legendre collocation discretization in time for some hamiltonian pdes is developed the scheme preserves the multisymplectic geometry structure exactly by satisfying the discrete multisymplectic conservation law moreover the discrete energy and momentum conservative properties of the multisymplectic integrator are also discussed in order to testify the superiority of the multisymplectic method it is applied to nonlinear dirac equation numerical experiments are given to illustrate the accuracy of the approximation and the conserved quantities

generalized multi symplectic integrators for a class of hamiltonian nonlinear wave pdes nonlinear wave equations such as burgers equation and compound kdv burgers equation are a class of partial differential equations pdes with dissipation in hamiltonian space the numerical method of which plays an important role in complex fluid analysis based on the multi symplectic idea a new theoretical framework named generalized multi symplectic integrator for a class of nonlinear wave pdes with small damping is proposed in this paper the generalized multi symplectic formulation is introduced and a twelve point generalized multi symplectic scheme which satisfies two discrete modified conservation laws approximately as well as the local momentum conservation law accurately is constructed to solve the first order pdes that derived from the compound kdv burgers equation to test the generalized multi symplectic scheme several numerical experiments on the travelling front solution are carried out the results of which imply that the generalized multi symplectic scheme can simulate the travelling front solution accurately and satisfy the modified conservation laws well when step sizes and the damping parameter satisfy the inequality 41 it is more remarkable that the scheme 36 can be used to capture the shock wave structure of the compound kdv burgers equation within one data point which can further illustrate the good structure preserving property of the generalized multi symplectic scheme 36 from the results of this paper we can conclude that similar to the multi symplectic scheme the generalized multi symplectic scheme also has two remarkable advantages the excellent long time numerical behavior and the good conservation property for the existing of the excellent numerical properties the generalized multi symplectic method can be used to exposit some specific phenomena in the complex fluid

the multi symplectic fourier pseudospectral method for solving two dimensional hamiltonian pdes in this paper the multi symplectic fourier pseudospectral msfp method is generalized to solve two dimensional hamiltonian pdes with periodic boundary conditions using the fourier pseudospectral method in the space of the two dimensional hamiltonian pde 2d hpde the semi discrete system obtained is proved to have semi discrete multi symplectic conservation laws and a global symplecticity conservation law then the implicit midpoint rule is employed for time integration to obtain the msfp method for the 2d hpde the fully discrete multi symplectic conservation laws are also obtained in addition the proposed method is applied to solve the zakharov kuznetsov zk equation and the kadomtsev petviashvili kp equation numerical experiments on soliton solutions of the zk equation and the kp equation show the high accuracy and effectiveness of the proposed method

a survey on symplectic and multi symplectic algorithms we simply review symplectic and multi symplectic algorithms in the paper these algorithms have many superiorities to general non symplectic algorithms such as they have long time behavior they can preserve the symplectic geometry structure of the hamiltonian system exactly and they can simulate original physical phenomena well some commonly used methods to construct symplectic and multi symplectic schemes are simply surveyed numerical examples are also enumerated to illustrate their performances

existence of optimal controls via continuous dependence on parameters abstract we consider a system of elliptic equations and derive sufficient conditions for solutions of the system to depend continuously on parameters these conditions are employed to obtain an existence result for an optimal control problem

energy conservation issues in the numerical solution of the semilinear wave equation in this paper we discuss energy conservation issues related to the numerical solution of the semilinear wave equation as is well known this problem can be cast as a hamiltonian system that may be autonomous or not depending on the prescribed boundary conditions we relate the conservation properties of the original problem to those of its semi discrete version obtained by the method of lines subsequently we show that the very same properties can be transferred to the solutions of the fully discretized problem obtained by using energy conserving methods in the hbvms hamiltonian boundary value methods class similar arguments hold true for different types of hamiltonian partial differential equations e g the nonlinear schrodinger equation

optimal reconstruction of bandlimited bounded signals a formula is presented for restoring bounded bandlimited signals from a finite number of samples the restoration is linear in the samples used and the error bound of the method is the theoretical minimum

on the degrees of freedom of scattered fields starting from the observation that fields differing less than a prescribed error cannot be resolved as distinct entities the degrees of freedom of the scattered field are introduced and then computed the degrees of freedom are shown to be practically equal to the nyquist number appropriate to the effective spatial bandwidth of the scattered field and to the extension of the observation domain accordingly a finite number of elements of information can be used to determine the scattered field to a prescribed approximation error it is also shown that the field representation can be made in terms of field values and simple sampling functions provided that a marginal increase in the approximation error is tolerated the results not only completely justify the use of sampling interpolation for representing scattering fields but also demonstrate that such representation is practically an optimal one an algorithm for the reconstruction of scattered fields given the maximum allowed error is then produced

a note on multisymplectic fourier pseudospectral discretization for the nonlinear schrodinger equation based on the multisymplectic fourier pseudospectral scheme for the nonlinear schrodinger nls equation we investigate some discrete properties corresponding to local conservation laws of the original equation the discrete normal conservation law is proved and the error estimation of local and global energy conservation laws are also obtained numerical experiments for cubic nls equation are provided to demonstrate the consistency between the theoretical analysis and the numerical results

multi symplectic quasi interpolation method for hamiltonian partial differential equations abstract in this paper we propose a multi symplectic quasi interpolation method for solving multi symplectic hamiltonian partial differential equations based on the method of lines we first discretize the multi symplectic pdes using quasi interpolation method and then employ appropriate time integrators to obtain the full discrete system the local conservation properties including multi symplectic conservation laws energy conservation laws and momentum conservation laws are discussed in detail for illustration we provide two concrete examples the nonlinear wave equation and the nonlinear schrodinger equation the salient feature of our multi symplectic quasi interpolation method is that it is valid both on uniform grids and nonuniform grids the numerical results show the good accuracy and excellent conservation properties of the proposed method

convolution based trigonometric interpolation of band limited signals this paper presents a method to obtain a trigonometric polynomial that accurately interpolates a given band limited signal from a finite sequence of samples the polynomial delivers accurate approximations in the range covered by the sequence except for a short frame close to the range limits besides its accuracy increases exponentially with the frame width the method is based on using a band limited window in order to reduce the truncation error of a convolution series it is shown that the polynomial can be efficiently constructed and evaluated using algorithms designed for the discrete fourier transform dft specifically two basic procedures are presented one based on the fast fourier transform fft and another based on a recursive update algorithm for the short time fft the paper contains three applications the first is a variable fractional delay vfd filter which consists of a short time fft combined with the evaluation of a trigonometric polynomial this filter has low complexity and can be implemented using cordic rotations the second is the interpolation of nonuniform fourier summations where the proposed method eliminates the need to interpolate any kernel sample finally the third can be viewed as a generalization of the fft convolution algorithm and makes it possible to interpolate the output of an finite impulse response fir filter efficiently

a new method for avoiding the truncation error in near field antennas measurements a new method to avoid the truncation error in antenna near field measurements is presented the approach relies on the concept of information content of the field according to this point of view the truncation problem is solved by picking up the information that is lost due to the finite size of scanning area in points of the space reachable by the measurement system the method can be applied to any scanning geometry including the planar and cylindrical ones whenever the set up allows to vary the distance between the antenna under test and the probe during the scanning procedure application of the method to cylindrical near field scanning is numerically investigated assessing the effectiveness of the proposed technique

explicit multi symplectic methods for klein gordon schrodinger equations in this paper we propose explicit multi symplectic schemes for klein gordon schrodinger equation by concatenating suitable symplectic runge kutta type methods and symplectic runge kutta nystrom type methods for discretizing every partial derivative in each sub equation it is further shown that methods constructed in this way are multi symplectic and preserve exactly the discrete charge conservation law provided appropriate boundary conditions in the aim of the commonly practical applications a novel 2 order one parameter family of explicit multi symplectic schemes through such concatenation is constructed and the numerous numerical experiments and comparisons are presented to show the efficiency and some advantages of the our newly derived methods furthermore some high order explicit multi symplectic schemes of such category are given as well good performances and efficiencies and some significant advantages for preserving the important invariants are investigated by means of numerical experiments

multi symplectic spectral discretizations for the zakharov kuznetsov and shallow water equations abstract the time evolution of multi symplectic pdes with periodic boundary conditions in one and two space dimensions is considered we introduce the idea of a multi symplectic fourier transform for systems on periodic domains leading to a semi discretization on fourier space and to the concept of multi symplecticity on fourier space the spatial discretization leads to a discrete lattice model with certain discrete conservation laws imposed on it a discrete wave model a by product of the semi discretization is that it leads automatically to a system of hamiltonian odes in time when truncated we show that the one dimensional shallow water equations and the two dimensional zakharov kuznetsov equation are multi symplectic and derive spectral discretizations for these systems and present numerical experiments

general local energy preserving integrators for solving multi symplectic hamiltonian pdes in this paper we propose and investigate a general approach to constructing local energy preserving algorithms which can be of arbitrarily high order in time for solving hamiltonian pdes this approach is based on the temporal discretization using continuous runge kutta type methods and the spatial discretization using pseudospectral methods or gauss legendre collocation methods the local energy conservation law of our new schemes is analyzed in detail the effectiveness of the novel local energy preserving integrators is demonstrated by coupled nonlinear schrodinger equations and 2d nonlinear schrodinger equations with external fields our new schemes are compared with some classical multi symplectic and symplectic schemes in numerical experiments the numerical results show the remarkable long term behaviour of our new schemes

modeling of the acoustic wave equation with transform methods numerical methods are described for the simulation of wave phenomena with application to the modeling of seismic data two separate topics are studied the first deals with the solution of the acoustic wave equation the second topic treats wave phenomena whose direction of propagation is restricted within 90 degrees from a given axis in the numerical methods developed here the wave field is advanced in time by using standard time differencing schemes on the other hand expressions including space derivative terms are computed by fourier transform methods this approach to computing derivatives minimizes truncation errors another benefit of transform methods becomes evident when attempting to restrict propagation to upward moving waves e g to avoid multiple reflections constraints imposed on the direction of the wave propagation are accomplished most precisely in the wavenumber domain the error analysis of the algorithms shows that truncation errors are due mainly to time discretization such er

conservation laws of semidiscrete canonical hamiltonian equations there are many evolution partial differential equations which can be cast into hamiltonian form conservation laws of these equations are related to one parameter hamiltonian symmetries admitted by the pdes the same result holds for semidiscrete hamiltonian equations in this paper we consider semidiscrete canonical hamiltonian equations using symmetries we find conservation laws for the semidiscretized nonlinear wave equation and schrodinger equation

optimal interpolation of radiated fields over a sphere an optimal sampling interpolation algorithm is developed that allows the accurate recovery of scattered or radiated fields over a sphere from a minimum number of samples using the concept of the field equivalent spatial bandwidth a central interpolation scheme is developed to compute the field in theta phi coordinates starting from its samples the maximum allowable sample spacing and error upper bounds are also rigorously derived several simulated examples of pattern reconstruction are presented for both the cases of field and power pattern interpolation the interpolation error as a function of the retained sample number has been also evaluated and compared with the theoretical upper bounds the algorithm stability versus randomly distributed errors added to the exact samples is demonstrated

note on interpolational basis functions in the method of moments em scattering efficient basis sets for the method of moments may be obtained using quasi localized bandlimited interpolational functions that broadly speaking are defined relative to a mean sampling rate that is adjusted to curvature and proximity to edges thus reflecting the local spatial frequency bandwidth computed scattering data in a number of structures including perfectly conducting circular and elliptical two dimensional cylinders as well as a flat infinite strip indicate that reasonable accuracy can be obtained with an average rate of between 2 5 and 3 basis functions per wavelength average sampling rates need not correspond strictly to the bandwidth of the basis functions and there is considerable latitude with respect to random variation of sampling intervals although each basis function typically extends over several sample points required integrals can be obtained speedily by the use of standard sampling theoretical methods

on structure preserving discontinuous galerkin methods for hamiltonian partial differential equations energy conservation and multi symplecticity abstract in this paper we present and study discontinuous galerkin dg methods for one dimensional multi symplectic hamiltonian partial differential equations we particularly focus on semi discrete schemes with spatial discretization only and show that the proposed dg methods can simultaneously preserve the multi symplectic structure and energy conservation with a general class of numerical fluxes which includes the well known central and alternating fluxes applications to the wave equation the benjamin bona mahony equation the camassa holm equation the korteweg de vries equation and the nonlinear schrodinger equation are discussed some numerical results are provided to demonstrate the accuracy and long time behavior of the proposed methods numerically we observe that certain choices of numerical fluxes in the discussed class may help achieve better accuracy compared with the commonly used ones including the central fluxes

an efficient structure for the design of variable fractional delay filters based on the windowing method a variable fractional delay vfd filter is presented that constitutes an efficient alternative to the usual farrow structure the filter is constructed by multiplying the signal to interpolate by a window with proper truncation properties and then applying the sampling theorem the resulting interpolator involves less arithmetic operations than the existing variants of the farrow structure besides if its coefficients are numerically optimized its performance is close to that of the fir equi ripple filter the final part of the correspondence includes a description of the vfd filter implementation corresponding to this interpolator

error estimates of exponential wave integrator fourier pseudospectral methods for the nonlinear schrodinger equation abstract in this article several exponential wave integrator fourier pseudospectra methods are developed for solving the nonlinear schrodinger equation in two dimensions these numerical methods are based on the fourier pseudospectral discretization for spatial derivative first and then utilizing four types of numerical integration formulas including the gautschi type trapezoidal middle rectangle and simpson rules to approximate the time integral in phase space the resulting numerical methods cover two explicit and an implicit schemes and can be implemented effectively thanks to the fast discrete fourier transform additionally the error estimates of two explicit schemes are established by virtue of the mathematical induction and standard energy method finally extensive numerical results are reported to show the efficiency and accuracy of the proposed new methods and confirm our theoretical analysis

finite difference calculus invariant structure of a class of algorithms for the nonlinear klein gordon equation in a previous work the authors have presented a formalism for deriving systematically invariant symmetric finite difference algorithms for nonlinear evolution differential equations that admit conserved quantities this formalism is herein cast in the context of exact finite difference calculus the algorithms obtained from the proposed formalism are shown to derive exactly from discrete scalar potential functions using finite difference calculus in the same sense as that of the corresponding differential equation being derivable from its associated energy function a conserved quantity a clear ramification of this result is that the derived algorithms preserve certain discrete invariant quantities which are the consistent counterpart of the invariant quantities in the continuous case results on the nonlinear stability of a class of algorithms that are derived using the proposed formalism and that preserve energy or linear momentum are discussed in the context of finite difference calculus some

fast and accurate near field far field transformation by sampling interpolation of plane polar measurements an optimal sampling interpolation algorithm which allows the accurate recovery of plane rectangular near field samples from the knowledge of the plane polar ones is developed this enables the standard near field far field nf ff transformation which takes full advantage of the fast fourier transform fft algorithm to be applied to plane polar scanning the maximum allowable sample spacing is also rigorously derived and it is shown that it can be significantly greater than lambda 2 as the measurement place moves away from the source this allows a remarkable reduction of both measurement time and memory storage requirements the sampling approach is compared with that based on the bivariate lagrange interpolation bli method the sampling reconstruction agrees with the exact results significantly better than the bli in spite of the significantly lower number of required measurements