exact solutions to the acoustic source reconstruction problem in this paper we generalize the recently developed analytical solutions of the radiation modes problem to the determination of closed form expressions for the singular value expansion of a number of integral operators that map the boundary velocity of a baffled planar structure onto the acoustic pressure radiated in far field or intermediate regions exact solutions to this problem involve prolate spheroidal wave functions that correspond to a set of independent distributions with finite spatial support and maximal energy concentration in a given bandwidth of the transform domain a stable solution to the inverse source reconstruction problem is obtained by decomposing the unknown boundary velocity into a number of efficiently radiating singular velocity patterns that correspond to the number of degrees of freedom of the radiated field it is found that the degree of ill posedness of the inverse problem is significantly reduced when considering a hemi circular observation arc with respect to a linear array of sensors by a factor scaling on the small angular aperture subtended by the observation line estimates are derived from the spatial resolution limits that can be achieved in the source reconstruction problem from the dimension of the efficiently radiating subspace

pseudospectral method based on prolate spheroidal wave functions for frequency domain electromagnetic simulations we apply prolate spheroidal wave functions of order zero as basis functions in the pseudospectral method for frequency domain electromagnetic simulation problems like the traditional pseudospectral frequency domain psfd methods based on chebyshev and legendre polynomial series the prolate psfd method yields exponential order of accuracy in terms of the number of samples utilized per wavelength the prolate expansion is superior to the chebyshev and legendre polynomial series by a factor of spl pi 2 in addition the prolate psfd method employs a more uniform spatial grid achieving better resolution near the center of the domain

on hp convergence of prolate spheroidal wave functions and a new well conditioned prolate collocation scheme the first purpose of this paper is to provide further illustrations from both theoretical and numerical perspectives for the nonconvergence of h refinement in hp approximation by the prolate spheroidal wave functions pswfs a surprising convergence property that was first discovered by boyd et al 2013 3 the second purpose is to offer a new basis that leads to prolate collocation systems with condition numbers independent of c n the intrinsic bandwidth parameter and the number of collocation points we highlight that the collocation scheme together with a very practical rule for pairing up c n significantly outperforms the legendre polynomial based method and likewise other jacobi polynomial based methods in approximating highly oscillatory bandlimited functions

detection of attraction domains of non linear systems using bifurcation analysis and lyapunov functions the objective of this paper is to contribute to the problem of the domain of attraction doa of locally stable non linear systems determining exactly the attraction basin is in general a difficult open problem in our proposal the main original idea is to combine tools from bifurcation theory and from lyapunov theory it is believed that this interplay is the most adequate to get a nice picture of the essential objects affecting local stability the viability of all these ideas is confirmed by the successfull application to the furuta pendulum the strategies proposed here offer the advantage of a conceptually appealing approach which obtains a conservative ellipsoidal estimation of the doa this ellipsoid is reasonably good and the computational burden is drastically reduced compared to the pure lyapunov blind search the search is improved when bifurcation information is taken into account the synthesis problem is also considered and synthesis conditions are given in the form of feasible pa

mixed spectral boundary element embedding algorithms for the navier stokes equations in the vorticity stream function formulation an embedding approach based on fourier expansions and boundary integral equations is applied to the vorticity stream function formulation of the navier stokes equations the algorithm only requires efficient solvers of scalar elliptic equations and in an asymptotic version the boundary element method is only needed to solve the laplace equation the capabilities of this embedding method in both its full and asymptotic versions are pointed out by considering the classical problem of the flow between two eccentric cylinders

fast computation of the spectral differentiation by the fast multipole method abstract this paper is concerned with the application of the fast multipole method fmm to the pseudospectral methods the spectral differentiation in pseudospectral methods involves dense matrix vector products which if evaluated directly entail o n 2 operations the core of this work lies in the fact that the fmm leads to a significant speed up in cpu time with a major reduction in the amount of computer memory needed when performing matrix vector products the fmm is exploited to estimate the barycentric weights and the pseudospectral differentiation which reduces the complexity to o n furthermore we show the convergence analysis and present the applications to the second order boundary value problems numerical examples confirm the advantages of the method

a comparison of numerical algorithms for fourier extension of the first second and third kinds the range of fourier methods can be significantly increased by extending a nonperiodic function f x to a periodic function f on a larger interval when f x is analytically known on the extended interval the extension is straightforward when f x is unknown outside the physical interval there is no standard recipe worse still like a radarless aircraft groping through fog the algorithm may wreck on the mountain in fog problem a function f x which is perfectly well behaved on the physical interval may very well have singularities in the extended domain in this article we compare several algorithms for successfully extending a function f x into the fog even when the analytic extension is singular the best third kind extension requires singular value decomposition with iterative refinement but achieves accuracy close to machine precision

on the gibbs phenomenon i recovering exponential accuracy from the fourier partial sum of a nonperiodic analytic function it is well known that the fourier series of an analytic and periodic function truncated after 2n 1 terms converges exponentially with n even in the maximum norm it is also known that if the function is not periodic the rate of convergence deteriorates in particular there is no convergence in the maximum norm although the function is still analytic this is known as the gibbs phenomenon in this paper we show that the first 2n 1 fourier coefficients contain enough information about the function so that an exponentially convergent approximation in the maximum norm can be constructed the proof is a constructive one and makes use of the gegenbauer polynomials c n x it consists of two steps in the first step we show that the first m coefficients of the gegenbauer expansion based on c n x for 0 n m of any l2 function can be obtained within exponential accuracy provided that both and m are proportional to but smaller than n in the second step we construct the gegenbauer expansion based on c n 0 n m from the coefficients found in the first step we show that this series converges exponentially with n provided that the original function is analytic though nonperiodic thus we prove that the gibbs phenomenon can be completely overcome

some comments on fourier analysis uncertainty and modeling investigation of the problem of simultaneously concentrating a function and its fourier transform has led to some interesting special functions that have widespread applications in engineering they provide a means of proving a rigorous version of an engineering folk theorem called the 2wt theorem many generalizations of these ideas seem to possess a similar elegant mathematical structure a brief descriptive review is given of these developments

analytic solutions to the acoustic source reconstruction problem in this paper we generalize the recently developed analytical solutions of the radiation modes problem to the determination of closed form expressions for the singular value expansion of a number

eigenfunctions of the airyʼs integral transform properties numerical computations and asymptotic behaviors abstract this paper is devoted to the study of the spectrum of the integral operator with airy s kernel we provide the reader with some efficient methods of computing the eigenfunctions and the eigenvalues of this operator the first method is based on the use of a differential operator which commutes with this integral operator the second method is based on a gaussian quadrature method applied to the finite airy s transform the asymptotic behavior of the eigenfunctions is studied by the use of a wkb method some numerical examples are given to illustrate results of this work

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

adaptive radial basis function and hermite function pseudospectral methods for computing eigenvalues of the prolate spheroidal wave equation for very large bandwidth parameter asymptotic approximations show that the lowest modes of the prolate spheroidal wave equation are concentrated with an o 1 c length scale where c is the bandwidth parameter of the prolate differential equation accurate computation of the ground state eigenvalue by the long known legendre galerkin method requires roughly 3 8 c legendre polynomials some studies have therefore applied a grid with 20 000 points in conjunction with high order finite differences to achieve c 10 7 here we show that by adaptively applying either hermite functions or gaussian radial basis functions rbfs it is never necessary to use more than eighty degrees of freedom to calculate the lowest dozen eigenvalues of each symmetry class for small c the eigenmodes are not confined to a small portion of the domain 2 2 in latitude but are global we show that by periodizing the basis functions via imbricate series it is possible to apply hermite and rbfs even in the limit c 0 the legendre method is probably a little more efficient in this limit since the prolate functions tend to legendre polynomials in this limit a sideband truncation restricts the discretization to a small block taken from the large hermite galerkin matrix we show that sideband truncation with blocks as small as 5 5 is a very efficient way to compute high order modes in an appendix we prove a rigorous convergence theorem for the periodized hermite expansion

analysis of the truncation errors in the fast multipole method for scattering problems discretisation of the integral equations of acoustic scattering yields large dense systems of linear equations using the fast multipole method an approximate solution to these systems can be computed with a low operation count when implementing the method various infinite sums must be truncated in this paper sharp computable bounds on the errors of these truncations are derived which could form the basis for an automatic selection of truncation length this choice will guarantee a given solution accuracy whilst minimising the operation count of the fast multipole algorithm

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

the linear rational collocation method we introduce the collocation method based on linear rational interpolation for solving general hyperbolic problems prove its stability and its convergence in weighted norms and give numerical examples for its use

fast integral equation methods for rothe s method applied to the isotropic heat equation we present an efficient integral equation approach to solve the forced heat equation u t x email protected x f x u t in a two dimensional multiply connected domain with dirichlet boundary conditions instead of using an integral equation formulation based on the heat kernel we discretize in time first this approach known as rothe s method leads to a non homogeneous modified helmholtz equation that is solved at each time step we formulate the solution to this equation as a volume potential plus a double layer potential and both of these potentials are calculated with available tools accelerated by the fast multipole method for a total of n points in the discretization of the boundary and the domain the total computational cost per time step is o n we demonstrate our approach on the heat equation and the allen cahn equation

accuracy resolution and stability properties of a modified chebyshev method while the chebyshev pseudospectral method provides a spectrally accurate method integration of partial differential equations with spatial derivatives of order m requires time steps of approximately o n 2m for stable explicit solvers theoretically time steps may be increased to o n m with the use of a parameter alpha dependent mapped method introduced by kosloff and tal ezer em j comput phys 104 1993 pp 457 469 our analysis focuses on the utilization of this method for reasonable practical choices for n namely n lesssim 30 as may be needed for two or three dimensional modeling results presented confirm that spectral accuracy with increasing n is possible both for constant alpha hesthaven dinesen and lynov em j comput phys 155 1999 pp 287 306 and for alpha scaled with n alpha sufficiently different from 1 don and solomonoff em siam j sci comput 18 1997 pp 1040 1055 theoretical bounds however show that any realistic choice for alpha in which both resolution and accuracy considerations are imposed permits no more than a doubling of the time step for a stable explicit integrator in time much less than the o n improvement claimed by kosloff and tal ezer on the other hand by choosing alpha carefully it is possible to improve on the resolution of the chebyshev method in particular one may achieve satisfactory resolution with fewer than pi points per wavelength moreover this improvement is noted not only for waves with the minimal resolution but also for waves sampled up to about 8 points per wavelength our conclusions are verified by calculation of phase and amplitude errors for numerical solutions of first and second order one dimensional wave equations specifically while alpha can be chosen such that the mapped method improves the accuracy and resolution of the chebyshev method for practical choices of n it is not possible to achieve both single precision accuracy and gain the advantage of an o n m time step

asymptotic behaviors and numerical computations of the eigenfunctions and eigenvalues associated with the classical and circular prolate spheroidal wave functions abstract for a fixed bandwidth c the classical prolate spheroidal wave functions pswfs n c and the circular ones cpswfs n c form remarkable hilbertian bases for the spaces of fourier and hankel band limited functions with bandwidth c respectively the prolate spheroidal wave functions have already found many applications from various scientific area such as signal processing numerical analysis physics and random matrix theory these concrete applications of the pswfs and cpswfs require the accurate computation of these special functions to this end we develop in this work two approximate procedures the first procedure is based on pushing forward the wkb method and provide accurate approximations of the pswfs and the cpswfs in terms of some classical special functions the second procedure is based on efficient and accurate quadrature methods for the computation of the values and the eigenvalues associated with pswfs and the cpswfs both procedures are valid for small as well as for large values of the frequency band width c also we provide the reader with some numerical examples that illustrate the different results of this work

a new swing up law for the furuta pendulum in this paper the swing up problem for the furuta pendulum is solved applying fradkov s speed gradient sg method to a dimension 4 model of the system the new law is compared with the conventional astrom furuta strategy based on a dimension 2 model a comparative analysis including simulations and experiments whereby the advantages and effectiveness of the new law for swinging the pendulum up are shown is included

hybrid eulerian lagrangian semi implicit time integrators hybrid eulerian lagrangian semi implicit time integrators helsi are presented which use the standard semi implicit formulation as their starting point the advantage of such an approach is that existing models which employ eulerian semi implicit time integrators can easily be augmented to use the helsi approach that is provided that either a series of advection problems can be solved efficiently as in the operator integration factor splitting oifs method or that the grid and underlying numerics allow the construction of efficient search and interpolation stencils typically required of semi lagrangian methods once the advection problem is solved the approach then follows the standard semi implicit approach an important caveat of this approach is that the semi implicit time integrator must be constructed around the implicit backward difference formulas

an instability phenomenon in spectral methods the eigenvalues of chebyshev and legendre spectral differentiation matrices which determine the allowable time step in an explicit time integration are extraordinarily sensitive to rounding errors and other perturbations on a grid of n points per space dimension machine rounding leads to errors in the eigenvalues of size o n 2 this phenomenon may lead to inconsistency between predicted and observed time step restrictions one consequence of it is that spectral differentiation by interpolation in legendre points which has a favorable o n 1 time step restriction for the model problem u t u x in theory is subject to an o n 2 restriction in practice the same effect occurs with chebyshev points for the model problem u t xu x another consequence is that a spectral calculation with a fixed time step may be stable in double precision but unstable in single precision we know of no other examples in numerical computation of this kind of precision dependent stability

accuracy enhancement for higher derivatives using chebyshev collocation and a mapping technique a new method is investigated to reduce the roundoff error in computing derivatives using chebyshev collocation methods by using a grid mapping derived by kosloff and tal ezer and the proper choice of the parameter alpha the roundoff error of the k th derivative can be reduced from o n 2k to o n lge k where epsilon is the machine precision and n is the number of collocation points this drastic reduction of roundoff error makes mapped chebyshev methods competitive with any other algorithm in computing second or higher derivatives with large n several other aspects of the mapped chebyshev differentiation matrix are also studied revealing that the mapped chebyshev methods require much less than pi points to resolve a wave the eigenvalues are less sensitive to perturbation by roundoff error and larger time steps can be used for solving pdes all these advantages of the mapped chebyshev methods can be achieved while maintaining spectral accuracy

traps and snares in eigenvalue calculations with application to pseudospectral computations of ocean tides in a basin bounded by meridians we make several observations about eigenvalue problems using as examples laplace s tidal equations and the differential equation satisfied by the associated legendre functions whatever the discretization only some of the eigenvalues of then dimensional matrix eigenvalue problem will be good approximations to those of the differential equation usuallythen 2 eigenvalues of smallest magnitude for the tidal problem however the good eigenvalues are scattered so our first point is it is important to plot the drift of eigenvalues with changes in resolution we suggest plotting the difference between a low resolution eigenvalue and the nearest high resolution eigenvalue divided by the magnitude of the eigenvalue or the intermodal separation whichever is smaller second as a final safeguard it is important to look at the chebyshev coefficients of the mode we show a numerically computed anti kelvin wave which has little eigenvalue drift but is completely spurious as is obvious from its spectral series third inverting the roles of parameters can drastically modify the spectrum legendre s equation may have either an infinite number of discrete modes or only a handful depending on which parameter is the eigenvalue fourth when the modes are singular but decay to zero at the endpoints as is true of tides a tanh mapping can retrieve the usual exponential accuracy of spectral methods fifth the pseudospectral method is more reliable than deriving a banded galerkin matrix by means of recurrence relations the pseudospectral code is simple to check whereas it is easy to make an untestable mistake with the intricate algebra required for the galerkin method sixth we offer a brief cautionary tale about overlooked modes all these cautions are applicable toallforms of spatial discretization including finite difference and finite element methods however we limit our illustrations to spectral schemes where these difficulties are most easily resolved with a bit of care the pseudospectral method is a very robust and efficient method for solving differential eigenproblems even with singular eigenmodes

spectral methods based on prolate spheroidal wave functions for hyperbolic pdes we examine the merits of using prolate spheroidal wave functions pswfs as basis functions when solving hyperbolic pdes using pseudospectral methods the relevant approximation theory is reviewed and some new approximation results in sobolev spaces are established an optimal choice of the band limit parameter for pswfs is derived for single mode functions our conclusion is that one might gain from using the pswfs over the traditional chebyshev or legendre methods in terms of accuracy and efficiency for marginally resolved broadband solutions

the direct lyapunov method for the stabilisation of the furuta pendulum a nonlinear controller for the stabilisation of the furuta pendulum is presented the control strategy is based on a partial feedback linearisation in a first stage only the actuated coordinate of the furuta pendulum is linearised then the stabilising feedback controller is obtained by applying the lyapunov direct method that is using this method we prove local asymptotic stability and demonstrate that the closed loop system has a large region of attraction the stability analysis is carried out by means of lasalle s invariance principle to assess the controller effectiveness the results of the corresponding numerical simulations are presented

parallel algorithms with local fourier basis abstract we present a nonoverlapping domain decomposition method with local fourier basis applied to a model problem in liquid flames the introduction of domain decomposition techniques in this paper is for numerical and parallel efficiency purposes when one requires a large number of grid points to catch complex structures we obtain then a high order accurate domain decomposition method that allows us to generalize our previous work on the use of local fourier basis to solve combustion problems with nonperiodic boundary conditions m garbey and d tromeur dervout j comput phys 145 316 1998 local fourier basis methodology fully uses the superposition principle to split the searched solution in a numerically computed part and an analytically computed part our present methodology generalizes the israeli et al 1993 j sci comput 8 135 method which applies domain decomposition with local fourier basis to the helmholtz s problem in the present work several new difficulties occur first the problem is unsteady and nonlinear which makes the periodic extension delicate to construct in terms of stability and accuracy second we use a streamfunction biharmonic formulation of the incompressible navier stokes equation in two space dimensions the application of domain decomposition with local fourier basis to a fourth order operator is more difficult to achieve than for a second order operator a systematic investigation of the influence of the method s parameters on the accuracy is done a detail parallel mimd implementation is given we give an a priori estimate that allows the relaxation of the communication between processors for the interface problem treatment results on nonquasi planar complex frontal polymerization illustrate the capability of the method

spectral methods on arbitrary grids stable and spectrally accurate numerical methods are constructed on arbitrary grids for partial differential equations these new methods are equivalent to conventional spectral methods but do not rely on specific grid distributions specifically we show how to implement legendre galerkin legendre collocation and laguerre galerkin methodology on arbitrary grids

large mode number eigenvalues of the prolate spheroidal differential equation we show that the eigenvalues of the prolate spheroidal differential equation of zeroth order g n c where c is the so called bandwidth or ellipticity parameter are well approximated for large mode number n by a single function w in the form g n c email protected c c where c p 2 n 1 2 w is defined implicitly as the root of an algebraic equation e min 1 m 1 2 m 1 2 1 w where e is the usual incomplete integral of the second kind and email protected 2 w with c c c w has a weak singularity at c 1 proportional to c 1 log c 1 plus iterated logarithms we give chebyshev series for w accurate for email protected 0

integral equation methods for stokes flow and isotropic elasticity in the plane we present a class of integral equation methods for the solution of biharmonic boundary value problems with applications to two dimensional stokes flow and isotropic elasticity the domains may be multiply connected and finite infinite or semi infinite in extent our analytic formulation is based on complex variables and our fast multipole based iterative solution procedure requireso n operations wherenis the number of nodes in the discretization of the boundary the performance of the methods is illustrated with several large scale numerical examples

the optimization of convergence for chebyshev polynomial methods in an unbounded domain by using the method of steepest descent i have compared the suitability of three different methods for solving problems in a semi infinite or infinite domain using chebyshev polynomials exponential mappings are uniformyly bad domain truncation and algebraic mapping both work well but each is superior for a different category of problems when the solution is an entire function then domain truncation is best it is always simpler to apply than algebraic mapping and always at least as accurate much more accurate for functions which decay very rapidly for singular functions on the other hand algebraic mapping is better because it is less sensitive to the mapping scale factor l permitting a better compromise in resolving both the singularity and the exponential decay for both types of problems i give simple explicit estimates of both the optimum choise of domain size or mapping factor and of the attainable accuracy for the model entire function exp azk on a semi infinite interval 0 the optimum domain size is l 0 896 na 1k and the smallest attainable error is roughly e n 0 896 k where n is the number of chebyshev polynomals used similar formulas for singular functions handled via algebraic mapping are given in 3 41 to 3 45 below

fast integral equation methods for the modified helmholtz equation we present integral equation methods for the solution to the two dimensional modified helmholtz equation u x email protected email protected x 0 in bounded or unbounded multiply connected domains we consider both dirichlet and neumann problems we derive well conditioned fredholm integral equations of the second kind which are discretized using high order hybrid gauss trapezoid rules our fast multipole based iterative solution procedure requires only o n operations where n is the number of nodes in the discretization of the boundary we demonstrate the performance of our methods on several numerical examples and we show that they have both the ability to handle highly complex geometry and the potential to solve large scale problems

sampling theory approach to prolate spheroidal wavefunctions we use the whittaker shannon sampling theorem to show that the eigenvalue problem for the sinc kernel is equivalent to a discrete eigenvalue problem the well known eigenfunctions namely the prolate spheroidal wavefunctions their corresponding eigenvalues and the orthogonality and completeness properties are determined without invoking the prolate spheroidal differential equation this analysis based on the sampling theorem may be used for calculating the eigenvalues and eigenfunctions of bandlimited kernels in general as we illustrate with an additional example of the sinc2 kernel

the prolate spheroidal wave functions as invariants of the time reversal operator for an extended scatterer in the fraunhofer approximation the decomposition of the time reversal operator known by the french acronym dort is widely used to detect locate and focus on scatterers in various domains such as underwater acoustics medical ultrasound and nondestructive evaluation in the case of point scatterers the theory is well understood the number of nonzero eigenvalues is equal to the number of scatterers and the eigenvectors correspond to the scatterers green s function in the case of extended objects however the formalism is not as simple it is shown here that in the fraunhofer approximation analytical solutions can be found and that the solutions are functions called prolate spheroidal wave functions these functions have been studied in information theory as a basis of band limited and time limited signals they also arise in optics the theoretical solutions are compared to simulation results most importantly the intuition that for an extended objects the number of nonzero eigenvalues is proportional to the number of resol

time dependent hermite galerkin spectral method and its applications a time dependent hermite galerkin spectral method thgsm is investigated in this paper for the nonlinear convection diffusion equations in the unbounded domains the time dependent scaling factor and translating factor are introduced in the definition of the generalized hermite functions ghf as a consequence the thgsm based on these ghf has many advantages not only in theoretical proofs but also in numerical implementations the stability and spectral convergence of our proposed method have been established in this paper the korteweg de vries burgers kdvb equation and its special cases including the heat equation and the burgers equation as the examples have been numerically solved by our method the numerical results are presented and it surpasses the existing methods in accuracy our theoretical proof of the spectral convergence has been supported by the numerical results

numerical perturbative and chebyshev inversion of the incomplete elliptic integral of the second kind abstract the incomplete elliptic integral of the second kind e sin t m 0 t 1 m sin 2 t dt where m 0 1 is the elliptic modulus can be inverted with respect to angle t by solving the transcendental equation e sin t m z 0 we show that newton s iteration t n 1 t n e sin t m z 1 m sin 2 t always converges to t z m e 1 z m within a relative error of less than 10 10 in three iterations or less from the first guess t 0 z m 2 r 2 where defining 1 z e 1 m r 1 m 2 2 and atan 1 m we briefly discuss three alternative initialization strategies homotopy initialization t 0 z m 1 m z 0 mt z m m 1 perturbation series in powers of m and inversion of the chebyshev interpolant of the incomplete elliptic integral although all work well and are general strategies applicable to a very wide range of problems none of these three alternatives is as efficient as the empirical initialization which is completely problem specific this illustrates tai tsun wu s maxim usefulness is often inversely proportional to generality

high order corrected trapezoidal quadrature rules for singular functions a class of quadrature formulae is presented applicable to both nonsingular and singular functions generalizing the classical endpoint corrected trapezoidal quadrature rules while the latter rules are usually derived by means of the euler maclaurin formula their generalizations are obtained as solutions of certain systems of linear algebraic equations a procedure is developed for the construction of very high order quadrature rules applicable to functions with a priori specified singularities and relaxing the requirements on the distribution of nodes the scheme applies not only to nonsingular functions but also to a wide class of functions with monotonic singularities numerical experiments are presented demonstrating the practical usefulness of the new class of quadratures tables of quadrature weights are included for singularities of the form s x x lambda for a variety of values of lambda and s x log x

on spectral approximations in elliptical geometries using mathieu functions we consider in this paper approximation properties and applications of mathieu functions a first set of optimal error estimates are derived for the approximation of periodic functions by using angular mathieu functions these approximation results are applied to study the mathieu legendre approximation to the modified helmholtz equation and helmholtz equation illustrative numerical results consistent with the theoretical analysis are also presented

mappings and accuracy for chebyshev pseudo spectral approximations abstract the effect of mappings on the approximation by chebyshev collocation of functions which exhibit localized regions of rapid variation is studied a general strategy is introduced whereby mappings are adaptively constructed which map specified classes of rapidly varying functions into low order polynomials a particular family of mappings constructed in this way is tested on a variety of rapidly varying functions similar to those occurring in applications it is shown that the mapped function can be approximated much more accurately by chebyshev polynomial approximations than the original function the effect on the approximation of introducing subdomains is also studied the accuracy of the pseudo spectral approximation is very sensitive to the location of the interface although this sensitivity is reduced when mappings are employed within the subdomains

accelerating the convergence of spectral deferred correction methods in the recent paper by dutt greengard and rokhlin a variant of deferred or defect correction methods is presented which couples gaussian quadrature with the picard integral equation formulation of the initial value ordinary differential equation the resulting spectral deferred correction sdc methods have been shown to possess favorable accuracy and stability properties even for versions with very high order of accuracy in this paper we show that for linear problems the iterations in the sdc algorithm are equivalent to constructing a preconditioned neumann series expansion for the solution of the standard collocation discretization of the ode this observation is used to accelerate the convergence of sdc using the gmres krylov subspace method for nonlinear problems the gmres acceleration is coupled with a linear implicit approach stability and accuracy analyses show the accelerated scheme provides an improvement in the accuracy efficiency and stability of the original sdc approach furthermore preliminary numerical experiments show that accelerating the convergence of sdc methods can effectively eliminate the order reduction previously observed for stiff ode systems

stabilization of the furuta pendulum around its homoclinic orbit we present an energy based control approach to control the furuta pendulum a controller is proposed for swinging the pendulum and raise it to its uppermost unstable equilibrium position the passivity properties of the system are used as a guideline in the control strategy the stability analysis is carried out by using a lyapunov technique

nonlinear optimization quadrature and interpolation we present a nonlinear optimization procedure for the design of generalized gaussian quadratures for a fairly broad class of functions while some of the components of the algorithm have been published previously we introduce an improved procedure for the determination of an acceptable initial point for the continuation scheme that stabilizes the newton type process used to find the quadratures the resulting procedure never failed when applied to chebyshev systems for which the existence and uniqueness of generalized gaussian quadratures are well known it also worked for many non chebyshev systems for which the generalized gaussian quadratures are not guaranteed to exist the performance of the algorithm is illustrated with several numerical examples some of the presented quadratures integrate efficiently large classes of singular functions

oval shaped droplet solutions in the saturation process of some pattern formation problems the first stage in the transition from a coarse structure to a fine structure in many pattern formation problems involves the change of a standard geometric object such as a round disc to a less standard geometric object such as an oval shaped set on a generic domain two small oval shaped sets are found as solutions to a nonlocal geometric problem this problem arises as the singular limit of both the ohta kawasaki theory for diblock copolymers and the gierer meinhardt theory for morphogenesis in cell development the two sets have the same center which is a global minimum of the diagonal of the regular part of a green s function this minimum point may be regarded as a kind of center of the domain moreover the second derivatives of the regular part of the green s function define a major axis and a minor axis for the domain one of the oval shaped solutions is stable and aligns itself along the major axis the other oval shaped solution is unstable and is parallel to the minor axis

a numerical framework for singular limits of a class of reaction diffusion problems we present a numerical framework for solving localized pattern structures of reaction diffusion type far from the turing regime we exploit asymptotic structure in a set of well established pattern formation problems to analyze a singular limit model that avoids time and space adaptation typically associated to full numerical simulations of the same problems the singular model involves the motion of a curve on which one of the chemical species is concentrated the curve motion is non local with an integral equation that has a logarithmic singularity we generalize our scheme for various reaction terms and show its robustness to other models with logarithmic singularity structures one such model is the 2d mullins sekerka flow which we implement as a test case of the method we then analyze a specific model problem the saturated gierer meinhardt problem where we demonstrate dynamic patterns for a variety of parameters and curve geometries

an improved estimate of pswf approximation and approximation by mathieu functions abstract in this paper an error estimate of spectral approximations by prolate spheroidal wave functions pswfs with explicit dependence on the bandwidth parameter and optimal order of convergence is derived which improves the existing result in chen et al spectral methods based on prolate spheroidal wave functions for hyperbolic pdes siam j numer anal 43 5 2005 1912 1933 the underlying argument is applied to analyze spectral approximations of periodic functions by mathieu functions which leads to new estimates featured with explicit dependence on the intrinsic parameter

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

total energy shaping control of mechanical systems simplifying the matching equations via coordinate changes total energy shaping is a controller design methodology that achieves asymptotic stabilization of mechanical systems endowing the closed loop system with a lagrangian or hamiltonian structure with a desired energy function that qualifies as lyapunov function for the desired equilibrium the success of the method relies on the possibility of solving two pdes which identify the kinetic and potential energy functions that can be assigned to the closed loop particularly troublesome is the partial differential equation pde associated to the kinetic energy which is nonlinear and inhomogeneous and the solution that defines the desired inertia matrix must be positive definite in this note we prove that we can eliminate or simplify the forcing term in this pde by modifying the target dynamics and introducing a change of coordinates in the original system furthermore it is shown that in the particular case of transformation to the lagrangian coordinates the possibility of simplifying the pdes is determined by the interaction between the coriolis and centrifugal forces and the actuation structure the examples of pendulum on a cart and furuta s pendulum are used to illustrate the results

pseudospectral methods on a semi infinite interval with application to the hydrogen atom a comparison of the mapped fourier sine method with laguerre series and rational chebyshev expansions the fourier sine with mapping pseudospectral algorithm of fattal et al phys rev e 53 1996 1217 has been applied in several quantum physics problems here we compare it with pseudospectral methods using laguerre functions and rational chebyshev functions we show that laguerre and chebyshev expansions are better suited for solving problems in the interval r 0 for example the coulomb schrodinger equation than the fourier sinemapping scheme all three methods give similar accuracy for the hydrogen atom when the scaling parameter l is optimum but the laguerre and chebyshev methods are less sensitive to variations in l we introduce a new variant of rational chebyshev functions which has a more uniform spacing of grid points for large r and gives somewhat better results than the rational chebyshev functions of boyd j comp phys 70 1987 63

fourth order accurate evaluation of integrals in potential theory on exterior 3d regions we present two methods for the rapid high order accurate evaluation of integrals in potential theory on general unbounded 3d regions our methods allow for direct calculation of derivatives of the integrals as well one of the methods uses a fourth order compact stencil and the other uses a nonstandard variant of richardson extrapolation both methods involve calculation of discontinuities in high order derivatives of the integrals across the boundary of the integration region the extrapolation method in addition involves correction for the discontinuities in truncation error the number of operations required for the methods is essentially equal to twice the number of operations needed to solve poisson s equation on a regular grid both methods avoid problems associated with using quadrature methods to evaluate integrals with singular kernels numerical results are presented for experiments on a variety of geometries in free space

rapid prolate pseudospectral differentiation and interpolation with the fast multipole method pseudospectral methods utilizing prolate spheroidal wave functions as basis functions have been shown to possess advantages over the conventional pseudospectral methods based on trigonometric and orthogonal polynomials however the spectral differentiation and interpolation steps of the prolate pseudospectral method involve matrix vector products which if evaluated directly entail o n2 memory requirement and computational complexity where n is the number of unknowns utilized for discretization and interpolation in this work we show that the fast multipole method fmm can be used to reduce the memory requirement and computational complexity of the prolate pseudospectral method to o n example simulation results demonstrate the speed and accuracy of the resulting fast prolate pseudospectral solver

a new method of imposing boundary conditions in pseudospectral approximations of hyperbolic equations a new method to impose boundary conditions for pseudospectral approximations to hyperbolic equations is suggested this method involves the collocation of the equation at the boundary nodes as well as satisfying boundary conditions stability and convergence results are proven for the chebyshev approximation of linear scalar hyperbolic equations the eigenvalues of this method applied to parabolic equations are shown to be real and negative

front migration in the nonlinear cahn hilliard equation the method of matched asymptotic expansions is used to describe solutions of the nonlinear cahn hilliard equation for phase separation in n 1 space dimensions the expansion is formally valid when the thickness of internal transition layers is small compared with the distance separating layers and with their radii of curvature on the dominant slowest timescale the interface velocity is determined by the mean curvature of the interface by a non local relation which is identical to that in a well known quasi static model of solidification which exhibits a shape instability discovered by mullins sekerka j appl phys 34 323 329 1963 on a faster timescale the cahn hilliard equation regularizes a classic two phase stefan problem similarity solutions of the two phase stefan problem should describe boundary layers existence and uniqueness of such similarity solutions which admit metastable states is proved rigorously in an appendix

orthogonal rational functions on a semi infinite interval abstract by applying a mapping to the chebyshev polynomials we define a new spectral basis the rational chebyshev functions on the semi infinite interval denoted by tl n y continuing earlier work by the author and by grosch and orszag we show that these rational functions inherit most of the good numerical characteristics of the chebyshev polynomials orthogonality completeness exponential or infinite order convergence matrix sparsity for equations with polynomial coefficients and simplicity seven numerical examples illustrate their versatility the charney stability problem of meteorology for example is solved to show the feasibility of applying spectral methods to a semi infinite atmosphere for functions that are singular at both endpoints such as k 1 y one may combine rational chebyshev functions with a preliminary mapping to obtain a single exponentially convergent expansion on y 0 finally we successfully generalize the wkb method to obtain for the j 0 bessel function an amplitude phase approximation which is convergent rather than asymptotic and is accurate not merely for large y but for all y even the origin

generalized holomorphic szego kernel in 3d spheroids monogenic orthogonal polynomials over 3d prolate spheroids were previously introduced and shown to have some remarkable properties in particular the underlying functions take values in the quaternions identified with r 4 and are generally assumed to be nullsolutions of the well known moisil theodoresco system in this paper we show that these polynomial functions play an important role in defining the szego kernel function over the surface of 3d prolate spheroids as a concrete application we prove an explicit expression of the monogenic szego kernel function over 3d prolate spheroids and present two numerical examples

a direct adaptive poisson solver of arbitrary order accuracy we present a direct adaptive solver for the poisson equation which can achieve any prescribed order of accuracy it is based on a domain decomposition approach using local spectral approximation as well as potential theory and the fast multipole method in two space dimensions the algorithm requireso nk work wherenis the number of discretization points andkis the desired order of accuracy

spectral methods using rational basis functions on an infinite interval by using the map y l cot t where l is a constant differential equations on the interval y can be transformed into t 0 and solved by an ordinary fourier series in this article earlier work by grosch and orszag j comput phys 25 273 1977 cain ferziger and reynolds j comput phys 56 272 1984 and boyd j comput phys 25 43 1982 57 454 1985 siam j numer anal 1987 is extended in several ways first the series of orthogonal rational functions converge on the exterior of bipolar coordinate surfaces in the complex y plane second galerkin s method will convert differential equations with polynomial or rational coefficients into banded matrix problems third with orthogonal rational functions it is possible to obtain exponential convergence even for u y that asymptote to a constant although this behavior would wreck alternatives such as hermite or sinc expansions fourth boundary conditions are usually natural rather than essential in the sense that the singularities of the differential equation will force the numerical solution to have the correct behavior at infinity even if no constraints are imposed on the basis functions fifth mapping a finite interval to an infinite one and then applying the rational chebyshev functions gives an exponentially convergent method for functions with bounded endpoint singularities these concepts are illustrated by five numerical examples

analysis of spectral approximations using prolate spheroidal wave functions in this paper the approximation properties of the prolate spheroidal wave functions of order zero pswfs are studied and a set of optimal error estimates are derived for the pswf approximation of non periodic functions in sobolev spaces these results serve as an indispensable tool for the analysis of pswf spectral methods a pswf spectral galerkin method is proposed and analyzed for elliptic type equations illustrative numerical results consistent with the theoretical analysis are also presented

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

new quadrature formulas from conformal maps gauss and clenshaw curtis quadrature like legendre and chebyshev spectral methods make use of grids strongly clustered at boundaries from the viewpoint of polynomial approximation this seems necessary and indeed in certain respects optimal nevertheless such methods may waste a factor of pi 2 with respect to each space dimension we propose new nonpolynomial quadrature methods that avoid this effect by conformally mapping the usual ellipse of convergence to an infinite strip or another approximately straight sided domain the new methods are compared with related ideas of bakhvalov kosloff and tal ezer rokhlin and alpert and others an advantage of the conformal mapping approach is that it leads to theorems guaranteeing geometric rates of convergence for analytic integrands for example one of the formulas presented is proved to converge 50 faster than gauss quadrature for functions analytic in an varepsilon neighborhood of 1 1

the dynamics of multispike solutions to the one dimensional gierer meinhardt model the dynamical behavior of spike type solutions to a simplified form of the gierer meinhardt activator inhibitor model in a one dimensional domain is studied asymptotically and numerically in the limit of small activator diffusivity varepsilon in the limit varepsilon to 0 a quasi equilibrium solution for the activator concentration that has n localized peaks or spikes is constructed asymptotically using the method of matched asymptotic expansions for an initial condition of this form a differential algebraicsystem of equations describing the evolution of the spike locations is derived the equilibrium solutions for this system are discussed the spikes are shown to evolve on a slow time scale tau varepsilon 2 t towards a stable equilibrium provided that the inhibitor diffusivity d is below some threshold and that a certain stability criterion on the quasi equilibrium solution is satisfied throughout the slow dynamics if this stability condition is not satisfied initially or else is no l

inverse methods for reconstruction of channel taps in ofdm systems we describe a novel pilot aided method for estimation of doubly selective wireless channels in ofdm systems we compute the first few fourier coefficients of each channel tap from the pilot information we then estimate the bem coefficients of the channel taps from their respective fourier coefficients using a recently developed inverse reconstruction method for a system with l channel taps the proposed method uses o l log l operations and o l memory per ofdm symbol we validate our method by simulating a system conforming to the ieee 802 16e standard

a numerical comparison of seven grids for polynomial interpolation on the interval abstract seven types of chebyshev like grids in one dimension are compared according to four different criteria for accuracy the grid which minimizes the lebesgue constant is the best because it performs fairly well by all four criteria for the same reason minimizing the lebesgue constant seems to be the most useful measure of optimality because grids that are good according to this criterion are good when measured by other criteria too cfm optimality which is the property that all cardinal functions lagrange fundamental polynomials have maxima at the interpolation points seems to be the least discriminating criterion because all seven grids generate cardinal functions that have maxima at or very near the interpolation points the difference between these grids on all four criteria always less than a factor of two and usually much smaller are sufficiently modest so that the final choice between grids should probably not be made because of accuracy but rather based on other criteria such as ease of programming analytical simplicity conformality with other approximations and timestep in applications to partial differential equations to explore such nonaccuracy issues we also compared six of the grids on the basis of maximum allowed timestep when the grid is used to discretize the spatial coordinate and an explicit scheme is used for time marching the lebesgue optimal grid is also nearly optimal in the sense of allowing the longest timestep with a pseudospectral collocation algorithm the lebesgue grid allows a timestep three halves as long as that of the legendre lobatto spectral element method

comparison of interpolating functions and interpolating points in full wave multilevel green s function interpolation method the difficulty for the solution of full wave electromagnetic problems using multilevel green s function interpolation method mlgfim lies in applying interpolating approaches to efficiently and accurately approximate green s function with rapidly changing phase term we compare various interpolating schemes when radial basis function rbf is employed for the interpolation of scattered data of green s function we show that the infinitely smooth gaussian ga rbf has the best interpolation accuracy in order to improve the interpolation efficiency a new kind of staggered tartan grid is proposed a good calculation method for the shape parameter in ga rbf is given to solve its sensitivity to the group size and the number of interpolation points based on the analysis of variation of the number of interpolation points with electric length of the group adaptive choice of the types of interpolation functions and interpolation points are employed numerical examples show that the computational efficiency of this new interpolation scheme is much improved over the previously reported ones

convergent cartesian grid methods for maxwell s equations in complex geometries a convergent second order cartesian grid finite difference scheme for the solution of maxwell s equations is presented the scheme employs a staggered grid in space and represents the physical location of the material and metallic boundaries correctly hence eliminating problems caused by staircasing and contrary to the popular yee scheme enforces the correct jump conditions on the field components across material interfaces a detailed analysis of the accuracy of the new embedding scheme is presented confirming its second order global accuracy furthermore the scheme is proven to be a bounded error scheme and thus convergent conditions for fully discrete stability is furthermore established this enables the derivation of bounds for fully discrete stability with cfl restrictions being almost identical to those of the much simpler yee scheme the analysis exposes that the effects of staircasing as well as a lack of properly enforced jump conditions on the field components have significant consequences for the global accuracy it is among other things shown that for cases in which a field component is discontinuous along a grid line as happens at general two and three dimensional material interfaces the yee scheme may exhibit local divergence and loss of global convergence to validate the analysis several one and two dimensional test cases are presented showing an improvement of typically 1 to 2 orders of accuracy at little or no additional computational cost over the yee scheme which in most cases exhibits first order accuracy c 2001 academic press

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

a new fast multipole accelerated poisson solver in two dimensions we present an adaptive fast multipole method for solving the poisson equation in two dimensions the algorithm is direct assumes that the source distribution is discretized using an adaptive quad tree and allows for dirichlet neumann periodic and free space conditions to be imposed on the boundary of a square the amount of work per grid point is comparable to that of classical fast solvers even for highly nonuniform grids

regular article spectral collocation time domain modeling of diffractive optical elements a spectral collocation multi domain scheme is developed for the accurate and efficient time domain solution of maxwell s equations within multi layered diffractive optical elements special attention is being paid to the modeling of out of plane waveguide couplers emphasis is given to the proper construction of high order schemes with the ability to handle very general problems of considerable geometric and material complexity central questions regarding efficient absorbing boundary conditions and time stepping issues are also addressed the efficacy of the overall scheme for the time domain modeling of electrically large and computationally challenging problems is illustrated by solving a number of plane as well as non plane waveguide problems

prolate spheroidal wavefunctions as an alternative to chebyshev and legendre polynomials for spectral element and pseudospectral algorithms prolate spheroidal functions of order zero are generalizations of legendre polynomials which when the bandwidth parameter c 0 oscillate more uniformly on x 1 1 than either chebyshev or legendre polynomials this suggests that compared to these polynomials prolate functions give more uniform spatial resolution further when used as the spatial discretization for time dependent partial differential equations in combination with explicit time marching prolate functions allow a longer stable timestep than legendre polynomials we show that these advantages are real and further that it is almost trivial to modify existing pseudospectral and spectral element codes to use the prolate basis however improvements in spatial resolution are at most a factor of 2 approached slowly as n the timestep can be lengthened by several times but not by a factor that grows rapidly with n the prolate basis is not likely to radically expand the range of problems that can be done on a workstation however for production runs on the bleeding edge edge of arithmurgy such as numerical weather prediction the rewards for switching to a prolate basis may be considerable

generalized gaussian quadrature rules for systems of arbitrary functions a numerical algorithm is presented for the construction of generalized gaussian quadrature rules originally introduced by s karlin and w studden over three decades ago the quadrature rules to be discussed possess most of the desirable properties of the classical gaussian integration formulae such as positivity of the weights rapid convergence mathematical elegance etc the algorithm is applicable to a wide class of functions including smooth functions not necessarily polynomials as well as functions with end point singularities such as those encountered in the solution of integral equations complex analysis potential theory and several other areas the performance of the algorithm is illustrated with several numerical examples

the eps method a new method for constructing pseudospectral derivative operators we develop a new type of derivative matrix for pseudospectral methods the norm of these matrices grows at the optimal rate o n 2 for n by n matrices in contrast to standard pseudospectral constructions that result in o n 4 growth of the norm the smaller norm has a big advantage when using the derivative matrix for solving time dependent problems such as wave propagation the construction is based on representing the derivative operator as an integral kernel and does not rely on the interpolating polynomials in particular we construct second derivative matrices that incorporate dirichlet or neumann boundary conditions on an interval and on the disk but the method can be used to construct a wide variety of commonly used operators for solving pdes and integral equations the construction can be used with any quadrature including traditional gauss legendre quadratures but we have found that by using quadratures based on prolate spheroidal wave functions we can achieve a near optimal sampling rate close to two points per wavelength even for non periodic problems we provide numerical results for the new construction and demonstrate that the construction achieves similar or better accuracy than traditional pseudospectral derivative matrices while resulting in a norm that is orders of magnitude smaller than the standard construction to demonstrate the advantage of the new construction we apply the method for solving the wave equation in constant and discontinuous media and for solving pdes on the unit disk we also present two compression algorithms for applying the derivative matrices in o nlogn operations

spectral analysis of the finite hankel transform and circular prolate spheroidal wave functions in this paper we develop two practical methods for the computation of the eigenvalues as well as the eigenfunctions of the finite hankel transform operator these different eigenfunctions are called circular prolate spheroidal wave functions cpswfs this work is motivated by the potential applications of the cpswfs as well as the development of practical methods for computing their values also in this work we should prove that the cpswfs form an orthonormal basis of the space of hankel band limited functions an orthogonal basis of l 2 0 1 and an orthonormal system of l 2 0 our computation of the cpswfs and their associated eigenvalues is done by the use of two different methods the first method is based on a suitable matrix representation of the finite hankel transform operator the second method is based on the use of an efficient quadrature method based on a special family of orthogonal polynomials also we give two maple programs that implement the previous two methods finally we present some numerical results that illustrate the results of this work

efficient spectral galerkin method i direct solvers of second and fourth order equations using legendre polynomials this paper presents some efficient algorithms based on the legendre galerkin approximations for the direct solution of the second and fourth order elliptic equations the key to the efficiency of these algorithms is to construct appropriate base functions which lead to systems with sparse matrices for the discrete variational formulations the complexities of the algorithms are a small multiple of n d 1 operations for a d dimensional domain with n 1 d unknowns while the convergence rates of the algorithms are exponential for problems with smooth solutions in addition the algorithms can be effectively parallelized since the bottlenecks of the algorithms are matrix matrix multiplications

numerical algorithms based on galerkin methods for the modeling of reactive interfaces in photoelectrochemical pec solar cells abstract this work concerns the numerical solution of a coupled system of self consistent reaction drift diffusion poisson equations that describes the macroscopic dynamics of charge transport in photoelectrochemical pec solar cells with reactive semiconductor and electrolyte interfaces we present three numerical algorithms mainly based on a mixed finite element and a local discontinuous galerkin method for spatial discretization with carefully chosen numerical fluxes and implicit explicit time stepping techniques for solving the time dependent nonlinear systems of partial differential equations we perform computational simulations under various model parameters to demonstrate the performance of the proposed numerical algorithms as well as the impact of these parameters on the solution to the model

an adaptive algorithm for spectral computations on unbounded domains abstract we consider spectral methods for solving differential equations on unbounded domains in particular we consider a spectral rational function method and a method based on domain truncation combined with fourier series both methods contain a free parameter which determines the accuracy of the spectral method when solving evolution equations the optimal parameter may change with time resulting in a significant loss of accuracy if the parameter is kept fixed we develop an automatic algorithm which adapts the parameter at regular time levels to maintain optimum accuracy in numerical tests we found that the proposed method is robust and efficient