injection induced flows in porous walled ducts analyse theorique de l ecoulement dans des tubes et canaux a paroi poreuse on tient compte des effets de compressibilite de conditions aux limites non ideales et de transition turbulente sur le developpement de l ecoulement on utilise un modele de turbulence a contrainte de reynolds et un schema de differences finies implicite pour resoudre le systeme d equations paraboliques resultant

spatiospectral concentration on a sphere we pose and solve the analogue of slepian s time frequency concentration problem on the surface of the unit sphere to determine an orthogonal family of strictly bandlimited functions that are optimally concentrated within a closed region of the sphere or alternatively of strictly spacelimited functions that are optimally concentrated in the spherical harmonic domain such a basis of simultaneously spatially and spectrally concentrated functions should be a useful data analysis and representation tool in a variety of geophysical and planetary applications as well as in medical imaging computer science cosmology and numerical analysis the spherical slepian functions can be found by solving either an algebraic eigenvalue problem in the spectral domain or a fredholm integral equation in the spatial domain the associated eigenvalues are a measure of the spatiospectral concentration when the concentration region is an axisymmetric polar cap the spatiospectral projection operator commutes with a sturm liouville operator this enables the eigenfunctions to be computed extremely accurately and efficiently even when their area bandwidth product or shannon number is large in the asymptotic limit of a small spatial region and a large spherical harmonic bandwidth the spherical concentration problem reduces to its planar equivalent which exhibits self similarity when the shannon number is kept invariant

super resolution on the sphere using convex optimization this paper considers the problem of recovering an ensemble of diracs on a sphere from its low resolution measurements the diracs can be located at any location on the sphere not necessarily on a grid we show that under a separation condition one can recover the ensemble with high precision by a three stage algorithm which consists of solving a semi definite program root finding and least square fitting the algorithm s computation time depends solely on the number of measurements and not on the required solution accuracy we also show that in the special case of non negative ensembles a sparsity condition is sufficient for recovery furthermore in the discrete setting we estimate the recovery error in the presence of noise as a function of the noise level and the super resolution factor

a generalized one dimensional fast multipole method with application to filtering of spherical harmonics the need to filter functions defined on the sphere arises in a number of applications such as climate modeling electromagnetic and acoustic scattering and several other areas recently it has been observed that the problem of uniform resolution filtering on the sphere can be performed efficiently via the fast multipole method fmm in one dimension in this paper we introduce a generalization of the fmm that leads to an accelerated version of the filtering process instead of multipole expansions the scheme uses special purpose bases constructed via the singular value decomposition of appropriately chosen submatrices of the filtering matrix the algorithm is applicable to a fairly wide class of projection operators its performance is illustrated with several numerical examples

double fourier series on a sphere the solution of elliptic and vorticity equations on a sphere is studied using double fourier series as orthogonal basis functions the basis functions incorporate sine series weighted by cosine latitude as meridional basis functions for even zonal wavenumbers other than zero to meet the pole condition as to the solution of poisson s equation it is found that the new method gives improved accuracy compared to the method of yee mon weather rev 109 501 1981 due to the absence of constraints imposed on spectral coefficients with the operation number being slightly increased the new method is applied to the vorticity equation along with the use of fourier and spherical harmonics filters and its accuracy is tested for the rotated rossby haurwitz wave it is shown that the basis functions adopted here provide high accuracy for all tests used numerical integration without spherical harmonics filters indicates that they are necessary for stable and accurate time integration comparison with the spherical harmonics model reveals that the present method is more accurate by a factor of order 112 for the test case further application to the advection equation is carried out the error measure for the strong advection of the cosine bell with various rotation angles indicates that the present method is capable of producing accurate and stable calculations without the pole problem suggesting that it could be applied to the numerical weather prediction model including shallow water equations without difficulty extension to the shallow water equations with accuracy tests as in williamson et al j comput phys 102 211 1992 will be given in the future additional time could be saved by introducing the reduced grids near poles in the present method besides the advantage of applying fft to both longitudinal and latitudinal directions

on steady rotational high speed flows the compressible taylor culick profile we consider the compressible flow analogue of the well known taylor culick profile we first present the compressible euler equations for steady axisymmetric isentropic flow assuming uniform injection of a calorically perfect gas in a porous chamber we then apply the rayleigh janzen expansion in powers of m 2 w where mw is the wall mach number we solve the ensuing equations to the order of m 4 w and apply the results up to the sonic point in a nozzleless chamber area averaging is also performed to reconcile with one dimensional theory our solution agrees with the existing theory to the extent that it faithfully captures the steepening of the taylor culick profile with downstream movement based on the closed form expressions that we obtain the main flow attributes are quantified parametrically and compared to the existing incompressible and quasi one dimensional theories verification by computational fluid dynamics is also undertaken comparison with two turbulent flow models shows excellent agreement particularly in retracing the streamwise evolution of the velocity regardless of the mach number we observe nearly identical trends in chambers that are rescaled by the critical sonic length ls using a suitable transformation we prove the attendant similarity and provide universal criteria that can be used to assess the relative importance of gas compression in solid rocket motors owing to sharper velocity gradients at the wall we find that an incompressible model underestimates the skin friction along the wall and underpredicts the centreline speed by as much as 13 at the sonic point in practice such deviations become appreciable at high injection rates or chamber aspect ratios

an efficient implementation of a high order filter for a cubed sphere spectral element model abstract a parallel scalable isotropic scale selective spatial filter was developed for the cubed sphere spectral element model on the sphere the filter equation is a high order elliptic helmholtz equation based on the spherical laplacian operator which is transformed into cubed sphere local coordinates the laplacian operator is discretized on the computational domain i e on each cell by the spectral element method with gauss lobatto lagrange interpolating polynomials gllips as the orthogonal basis functions on the global domain the discrete filter equation yielded a linear system represented by a highly sparse matrix the density of this matrix increases quadratically linearly with the order of gllip order of the filter and the linear system is solved in only o n g operations where n g is the total number of grid points the solution obtained by a row reduction method demonstrated the typical accuracy and convergence rate of the cubed sphere spectral element method to achieve computational efficiency on parallel computers the linear system was treated by an inverse matrix method a sparse matrix vector multiplication the density of the inverse matrix was lowered to only a few times of the original sparse matrix without degrading the accuracy of the solution for better computational efficiency a local domain high order filter was introduced the filter equation is applied to multiple cells and then the central cell was only used to reconstruct the filtered field the parallel efficiency of applying the inverse matrix method to the global and local domain filter was evaluated by the scalability on a distributed memory parallel computer the scale selective performance of the filter was demonstrated on earth topography the usefulness of the filter as a hyper viscosity for the vorticity equation was also demonstrated

on the analytical summation of fourier series and its relation to the asymptotic behaviour of fourier transforms forms of the poisson summation formula psf appropriate for the summation of semi infinite and infinite fourier series are derived application of these results to the acceleration of convergence of various types of series with monotonically decreasing coefficient functions yields transformed series with terms that decay either exponentially or with the inverse first or second power of the index variable these two very different convergence properties are explained in terms of the asymptotic properties of the relevant fourier transforms which are in turn related to the power series expansions of the summand functions in the original fourier series the result is that the poisson summation formula works best for fourier cosine series in which the summand functions are expansible in even powers and for fourier sine series in which the summand functions have power series with odd powers here application of the psf produces series of terms that decay exponentially with increasing argument x in contrast application of the semi infinite version of the psf to fourier cosine series of terms with odd power expansions or to fourier sine series of terms with even power expansions yields transformed series involving functions of the form which decay approximately as if the summand function in the fourier series has a power series with both even and odd powers the transformed series involves sine and cosine integral functions which decay approximately as fourier series of these last three types in general require additional acceleration for example by application of the kummer transformation

expansion maximization compression algorithm with spherical harmonics for single particle imaging with x ray lasers in three dimensional 3d single particle imaging with x ray free electron lasers particle orientation is not recorded during measurement but is instead recovered as a necessary step in the reconstruction of a 3d image from the diffraction data here we use harmonic analysis on the sphere to cleanly separate the angular and radial degrees of freedom of this problem providing new opportunities to efficiently use data and computational resources we develop the expansion maximization compression algorithm into a shell by shell approach and implement an angular bandwidth limit that can be gradually raised during the reconstruction we study the minimum number of patterns and minimum rotation sampling required for a desired angular and radial resolution these extensions provide new avenues to improve computational efficiency and speed of convergence which are critically important considering the very large datasets expected from experiment

a performance comparison of associated legendre projections the associated legendre projection provides a means for accelerating the dynamical core of a global weather or climate model therefore the goal is to determine the fastest possible projection algorithm of which this paper compares six the standard method which computes the projection using a forward and backward associated legendre transform the direct method which uses a single projection matrix when this approach results in fewer operations the fast multipole method the weighted orthogonal complement method the seminaive method and a projection based on transforms proposed by mohlenkamp timing results indicate that all the projections behave like o n3 algorithms up to at least n 200 spectral truncation for this range of resolutions the weighted orthogonal complement has the lowest operation count best cache utilization and best overall timings

exact wavelets on the ball we develop an exact wavelet transform on the three dimensional ball i e on the solid sphere which we name the flaglet transform for this purpose we first construct an exact transform on the radial half line using damped laguerre polynomials and develop a corresponding quadrature rule combined with the spherical harmonic transform this approach leads to a sampling theorem on the ball and a novel three dimensional decomposition which we call the fourier laguerre transform we relate this new transform to the well known fourier bessel decomposition and show that band limitedness in the fourier laguerre basis is a sufficient condition to compute the fourier bessel decomposition exactly we then construct the flaglet transform on the ball through a harmonic tiling which is exact thanks to the exactness of the fourier laguerre transform from which the name flaglets is coined the corresponding wavelet kernels are well localised in real and fourier laguerre spaces and their angular aperture is invariant under radial translation we introduce a multiresolution algorithm to perform the flaglet transform rapidly while capturing all information at each wavelet scale in the minimal number of samples on the ball our implementation of these new tools achieves floating point precision and is made publicly available we perform numerical experiments demonstrating the speed and accuracy of these libraries and illustrate their capabilities on a simple denoising example

spherical harmonic projectors the harmonic projection hp which is implicit in the numerous harmonic transforms between physical and spectral spaces is responsible for the reliability of the spectral method for modeling geophysical phenomena as currently configured the hp consists of a forward transform from physical to spectral space harmonic analysis immediately followed by a harmonic synthesis back to physical space unlike its fourier counterpart in cartesian coordinates the hp does not identically reconstruct the original function on the surface of the sphere but rather replaces it with a weighted least squares approximation the importance of the hp is that it uniformly resolves waves on the surface of the sphere and therefore eliminates high frequencies that are artificially induced by the clustering of grid points in the neighborhood of the poles the hp also maintains spectral accuracy when combined with the double fourier method originally the hp required o n 3 storage where n is the number of latitudinal points however this was recently reduced to o n 2 using an algorithm that also provided a savings of up to 50 percent in compute time the hp was also generalized to an arbitrary latitudinal distribution of points however the hp as a composite of analysis and synthesis can be subject to considerable error depending on the point distribution here we define a variant of the traditional hp that is well conditioned with condition number 1 for any point distribution in addition storage requirements are further reduced because the projections corresponding to all longitudinal wave numbers m are expressed in terms of a single orthogonal matrix

an empirical bayes approach to directional data and efficient computation on the sphere this paper proposes a consistent nonparametric empirical bayes estimator of the prior density for directional data the methodology is to use fourier analysis on s 2 to adapt euclidean techniques to this non euclidean environment general consistency results are obtained in addition a discussion of efficient numerical computation of fourier transforms on s 2 is given and their applications to the methods suggested in this paper are sketched

wavelet based segmentation on the sphere abstract segmentation a useful powerful technique in pattern recognition is the process of identifying object outlines within images there are a number of efficient algorithms for segmentation in euclidean space that depend on the variational approach and partial differential equation modelling wavelets have been used successfully in various problems in image processing including segmentation inpainting noise removal super resolution image restoration and many others wavelets on the sphere have been developed to solve such problems for data defined on the sphere which arise in numerous fields such as cosmology and geophysics in this work we propose a wavelet based method to segment images on the sphere accounting for the underlying geometry of spherical data our method is a direct extension of the tight frame based segmentation method used to automatically identify tube like structures such as blood vessels in medical imaging it is compatible with any arbitrary type of wavelet frame defined on the sphere such as axisymmetric wavelets directional wavelets curvelets and hybrid wavelet constructions such an approach allows the desirable properties of wavelets to be naturally inherited in the segmentation process in particular directional wavelets and curvelets which were designed to efficiently capture directional signal content provide additional advantages in segmenting images containing prominent directional and curvilinear features we present several numerical experiments applying our wavelet based segmentation method as well as the common k means method on real world spherical images including an earth topographic map a light probe image solar data sets and spherical retina images these experiments demonstrate the superiority of our method and show that it is capable of segmenting different kinds of spherical images including those with prominent directional features moreover our algorithm is efficient with convergence usually within a few iterations

spin weighted spherical wavelets derived from approximate identities abstract spin weighted functions introduced in 1966 by newman and penrose are used primarily for description of the cosmic microwave background radiation in the present paper an analysis tool for spin weighted functions is proposed namely spin weighted spherical wavelets based on approximate identities in the spin 0 case there exist two different definitions of continuous wavelet transforms the group theoretical one introduced by antoine et al and that based on approximate identities pioneered by freeden et al the first one has been recently extended to the case of spin weighted functions by mcewen et al and the present research is the approximate identities counterpart of their work wavelets presented here can be constructed for and the wavelet transform can be applied to functions of each integer spin weight further the set of parameters scale and position is continuous a big advantage of the construction is that the wavelet transform is invertible directly by an integral last but not least wavelets can reveal directional features such that the directional intensity of a signal can be probed

a standard test set for numerical approximations to the shallow water equations in spherical geometry a suite of seven test cases is proposed for the evaluation of numerical methods intended for the solution of the shallow water equations in spherical geometry the shallow water equations exhibit the major difficulties associated with the horizontal dynamical aspects of atmospheric modeling on the spherical earth these cases are designed for use in the evaluation of numerical methods proposed for climate modeling and to identify the potential trade offs which must always be made in numerical modeling before a proposed scheme is applied to a full baroclinic atmospheric model it must perform well on these problems in comparison with other currently accepted numerical methods the cases are presented in order of complexity they consist of advection across the poles steady state geostrophically balanced flow of both global and local scales forced nonlinear advection of an isolated low zonal flow impinging on an isolated mountain rossby haurwitz waves and observed atmospheric states one of the cases is also identified as a computer performance algorithm efficiency benchmark for assessing the performance of algorithms adapted to massively parallel computers

scattering theory in the energy space for a class of non linear wave equations we study the asymptotic behaviour in time of the solutions and the theory of scattering in the energy space for the non linear wave equation square varphi f varphi 0 in n n 3 we prove the existence of the wave operators asymptotic completeness for small initial data and forn 4 asymptotic completeness for arbitrarily large data the assumptions onf cover the case wheref behaves slightly better than a single powerp 1 4 n 2 both near zero and at infinity see 1 5 1 6 and 1 8

a test of a modified algorithm for computing spherical harmonic coefficients using an fft this note describes the results of testing a modified method based on an algorithm by dilts 1 for computing the spherical harmonic expansion coefficients for a function on a sphere using a two dimensional fft the new version of the dilts program eliminates problems of overflow and large storage requirements encountered when harmonic degree values of l 16 are computed thus allowing the determination of coefficients for high l values l 300 however results from timing tests indicate that the new version of the dilts program is not practical for computing spherical harmonic expansion coefficients for large l due to the very long cpu times required for execution of the program the most effective technique remains the one pioneered by brown 2

sampling sparse signals on the sphere algorithms and applications we propose a sampling scheme that can perfectly reconstruct a collection of spikes on the sphere from samples of their lowpass filtered observations central to our algorithm is a generalization of the annihilating filter method a tool widely used in array signal processing and finite rate of innovation fri sampling the proposed algorithm can reconstruct k spikes from k sqrt k 2 spatial samples for large k this sampling requirement improves over previously known fri sampling schemes on the sphere by a factor of four we showcase the versatility of the proposed algorithm by applying it to three problems 1 sampling diffusion processes induced by localized sources on the sphere 2 shot noise removal and 3 sound source localization ssl by a spherical microphone array in particular we show how ssl can be reformulated as a spherical sparse sampling problem

a united approach to accelerating trigonometric expansions abstract the fourier series of a smooth function on a compact interval usually has slow convergence due to the fact that the periodic extension of the function has jumps at the interval endpoints for various symmetry conditions polynomial interpolation methods have been developed for performing a boundary correction the resulting variants of krylov approximants are a sum of a correction polynomial and a fourier sum of the corrected function 1 8 in this paper we review these methods and derive estimates in the maximum norm we further show that derivatives of the krylov approximants are again krylov approximants of derivatives of the considered function this enables us to give a unified treatment of the problem of simultaneous approximation

an algorithm for the numerical evaluation of the associated legendre functions that runs in time independent of degree and order abstract we describe a method for the numerical evaluation of normalized versions of the associated legendre functions p and q of degrees 0 1 000 000 and orders for arguments in the interval 1 1 our algorithm which runs in time independent of and is based on the fact that while the associated legendre functions themselves are extremely expensive to represent via polynomial expansions the logarithms of certain solutions of the differential equation defining them are not we exploit this by numerically precomputing the logarithms of carefully chosen solutions of the associated legendre differential equation and representing them via piecewise trivariate chebyshev expansions these precomputed expansions which allow for the rapid evaluation of the associated legendre functions over a large swath of parameter domain mentioned above are supplemented with asymptotic and series expansions in order to cover it entirely the results of numerical experiments demonstrating the efficacy of our approach are presented and our code for evaluating the associated legendre functions is publicly available

a phase sensitive approach to filtering on the sphere this paper examines filtering on a sphere by first examining the roles of spherical harmonic magnitude and phase we show that phase is more important than magnitude in determining the structure of a spherical function we examine the properties of linear phase shifts in the spherical harmonic domain which suggest a mechanism for constructing finite impulse response fir filters we show that those filters have desirable properties such as being associative mapping spherical functions to spherical functions allowing directional filtering and being defined by relatively simple equations we provide examples of the filters for both spherical and manifold data

an integral transform approach for a mixed boundary problem involving a flowing partially penetrating well with infinitesimal well skin 1 a flowing partially penetrating well with infinitesimal well skin is a mixed boundary because a cauchy condition is prescribed along the screen length and a neumann condition of no flux is stipulated over the remaining unscreened part an analytical approach based on the integral transform technique is developed to determine the laplace domain solution for such a mixed boundary problem in a confined aquifer of finite thickness first the mixed boundary is changed into a homogeneous neumann boundary by substituting the cauchy condition with a neumann condition in terms of well bore flux that varies along the screen length and is time dependent despite the well bore flux being unknown a priori the modified model containing this homogeneous neumann boundary can be solved with the laplace and the finite fourier cosine transforms to determine well bore flux screen length is discretized into a finite number of segments to which the cauchy condition is reinstated this reinstatement also restores the relation between the original model and the solutions obtained for a given time the numerical inversion of the laplace domain solution yields the drawdown distributions well bore flux and the well discharge this analytical approach provides an alternative for dealing with the mixed boundary problems especially when aquifer thickness is assumed to be finite

sampling signals with a finite rate of innovation on the sphere the state of the art in sampling theory now contains several theorems for signals that are non bandlimited for signals on the sphere however most theorems still require the assumptions of bandlimitedness in this work we show that a particular class of non bandlimited signals which have a finite rate of innovation can be exactly recovered using a finite number of samples we consider a sampling scheme where k weighted diracs are convolved with a kernel on the rotation group we prove that if the sampling kernel has a bandlimit l 2k then 2k 1 4k 1 1 equiangular samples are sufficient for exact reconstruction if the samples are uniformly distributed on the sphere we argue that the signal can be accurately reconstructed using 4k2 samples and validate our claim through numerical simulations to further reduce the number of samples required we design an optimal sampling kernel that achieves accurate reconstruction of the signal using only 3k samples the number of parameters of the weighted diracs in addition to weighted diracs we show that our results can be extended to sample diracs integrated along the azimuth finally we consider kernels with antipodal symmetry which are common in applications such as diffusion magnetic resonance imaging

a fourier spectral element algorithm for thermal convection in rotating axisymmetric containers we present a fourier spectral element approach for modeling thermal convection in a rotating axisymmetric container following the theory detailed in bernardi et al c bernardi m dauge y maday spectral methods for axisymmetric domains gauthier villars paris 1999 a fourier expansion of the field variables is performed in the periodic direction and the resulting collection of meridional problems is discretized by means of a parallel spectral element method a gauss lobatto jacobi 0 1 quadrature which incorporates the cylindrical radius in its weight is introduced to avoid a potential degeneracy of the discrete set of equations due to those nodes located on the axis of symmetry a second order timestepping scheme is presented which treats the coriolis and viscous forces implicitly numerical comparisons with analytical and published numerical solutions in spherical and cylindrical geometries are presented which highlight the accuracy of the model and demonstrate its spectral convergence properties

generalized solutions of a nonlinear parabolic equation with generalized functions as initial data abstract in h brezis a friedman nonlinear parabolic equations involving measures as initial conditions j math pure appl 9 1983 73 97 brezis and friedman prove that certain nonlinear parabolic equations with the measure as initial data have no solution however in j f colombeau m langlais generalized solutions of nonlinear parabolic equations with distributions as initial conditions j math anal appl 1990 186 196 colombeau and langlais prove that these equations have a unique solution even if the measure is substituted by any colombeau generalized function of compact support here we generalize colombeau and langlais result proving that we may take any generalized function as the initial data our approach relies on recent algebraic and topological developments of the theory of colombeau generalized functions and results from j aragona colombeau generalized functions on quasi regular sets publ math debrecen 2006 371 399

further study on the high order double fourier series spectral filtering on a sphere a high order harmonic spectral filter hsf is further studied using double fourier series dfs which performs filtering in terms of successive inversion of tridiagonal matrices with complex valued elements the high order harmonics filter equation is split into multiple helmholtz equations it is found that the filter provides the same order of accuracy as the spectral filter in j comput phys 177 2002 313 that consists of the pentadiagonal matrices with real valued elements the advantage of the filter over the previous one lies on the simplicity and easiness of numerical implementation or computer coding just requiring the same complexity as poisson s equation solver however the operation count associated with the filter increases by a factor of about 2 to circumvent the inefficiency while preserving the simplicity an easy way to construct pentadiagonal matrices associated with the biharmonic equation is presented in which the tridiagonal matrices related with poisson s equation are manipulated computational efficiency of the spectral filter is discussed in terms of the relative computing time to the spectral transform it is revealed that the computing cost requiring o n2 operations with n being the truncation for the spectral filtering even with the complex valued matrices is not significant in the dfs spectral model that is characterized by o n2 log2 n operations filtering with different dfs expansions is discussed with a focus on the accuracy and pole condition it is shown that the dfs violating the pole conditions produces a discontinuity at poles in case of wave truncation and its influence spreads over the globe the spectral filter is applied to two kinds of uniform grid data the regular and the shifted grids and the results are compared with each other the operator splitting or spherical harmonics factorization makes it feasible to apply the finite difference method to the high order harmonics filter with ease because only the five point stencil computations are required the application could also be extended to other numerical methods only if the helmholtz equation solver is available

generalized sampling expansion for functions on the sphere functions on the sphere appear in several applications including geodesics imaging and acoustics sampling of these functions may result in aliasing if the sampling condition is not met the generalized sampling expansion introduced by papoulis enables the reconstruction of a band limited function sampled at a frequency lower than the nyquist frequency using the outputs of several linear time invariant systems this paper formulates the generalized sampling expansion for functions on the sphere using spherical harmonics decomposition facilitating sub nyquit sampling without aliasing error an analysis of linear systems on the sphere and the aliasing phenomenon in the spherical harmonics domain is presented examples demonstrating the performance of the method and its limitations are studied

sparse image reconstruction on the sphere implications of a new sampling theorem we study the impact of sampling theorems on the fidelity of sparse image reconstruction on the sphere we discuss how a reduction in the number of samples required to represent all information content of a band limited signal acts to improve the fidelity of sparse image reconstruction through both the dimensionality and sparsity of signals to demonstrate this result we consider a simple inpainting problem on the sphere and consider images sparse in the magnitude of their gradient we develop a framework for total variation inpainting on the sphere including fast methods to render the inpainting problem computationally feasible at high resolution recently a new sampling theorem on the sphere was developed reducing the required number of samples by a factor of two for equiangular sampling schemes through numerical simulations we verify the enhanced fidelity of sparse image reconstruction due to the more efficient sampling of the sphere provided by the new sampling theorem

efficient calculation of the free space periodic green s function electromagnetic scattering from periodic structures can be formulated in terms of an integral equation that has as its kernel a periodic green s function the periodic green s function can be derived as a response to an array of line point sources spatial domain or as a response to series of current sheets spectral domain these responses are a fourier transform pair and are slowly convergent summations the convergence problems in each domain arise from unavoidable singularities in the reciprocal domain a method is discussed to overcome the slow convergence by using the poisson summation formula and summing in a combination of spectral and spatial domains a parameter study is performed to determine an optimum way to weigh the combination of domains simple examples of scattering from a one dimensional array of strips and two dimensional array of plates are used to illustrate the concepts

on the lagrangian optimization of wall injected flows from the hart mcclure potential to the taylor culick rotational motion the taylor culick solution for a porous cylinder is often used to describe the bulk gas motion in idealized representations of solid rocket motors however other approximate solutions may be found that satisfy the same fundamental constraints in this vein steeper or smoother profiles may be observed in either experimental or numerical tests particularly in the presence of intense levels of acoustic energy in this study we use the lagrangian optimization principle to arrive at multiple solutions that can help to explain the practically observed motions we then search for the extreme states that display either the least or the most kinetic energy these are derived and found to be dependent on the chamber s aspect ratio by assuming a slender case simple expressions are retrieved from which both the rotational taylor culick and the irrotational hart mcclure solutions are recovered as special cases at the outset a new family of flow approximations is derived extending from purely potential to highly rotational fields these are constructed verified and catalogued based on their kinetic energies to help understand the tendency of a motion to shift energy states a second law analysis is performed and used to rank these solutions based on their entropy content interestingly the taylor culick profile is found to represent an equilibrium state entailing the most energy and entropy among those starting from the rest a formal extension of kelvin s minimum energy theorem to an open region is then pursued followed by a direct implementation to the problem at hand three other flow motions with open boundaries are examined to further exemplify the application of kelvin s extended criteria our efforts illustrate the overarching harmony that exists between lagrange s minimization principle and kelvin s minimum energy theorem both converge on the potential solution as the least kinetic energy bearer even when the velocity at the open barrier is finite

generalized discrete spherical harmonic transforms two generalizations of the spherical harmonic transforms are provided first they are generalized to an arbitrary distribution of latitudinal points i this unifies transforms for gaussian and equally spaced distributions and provides transforms for other distributions commonly used to model geophysical phenomena the discrete associated legendre functions pnm i are shown to be orthogonal to within roundoff error with respect to a weighted inner product thus providing the forward transform to spectral space second the representation of the transforms is also generalized to rotations of the discrete basis set pnm i a discrete function basis is defined that provides an alternative to pnm i on a grid with n latitudes the new basis requires o n2 memory compared to the usual o n3 the resulting transforms differ in spectral space but provide identical results for certain applications for example a forward transform followed immediately by a backward transform projects the original discrete function in a manner identical to the existing transforms namely they both project the original function onto the same smooth least squares approximation without the high frequencies induced by the closeness of the points in the neighborhood of the poles finally a faster projection is developed based on the new transforms

multipole expansions and pseudospectral cardinal functions a new generalization of the fast fourier transform the polynomial or trigonometric interpolant of an arbitrary function f x may be presented as a critical function series whose coefficients are the values of f x at the interpolation points we show that the cardinal series is identical to the sum of the forces due to a set of n point charges with appropriate force laws it follows that the cardinal series can be summed via the fast multipole method fmm in o nlog sub 2 n operations which is much cheaper than the o n sup 2 cost of direct summation the fmm is slower than the fast fourier transform fft so the latter should always be used where applicable however the multipole expansion succeeds where the fft fails in particular the fmm can be used to evaluate fourier and chebyshev series on an irregular grid as is needed when adaptively regridding in a time integration also the multipole expansion can be applied to basis sets for which the fft is inapplicable even on the canonical grid including legendre polynomials hermite and laguerre functions spherical harmonics and zinc functions 12 refs

a semi lagrangian double fourier method for the shallow water equations on the sphere we describe a numerical method based on the semi lagrangian semi implicit approach for solving the shallow water equations swes in spherical coordinates the most popular spatial discretization method used in global atmospheric models is currently the spectral transform method which generates high order numerical solutions and provides an elegant solution to the pole problems induced by a spherical coordinate system however the standard spherical harmonic spectral transform method requires associated legendre transforms which for problems with resolutions of current interest have a computational complexity of o n3 where n is the number of spatial subintervals in one dimension thus the double fourier spectral method which uses trigonometric series may be a viable alternative the advantage of the double fourier method is that fast fourier transforms which have a computational complexity of o n2 log n can be used in both the longitudinal and latitudinal directions in this implementation the swes are discretized in time by means of the three time level semi lagrangian semi implicit method which integrates along fluid trajectories and allows large time steps while maintaining stability numerical results for the standard swes test suite are presented to demonstrate the stability and accuracy of the method

rotationally invariant quadratures for the sphere we construct near optimal quadratures for the sphere that are invariant under the icosahedral rotation group these quadratures integrate all n 1 2 linearly independent functions in a rotationally invariant subspace of maximal order and degree n the nodes of these quadratures are nearly uniformly distributed and the number of nodes is only marginally more than the optimal n 1 2 3 nodes using these quadratures we discretize the reproducing kernel on a rotationally invariant subspace to construct an analogue of lagrange interpolation on the sphere this representation uses function values at the quadrature nodes in addition the representation yields an expansion that uses a single function centred and mostly concentrated at nodes of the quadrature thus providing a much better localization than spherical harmonic expansions we show that this representation may be localized even further we also describe two algorithms of complexity for using these grids and representations finally we note that our approach is also applicable to other discrete rotation groups

fast and stable algorithms for discrete spherical fourier transforms abstract in this paper we propose an algorithm for the stable and efficient computation of fourier expansions of square integrable functions on the unit sphere s r 3 as well as for the evaluation of these fourier expansions at special knots the heart of the algorithm is an efficient realization of discrete legendre function transforms based on a modified and stabilized version of the driscoll healy algorithm

associated legendre functions on the cut abstract algorithms and a code are described for the computation of the associated legendre functions pvu cos pv u cos qvu cos and the normalized legendre polynomial p v u cos in the ranges 0 1 2 0 1 2 1 2 v the algorithms are based on power series expansions and recurrence relations they are executed in extended range arithmetic thereby admitting very extensive ranges of and v without causing overflow or underflow

acceleration of algebraically converging fourier series when the coefficients have series in powers of 1 n if the coefficients in a fourier cosine series f x f email protected n 0 a ncos nx decrease as a small negative power of n then one may need millions of terms to sum the series to high accuracy we show that if the a n are known analytically and have a power series in 1 n then it is straightforward to approximate f x as a series of what we shall the lanczos krylov lk functions we describe the similar methodology for sine series general fourier series are merely the sum of a cosine series with a sine series and thus are implicitly handled too for cosine coefficients that involve only even powers of n and sine coefficients that are functions of odd powers of n the lk functions may be expressed in terms of bernoulli polynomials the lk functions for cosine coefficients involving odd powers of n and for sine coefficients in even powers of n are not known explicitly these are also known as clausen functions we provide rapidly convergent series to compute these clausen functions to high accuracy our method includes the endpoint subtraction ideas of lanczos and krylov but is more general the sum n 1 1 n 1 1 email protected cos nx where l 0 is a constant arises in phase transitions in absorbed monolayers on metal surfaces it is easily summed by our method which correctly incorporates the logarithmic singularities at x email protected

spatially varying spectral filtering of signals on the unit sphere this paper presents a general framework for spatially varying spectral filtering of signals defined on the unit sphere as an analogy to joint time frequency filtering for this purpose we first map spherical signals from spatial domain into joint spatio spectral domain where a spatio spectral signal transformation or modification is introduced for mapping spatial signals into joint spatio spectral domain we use the spatially localized spherical harmonic transform slsht from the literature we then propose a suitable scheme to transform the modified signal from the spatio spectral domain back to an admissible signal in the spatial domain using the least squares approach we also show that the overall action of the slsht and spatio spectral signal modification can be described through a single transformation matrix which is useful in practice finally we discuss two specific and useful instances of spatially varying spectral filtering defined through multiplicative and convolutive modification of the slsht distribution and show through numerical examples their effectiveness in selective spectral filtering of different spatial regions of the signal

evaluation of the spherical bessel transform of a whittaker function an application of a difference equation method a simple analytic expression for the spherical bessel transform of the zero range bound state wave function with the coulomb interaction present for the lth partial wave expressed as a whittaker function is obtained the result is given in terms of polynomials of degree l the exponential function and a simple hypergeometric function which is independent of l transformations of this latter function are derived in terms of more rapidly convergent series the method presented has much wider application since it relies essentially only on the existence of differential difference equations for the functions involved and the solution of the inhomogeneous difference and differential equations satisfied by the transform

numerical comparisons of nonlinear convergence accelerators as part of a continuing program of numerical tests of convergence accelerators we have compared the iterated aitken s a2 method wynn s algorithm brezinski s 0 algorithm and levin s u transform on a broad range of test problems linearly convergence alternating monotone and irregular sign series logarithmically convergent series power method and bernoulli method sequences alternating and monotone asymptotic series and some perturba tion series arising in applications in each category either the algorithm or the u transform gives the best results of the four methods tested in some cases differences among methods are slight and in others they are quite striking 1 introduction we reported in 19 on a numerical test program that compared eleven different scalar convergence accelerators on a broad range of infinite series problems in an attempt to find an across the board accelerator that would handle both linear and logarithmic convergence and both alternating and monotone series or sequences the best candidates to emerge from that study were levin s u transform 14 and brezinski s 0 algorithm 6 we also provided some theoretical support for that conclusion in terms of the wide range of series summed exactly by these methods and by proving that 02 and the levin transforms all accelerate linear convergence the other levin transforms t and v produced essentially equivalent numerical results to u except that t fails on logarithmic convergence we have continued the test program started with 19 to include monotone linearly convergent series series with irregular sign patterns specifically fourier series and sequences whose terms do not depend explicitly on the previous term and or the index e g bernoulli s method since convergence accelerators are often known to transform divergent series to convergent ones we have also included in our test program divergent power series asymptotic series and a few perturbation series the methods considered in this paper are the u and 0 algorithms noted above and

application of double fourier series to the shallow water equations on a sphere the spectral transform method using a double fourier series as orthogonal basis functions as in cheong 2000 j comput phys 157 327 is extended to the solution of the shallow water equations on a sphere a spectral filter which mimics the implicit diffusion process with the third order laplacian operator is applied to the spectral components of predicted variables to prevent the aliasing error or nonlinear instability for a predicted variable the spectral filter needs only about 76n2 operations with n being the zonal and meridional wavenumber truncation the use of the filter even at every time step does not deteriorate the computational efficiency of the double fourier series model which comes from the availability of ffts the filter requires an additional memory for only 6n2 elements so the total memory space of o n2 is sufficient in the present model along with the incorporation of the polar filter the semi implicit time stepping procedure contributes to a significant increase in the time step size a standard test set proposed by williamson et al 1992 j comput phys 102 211 is used to evaluate the errors associated with the new method for various resolutions it is shown that as a whole the accuracy of the method is comparable to that of spherical harmonics model shm though the present method provides more accurate time integration for some cases but does not for other cases a long time integration far beyond the period specified in the standard test set also illustrates almost the same accuracy as that given by the shm the relative efficiency of the method to the shm appears from the resolution of 256 128 transform grids and it becomes significant for resolutions higher than 512 256 the computational efficiency is expected to increase further with an improved fft algorithm the test results suggest that the new method could be extended to three dimensional numerical models used for weather prediction

high order harmonic spectral filter with the double fourier series on a sphere a high order harmonic spectral filter hsf is implemented to smooth out a field variable defined on a spherical surface using the double fourier series dfs as orthogonal basis functions the filter consists of the solution of the high order harmonic diffusion equation with the implicit method where the high order harmonic operator is split into second or lower order harmonic operators the second order harmonic operator is replaced by a pentadiagonal matrix whose elements are the spectral coefficients first a biharmonic spectral filter bihsf the prototype of the high order hsf is developed where only the second order harmonic operator is included it is found that the computational error for the inversion of a pentadiagonal matrix remains in the order of machine rounding compared to the mixed order hsf with dfs used in the previous study which contains the second and third order harmonic operators the bihsf can provide a sharper cutoff of high wavenumbers as well as improved computational efficiency these advantages come from the fact that for each set of spectral coefficients the bihsf needs only a single inversion of the pentadiagonal matrix whereas the mixed order hsf requires triple inversions and an auxiliary operation of the tridiagonal matrix based on the bihsf the high order hsf up to the sixth order which is composed of a multiple inversion of tri or pentadiagonal matrices is designed tests with the rotated gaussian fields revealed that the hsf with dfs is isotropic application to various problems including a time dependent strongly nonlinear case and the observed flow indicates that the high order hsf is well capable of smoothing out selectively high wavenumber components without significantly affecting the conserved quantity such as total kinetic energy

the spectral element method for the shallow water equations on the sphere the spectral element method is implemented for the shallow water equations in spherical geometry and its performance is compared with other models this is the first step in evaluating the suitability of spectral elements for climate modeling the potential advantages and disadvantages of spectral elements over more conventional models used for climate studies are discussed the method requires that the sphere be tiled with rectangles for which we make use of the gnomonic projection to map the sphere onto the cube to measure the performance of the method relative to other models results are presented from a standard suite of shallow water test cases for the sphere these results confirm the spectral accuracy of the method

fast directional spatially localized spherical harmonic transform we propose a transform for signals defined on the sphere that reveals their localized directional content in the spatio spectral domain when used in conjunction with an asymmetric window function we call this transform the directional spatially localized spherical harmonic transform directional slsht which extends the slsht from the literature whose usefulness is limited to symmetric windows we present an inversion relation to synthesize the original signal from its directional slsht distribution for an arbitrary window function as an example of an asymmetric window the most concentrated band limited eigenfunction in an elliptical region on the sphere is proposed for directional spatio spectral analysis and its effectiveness is illustrated on the synthetic and mars topographic data sets finally since such typical data sets on the sphere are of considerable size and the directional slsht is intrinsically computationally demanding depending on the band limits of the signal and window a fast algorithm for the efficient computation of the transform is developed the floating point precision numerical accuracy of the fast algorithm is demonstrated and a full numerical complexity analysis is presented

spectral transform solutions to the shallow water test set abstract solutions to the test case suite proposed by williamson et al j comput phys 102 211 1992 for the shallow water equations in spherical geometry are presented the solutions have been generated using a conventional spectral transform technique combined with a semi implicit time differencing scheme for several of the test cases closed form solutions do not exist for these cases high resolution numerical integrations of the spectral transform model are used to provide reference solutions against which alternative numerical schemes and lower resolution spectral transform solutions can be evaluated the sensitivity of the high resolution numerical solutions associated with temporal truncation spatial truncation and internal dissipation are quantified in order to help bound their uncertainty in almost all of the test cases the spectral transform method proves to be a highly accurate solution technique this is particularly the case at resolutions typically associated with atmospheric general circulation models used to simulate the atmosphere s climate the most serious deficiency of the spectral transform method in the context of the test cases is the introduction of spurious minima and maxima into the solution caused by gibbs phenomenon when sharp gradients exist although this behavior is not necessarily a problem for accurately simulating fluid flow it can become a serious problem for atmospheric general circulation models if the spurious wave structures result in nonphysical states such as negative water vapor mixing ratio

spectral algorithms for vector elliptic equations in a spherical gap abstract this paper is devoted to the numerical solution of three dimensional elliptic equations of a vector field in the region contained between two concentric spheres the method of truncated series expansion in orthogonal functions is considered using chebyshev polynomials and vector spherical harmonics spectral algorithms are described for solving two point boundary value problems of the vector modes of the poisson helmholtz and biharmonic equations supplemented by boundary conditions typical of fluid dynamical applications the accuracy of the algorithms is illustrated by computing some analytical single mode examples the proposed algorithms when combined with efficient transform techniques can be used to solve multi mode nonlinear problems

the oscillatory channel flow with large wall injection in the presence of smallamplitude pressure oscillations the linearized navierstokes equations are solved to obtain an accurate description of the timedependent field in a channel having a rectangu

generalized solutions to a semilinear wave equation abstract semilinear wave equations in space dimension n 9 with singular data and various types of nonlinearities are considered we employ the framework of the algebra g l 2 of generalized functions in the general case the nonlinear term is regularized with respect to a small parameter e such that it becomes globally lipschitz for each such e this produces a family of cauchy problems to which we construct a family of solutions which in turn determines an element of g l 2 which serves as a generalized solution to the original equation for cubic nonlinear terms the equation is directly solved in g l 2 without regularizations finally in certain cases it is shown that the solution to the regularized equation coincides with the solution to the nonregularized one

on the efficient parallel computation of legendre transforms in this article we discuss a parallel implementation of efficient algorithms for computation of legendre polynomial transforms and other orthogonal polynomial transforms we develop an approach to the driscoll healy algorithm using polynomial arithmetic and present experimental results on the accuracy efficiency and scalability of our implementation the algorithms were implemented in ansi c using the bsplib communications library we also present a new algorithm for computing the cosine transform of two vectors at the same time

fast shallow water equation solvers in latitude longitude coordinates here we redirect attention to a fast pseudospectral method on the sphere developed by merilees in 1973 recently revived by fornberg in these works the required spatial derivatives are computed by the formal differentiation of one dimensional fourier series approximations to both scalar and vector functions on the surface of the sphere filters must be used to alleviate prohibitive time stepping restrictions and maintain stability on the non isotropic latitude longitude grids merilees original filter was eventually found to be unusable as it was unstable for longer runs in this paper we examine alternatives to merilees filter in particular we first use a harmonic filter that consists of a harmonic analysis followed directly by a synthesis the resulting stability and accuracy are identical to the traditional spectral transform method fewer legendre transforms are required since they are limited to the filter and not used to compute spatial derivatives in theory this approach can also be viewed as a fast spectral method since fast harmonic filters exist in the literature next we examine alternative fast fourier filters with intent to reproduce the accuracy and stability provided by the harmonic filter computational examples are provided with both high order difference and fourier derivative calculations in addition results are presented for both harmonic and fourier filters

on the associated orthogonal polynomials abstract by using the second order recurrence relation this paper gives some new results on associated orthogonal polynomials without referring to the continued fractions tool some results are very useful for obtaining the second order differential equation satisfied by the semi classical orthogonal polynomials hendriksen and van rossum 1985 maroni 1987 cf section 3 also the main formula derives from proposition 2 6 by which the fourth order differential equation satisfied by some laguerre hahn polynomials magnus 1984 is obtained cf belmehdi and ronveaux 1989 dini 1989 ronveaux 1990

a fast solver of the shallow water equations on a sphere using a combined compact difference scheme the spectral like three point combined compact difference sp ccd scheme is developed for the shallow water models on a sphere which is originally proposed by chu and fan j comput phys 140 1998 370 a standard test set proposed by williamson et al j comput phys 102 1992 211 is used to evaluate the accuracy of the new method it is shown that the accuracy of the method is comparable to that of the spherical harmonics model as a whole the relative efficiency of the method with respect to the spherical harmonics model appears from the resolution of 512 256 and it becomes significant for the higher resolutions the test results suggest that the method could be extended to numerical atmospheric models

a pseudospectral matrix method for time dependent tensor fields on a spherical shell we construct a pseudospectral method for the solution of time dependent non linear partial differential equations on a three dimensional spherical shell the problem we address is the treatment of tensor fields on the sphere as a test case we consider the evolution of a single black hole in numerical general relativity a natural strategy would be the expansion in tensor spherical harmonics in spherical coordinates instead we consider the simpler and potentially more efficient possibility of a double fourier expansion on the sphere for tensors in cartesian coordinates as usual for the double fourier method we employ a filter to address time step limitations and certain stability issues we find that a tensor filter based on spin weighted spherical harmonics is successful while two simplified non spin weighted filters do not lead to stable evolutions the derivatives and the filter are implemented by matrix multiplication for efficiency a key technical point is the construction of a matrix multiplication method for the spin weighted spherical harmonic filter as example for the efficient parallelization of the double fourier spin weighted filter method we discuss an implementation on a gpu which achieves a speed up of up to a factor of 20 compared to a single core cpu implementation

computation of spherical harmonic expansion coefficients via fft s abstract a method for numerically expanding an arbitrary function on the sphere in a series of spherical harmonics which makes use of the speed of a fast fourier transform is described discussions of the operation count storage requirements accuracy and an algebraic and a numerical example are included a comparison with straightforward integration is made throughout also a new method for evaluating the spherical harmonics is discussed

an optimal dimensionality sampling scheme on the sphere with fast spherical harmonic transforms we develop a sampling scheme on the sphere that permits accurate computation of the spherical harmonic transform and its inverse for signals band limited at l using only l 2 samples we obtain the optimal number of samples given by the degrees of freedom of the signal in harmonic space the number of samples required in our scheme is a factor of two or four fewer than existing techniques which require either 2l 2 or 4l 2 samples we note however that we do not recover a sampling theorem on the sphere where spherical harmonic transforms are theoretically exact nevertheless we achieve high accuracy even for very large band limits for our optimal dimensionality sampling scheme we develop a fast and accurate algorithm to compute the spherical harmonic transform and inverse with computational complexity comparable with existing schemes in practice we conduct numerical experiments to study in detail the stability accuracy and computational complexity of the proposed transforms we also highlight the advantages of the proposed sampling scheme and associated transforms in the context of potential applications

computational harmonic analysis for tensor fields on the two sphere in this paper we describe algorithms for the numerical computation of fourier transforms of tensor fields on the two sphere s2 these algorithms reduce the computation of an expansion on tensor spherical harmonics to expansions in scalar spherical harmonics and hence can take advantage of recent improvements in the efficiency of computation of scalar spherical harmonic transforms

a novel sampling theorem on the sphere we develop a novel sampling theorem on the sphere and corresponding fast algorithms by associating the sphere with the torus through a periodic extension the fundamental property of any sampling theorem is the number of samples required to represent a band limited signal to represent exactly a signal on the sphere band limited at l all sampling theorems on the sphere require o l2 samples however our sampling theorem requires less than half the number of samples of other equiangular sampling theorems on the sphere and an asymptotically identical but smaller number of samples than the gauss legendre sampling theorem the complexity of our algorithms scale as o l3 however the continual use of fast fourier transforms reduces the constant prefactor associated with the asymptotic scaling considerably resulting in algorithms that are fast furthermore we do not require any precomputation and our algorithms apply to both scalar and spin functions on the sphere without any change in computational complexity or computation time we make our implementation of these algorithms available publicly and perform numerical experiments demonstrating their speed and accuracy up to very high band limits finally we highlight the advantages of our sampling theorem in the context of potential applications notably in the field of compressive sampling

second generation curvelets on the sphere curvelets are efficient to represent highly anisotropic signal content such as a local linear and curvilinear structure first generation curvelets on the sphere however suffered from blocking artefacts we present a new second generation curvelet transform where scale discretized curvelets are constructed directly on the sphere scale discretized curvelets exhibit a parabolic scaling relation are well localized in both spatial and harmonic domains support the exact analysis and synthesis of both scalar and spin signals and are free of blocking artefacts we present fast algorithms to compute the exact curvelet transform reducing computational complexity from mathcal o l 5 to mathcal o l 3 log 2 l for signals band limited at l the implementation of these algorithms is made publicly available finally we present an illustrative application demonstrating the effectiveness of curvelets for representing directional curve like features in natural spherical images

a technique for the numerical verification of asymptotic expansions a technique is presented that uses numerical solutions to verify the order of the accuracy of an asymptotic expansion for several types of problems this technique may be introduced in any beginning course on asymptotic or perturbation methods but is equally suited to verifying advanced asymptotic results

multigrid methods for two and three dimensional poisson type equations on the sphere abstract the efficient solution of two and three dimensional poisson and helmholtz type equations on the sphere with multigrid methods is the subject of this work the strong anisotropic behavior of the laplacian in spherical coordinates poses severe requirements on the relaxation procedures in order to attain a full multigrid effectiveness the introduction of a new technique combined relaxation results in high efficiency even in cases with varying anisotropy direction where alternating line or plane relaxations would normally be required the combination of this new relaxation technique with semi coarsening strategies is also studied even for moderate grid sizes the multigrid solvers developed are shown to be competitive with two dimensional direct fast solvers in cases where the latter can be applied the optimal computational complexity of the multigrid methods their broad applicability in contrast to fast direct solvers and their high efficiency also in three spatial dimensions make these methods very interesting for applications

higher mean flow approximation for solid rocket motors with radially regressing walls the bulk gas motion in a circular port rocket motor is described using a rotational incompressible and viscous flow model that incorporates the effect of wall regression the mathematical idealization developed is also applicable to semi open porous tubes with expanding walls based on mass conservation a linear variation in the mean axial velocity is ascertained this relationship suggests investigating a spatial transformation of the proudman johnson form with the use of similar arguments a temporal transformation is also introduced when these transformations are applied in both space and time the navier stokes equations are reduced to a single nonlinear fourth order differential equation following this exact navier stokes reduction the resulting problem is solved using variation of parameters and small parameter perturbations the asymptotic solutions for the velocity pressure vorticity and shear are obtained as function of the injection reynolds number re and the dimensionless regression ratio a by way of verification it is shown that as re 0 yuan and finkelstein s solutions can be restored from ours similarly as re 0 culick s inviscid profile is recovered it is demonstrated that for a range of small re inviscid solutions are practical however for fast burning propellants under development the inviscid assumption deteriorates because it is applicable over a broader range of operating parameters the current analysis leads to a closed form mean flow solution that can be used instead of the inviscid profile to 1 prescribe an adjusted aeroacoustic field 2 describe the so called acoustic boundary layer 3 evaluate the viscous and rotational contributions to the acoustic stability growth rate factor 4 track the evolution of hydrodynamic instability and 5 accurately simulate the internal gasdynamics in rapidly regressing motors and cold flow experiments with medium to high levels of injection

convergence acceleration of fourier series by analytical and numerical application of poisson s formula a form of the poisson summation formula appropriate for the transformation of infinite fourier series is derived and shown to be applicable to slowly converging series arising in the solution of laplace s equation in a rectangular electrochemical cell the result is a rapidly convergent fourier series which is particularly useful in giving the potential close to the electrode surface results obtained from the analytical expression are found to be in close agreement with those resulting from numerical evaluation of the fourier integrals using simpson s rule and the euler transformation and direct summation by using goertzel s 1958 algorithm

weak solutions and development of singularities of the su 2 σ model pour toutes donnees initiales d energie finie le modele sigma o k a des solutions faibles globales en chaque dimension d espace temps

a fast spherical harmonics transform algorithm the spectral method with discrete spherical harmonics transform plays an important role in many applications in spite of its advantages the spherical harmonics transform has a drawback of high computational complexity which is determined by that of the associated legendre transform and the direct computation requires time of o n3 for cut off frequency n in this paper we propose a fast approximate algorithm for the associated legendre transform our algorithm evaluates the transform by means of polynomial interpolation accelerated by the fast multipole method fmm the divide and conquer approach with split legendre functions gives computational complexity o n2 log n experimental results show that our algorithm is stable and is faster than the direct computation for n 511

efficient computation of slepian functions for arbitrary regions on the sphere in this paper we develop a new method for the fast and memory efficient computation of slepian functions on the sphere slepian functions which arise as the solution of the slepian concentration problem on the sphere have desirable properties for applications where measurements are only available within a spatially limited region on the sphere and or a function is required to be analyzed over the spatially limited region slepian functions are currently not easily computed for large band limits for an arbitrary spatial region due to high computational and large memory storage requirements for the special case of a polar cap the symmetry of the region enables the decomposition of the slepian concentration problem into smaller subproblems and consequently the efficient computation of slepian functions for large band limits by exploiting the efficient computation of slepian functions for the polar cap region on the sphere we develop a formulation supported by a fast algorithm for the approximate computation of slepian functions for an arbitrary spatial region to enable the analysis of modern datasets that support large band limits for the proposed algorithm we carry out accuracy analysis of the approximation computational complexity analysis and review of memory storage requirements we illustrate through numerical experiments that the proposed method enables faster computation and has smaller storage requirements while allowing for sufficiently accurate computation of the slepian functions

the oscillatory pipe flow with arbitrary wall injection the linearized navierstokes equations play a central role in describing the unsteady motion of a viscous fluid inside a porous tube asymptotic solutions of these equations have been found and here

a fast spherical filter with uniform resolution this paper introduces a fast algorithm for obtaining a uniform resolution representation of a function known at a latitude longitude grid on the surface of a sphere equivalent to a triangular isotropic truncation of the spherical harmonic coefficients for the function the proposedspectral truncation method which is based on the fast multipole method and the fast fourier transform projects the function to a space with uniform resolution while avoiding surface harmonic transformations the method requireso n2logn operations foro n2 grid points as opposed too n3 operations for the standard spectral transform method providing a reduced complexity spectral method obviating the pole problem in the integration of time dependent partial differential equations on the sphere the filter s performance is demonstrated with numerical examples

vortical and acoustical mode coupling inside a porous tube with uniform wall suction this paper considers the oscillatory motion of gases inside a long porous tube of the closed open type in particular the focus is placed on describing an analytical solution for the internal acoustico vortical coupling that arises in the presence of appreciable wall suction this unsteady field is driven by longitudinal oscillatory waves that are triggered by small unavoidable fluctuations in the wall suction speed under the assumption of small amplitude oscillations the time dependent governing equations are linearized through a regular perturbation of the dependent variables further application of the helmholtz vector decomposition theorem enables us to discriminate between acoustical and vortical equations after solving the wave equation for the acoustical contribution the boundary driven vortical field is considered the method of matched asymptotic expansions is then used to obtain a closed form solution for the unsteady momentum equation developing from flow decomposition an exact series expansion is also derived and shown to coincide with the numerical solution for the problem the numerically verified end results suggest that the asymptotic scheme is capable of providing a sufficiently accurate solution this is due to the error associated with the matched asymptotic expansion being smaller than the error introduced in the navier stokes linearization a basis for comparison is established by examining the evolution of the oscillatory field in both space and time the corresponding boundary layer behavior is also characterized over a range of oscillation frequencies and wall suction velocities in general the current solution is found to exhibit features that are consistent with the laminar theory of periodic flows by comparison to the sexl profile in nonporous tubes the critically damped solution obtained here exhibits a slightly smaller overshoot and depth of penetration these features may be attributed to the suction effect that tends to attract the shear layers closer the wall