a new approach to inverse spectral theory i fundamental formalism we present a new approach distinct from gelfand levitan to the the orem of borg marchenko that the m function equivalently spectral measure for a finite interval or half line schrodinger operator determines the poten tial our approach is an analog of the continued fraction approach for the moment problem we prove there is a representation for the m function m 2 r b

a fast algorithm for simulating vesicle flows in three dimensions vesicles are locally inextensible fluid membranes that can sustain bending in this paper we extend the study of veerapaneni et al s k veerapaneni d gueyffier g biros d zorin a numerical method for simulating the dynamics of 3d axisymmetric vesicles suspended in viscous flows journal of computational physics 228 19 2009 7233 7249 to general non axisymmetric vesicle flows in three dimensions although the main components of the algorithm are similar in spirit to the axisymmetric case spectral approximation in space semi implicit time stepping scheme important new elements need to be introduced for a full 3d method in particular spatial quantities are discretized using spherical harmonics and quadrature rules for singular surface integrals need to be adapted to this case an algorithm for surface reparameterization is needed to ensure stability of the time stepping scheme and spectral filtering is introduced to maintain reasonable accuracy while minimizing computational costs to characterize the stability of the scheme and to construct preconditioners for the iterative linear system solvers used in the semi implicit time stepping scheme we perform a spectral analysis of the evolution operator on the unit sphere by introducing these algorithmic components we obtain a time stepping scheme that circumvents the stability constraint on the time step and achieves spectral accuracy in space we present results to analyze the cost and convergence rates of the overall scheme to illustrate the applicability of the new method we consider a few vesicle flow interaction problems a single vesicle in relaxation sedimentation shear flows and many vesicle flows

a computational method for solving the branched channel flow problem in the present work a method is suggested to find the solution of poisson s equation with dirichlet boundary conditions in polar coordinates this method is expressed over the unit circle and consequently over the upper half plane from which geometry we find the solution to poisson s equation over the branched channel domain the results seems very reasonable and the derived transformation from unit circle to y shape channel is accurate poisson s equation can be solved similarly over much other geometry the solution looks very encouraging and resembles that is experimentally known

evolution equation for short surface waves on water of finite depth abstract we address the question of determining the evolution equation for surface waves propagating in water whose depth is much larger than the typical wavelength of the surface disturbance we avoid making the usual approximation of supposing the evolution to be given in the form of a modulated wave packet we treat the problem by means of a conformal transformation allowing to explicitly find the dirichlet to neumann operator for the problem together with asymptotic expansions in parameters measuring the nonlinearity and depth this allows us to obtain an equation in physical variables valid in the weakly nonlinear deep water regime the equation is an integro differential equation which reduces to known cases for infinite depth we discuss solutions in a perturbative setting and show that the evolution equation describes stokes like waves

pseudospectral solution of the two dimensional navier stokes equations in a disk an efficient and accurate algorithm for solving the two dimensional 2d incompressible navier stokes equations on a disk with no slip boundary conditions is described the vorticity stream function formulation of these equations is used and spatially the vorticity and stream functions are expressed as fourier chebyshev expansions the poisson and helmholtz equations which arise from the implicit explicit time marching scheme are solved as banded systems using a postconditioned spectral tau method the polar coordinate singularity is handled by expanding fields radially over the entire diameter using a parity modified chebyshev series and building partial regularity into the vorticity the no slip boundary condition is enforced by transferring one of the two boundary conditions imposed on the stream function onto the vorticity via a solvability constraint significant gains in run times were realized by parallelizing the code in message passage interface mpi

accurate navier stokes investigation of transitional and turbulent flows in a circular pipe a new fast accurate and roundoff error robust numerical technique for integrating unsteady incompressible navier stokes equations in cylindrical coordinates is presented the algorithm is based on a special change of dependent variables which avoids the singularity problem and provides high accuracy and computational efficiency accuracy and stability of the method are thoroughly tested for the model problem of transitional and turbulent flows in an infinite circular pipe verification of the algorithm includes two issues first spectral characteristics of the hagen poiseuille flow stability problem are compared with those of the discrete linearized navier stokes operator secondly the results of direct navier stokes simulation of all stages of laminar turbulent transition in a circular pipe at reynolds number of 4000 are presented time evolution of finite amplitude disturbances of laminar flow was calculated until the statistically stationary turbulent flow regime was established in addition to common statistical analysis the possibility of turbulence description by means of velocity fields having certain symmetries is examined thus the algorithm presented seems to be a ready to use robust tool for accurate investigation and further parametric studies of both transition mechanisms and fully developed turbulent flow regimes

an efficient direct solver for multidimensional elliptic robin boundary value problems using a legendre spectral galerkin method in this paper a legendre galerkin method for solving second order elliptic differential equations subject to the most general nonhomogeneous robin boundary conditions is presented the homogeneous robin boundary conditions are satisfied exactly by expanding the unknown variable using a polynomial basis of functions which are built upon the legendre polynomials the direct solution algorithm here developed for the homogeneous robin problem in two dimensions relies upon a tensor product process nonhomogeneous robin data are taken into account by means of a lifting such a lifting is performed in two successive steps the first one to account for the data specified at the corners and the second one to account for the boundary values prescribed in the interior of the sides numerical results indicating the high accuracy and effectiveness of these algorithms are presented

compatibility conditions for time dependent partial differential equations and the rate of convergence of chebyshev and fourier spectral methods abstract compatibility conditions for partial differential equations pdes are an infinite set of relations between the initial conditions the pde and the boundary conditions which are necessary and sufficient for the solution to be c that is infinitely differentiable everywhere on the computational domain including the boundaries since the performance of chebyshev spectral and spectral element methods is dramatically reduced when the solution is not c one would expect that the compatibility conditions would be a major theme in the spectral literature instead it has been completely ignored therefore we pursue three goals here first we present a proof of the compatibility conditions in a simplified form that does not require functional analysis second we analyze the connection between the compatibility conditions and the rate of convergence of chebyshev methods lastly we describe strategies for slightly adjusting initial conditions so that the compatibility conditions are satisfied

on the spectrum of second order differential operators with complex coefficients the main objective of this paper is to extend the pioneering work of sims on second order linear differential equations with a complex coefficient in which he obtains an analogue of the titchmarsh weyl theory and classification the generalization considered exposes interesting features not visible in the special case in sims paper from 1957 an m function is constructed which is either unique or a point on a limit circle and the relationship between its properties and the spectrum of underlying m accretive differential operators analysed the paper is a contribution to the study of non self adjoint operators in general the spectral theory of such operators is rather fragmentary and further study is being driven by important physical applications to hydrodynamics electro magnetic theory and nuclear physics for instance

a well conditioned collocation method using a pseudospectral integration matrix in this paper a well conditioned collocation method is constructed for solving general p th order linear differential equations with various types of boundary conditions based on a suitable birkhoff interpolation we obtain a new set of polynomial basis functions that results in a collocation scheme with two important features the condition number of the linear system is independent of the number of collocation points and the underlying boundary conditions are imposed exactly moreover the new basis leads to an exact inverse of the pseudospectral differentiation matrix of the highest derivative at interior collocation points which is therefore called the pseudospectral integration matrix psim we show that psim produces the optimal integration preconditioner and stable collocation solutions with even thousands of points

a proof of the local borg marchenko theorem in 2 5 a local version of a basic uniqueness theorem in inverse spectral theory is given this paper gives a simple proof of this theorem

an optimal galerkin scheme to solve the kinematic dynamo eigenvalue problem in a full sphere the kinematic dynamo approximation describes the generation of magnetic field in a prescribed flow of electrically conducting liquid one of its main uses is as a proof of concept tool to test hypotheses about self exciting dynamo action indeed it provided the very first quantitative evidence for the possibility of the geodynamo despite its utility due to the requirement of resolving fine structures historically numerical work has proven difficult and reported solutions were often plagued by poor convergence in this paper we demonstrate the numerical superiority of a galerkin scheme in solving the kinematic dynamo eigenvalue problem in a full sphere after adopting a poloidal toroidal decomposition and expanding in spherical harmonics we express the radial dependence in terms of a basis of exponentially convergent orthogonal polynomials each basis function is constructed from a terse sum of one sided jacobi polynomials that not only satisfies the boundary conditions of matching to an electrically insulating exterior but is everywhere infinitely differentiable including at the origin this galerkin method exhibits more rapid convergence for a given problem size than any other scheme hitherto reported as demonstrated by a benchmark of the magnetic diffusion problem and by comparison to numerous kinematic dynamos from the literature in the axisymmetric flows we consider in this paper at a magnetic reynolds number of o 100 a convergence of 9 significant figures in the most unstable eigenvalue requires only 40 radial basis functions alternatively 4 significant figures requires 20 radial functions the terse radial discretization becomes particularly advantageous when considering flows whose associated numerical solution requires a large number of coupled spherical harmonics we exploit this new method to confirm the tentatively proposed positive growth rate of the planar flow of bachtiar et al 4 thereby verifying a counter example to the zel dovich anti dynamo theorem in a spherical geometry

tensor calculus in polar coordinates using jacobi polynomials spectral methods are an efficient way to solve partial differential equations on domains possessing certain symmetries the utility of a method depends strongly on the choice of spectral basis in this paper we describe a set of bases built out of jacobi polynomials and associated operators for solving scalar vector and tensor partial differential equations in polar coordinates on a unit disk by construction the bases satisfy regularity conditions at r 0 for any tensorial field the coordinate singularity in a disk is a prototypical case for many coordinate singularities the work presented here extends to other geometries the operators represent covariant derivatives multiplication by azimuthally symmetric functions and the tensorial relationship between fields these arise naturally from relations between classical orthogonal polynomials and form a heisenberg algebra other past work uses more specific polynomial bases for solving equations in polar coordinates the main innovation in this paper is to use a larger set of possible bases to achieve maximum bandedness of linear operations we provide a series of applications of the methods illustrating their ease of use and accuracy

an accurate second order approximation factorization method for time dependent incompressible navier stokes equations in spherical polar coordinates a finite difference method for solving three dimensional time dependent incompressible navier stokes equations in spherical polar coordinates is presented in detail a new algorithm which is second order accurate in time and space is considered and decoupling between the velocity and the pressure is achieved by this algorithm further the numerical method is tested by computing the spherical couette flow between two concentric spheres with the inner one rotating a comparison of the numerical solutions with available numerical results and experimental measurements is made it is demonstrated that the numerical code is valid for solving three dimensional unsteady incompressible navier stokes equations in spherical polar coordinates

water waves over a strongly undulating bottom two dimensional free surface potential flows of an ideal fluid over a strongly inhomogeneous bottom are investigated with the help of conformal mappings weakly nonlinear and exact nonlinear equations of motion are derived by the variational method for an arbitrary seabed shape parametrized by an analytical function as applications of this theory the band structure of linear waves over periodic bottoms is calculated and the evolution of strong solitary waves running from a deep region to a shallow region is numerically simulated

a divergence free spectral expansions method for three dimensional flows in spherical gap geometries abstract a spectral method for the solution of the incompressible navier stokes equations in spherical gap geometries is presented the method uses divergence free vector expansions which inherently satisfy the boundary conditions 1 basis and test functions are constructed from chebyshev polynomials and vector spherical harmonics vsh yielding a petrov galerkin weighted residual method that produces spectral convergence no rotational nor equatorial symmetry of the flow field is implicitly imposed the approach makes extensive use of the convenient properties of the vsh which are presented in a computationally suitable form whenever possible the alias free implementation of the method rests upon a standard explicit implicit time integration technique a vsh chebyshev vector transform with two fast directions is also developed and briefly presented several test cases are used to validate the resulting initial boundary value code axisymmetric basic spherical couette flow computations are compared with available numerical results while a three dimensional spiral taylor gortler vortex flow simulation is tested against experimental measurements very good agreement is found in all cases

variational data assimilation for the initial value dynamo problem the secular variation of the geomagnetic field as observed at the earth s surface results from the complex magnetohydrodynamics taking place in the fluid core of the earth one way to analyze this system is to use the data in concert with an underlying dynamical model of the system through the technique of variational data assimilation in much the same way as is employed in meteorology and oceanography the aim is to discover an optimal initial condition that leads to a trajectory of the system in agreement with observations taking the earth s core to be an electrically conducting fluid sphere in which convection takes place we develop the continuous adjoint forms of the magnetohydrodynamic equations that govern the dynamical system together with the corresponding numerical algorithms appropriate for a fully spectral method these adjoint equations enable a computationally fast iterative improvement of the initial condition that determines the system evolution the initial condition depends on the three dimensional form of quantities such as the magnetic field in the entire sphere for the magnetic field conservation of the divergence free condition for the adjoint magnetic field requires the introduction of an adjoint pressure term satisfying a zero boundary condition we thus find that solving the forward and adjoint dynamo system requires different numerical algorithms in this paper an efficient algorithm for numerically solving this problem is developed and tested for two illustrative problems in a whole sphere one is a kinematic problem with prescribed velocity field and the second is associated with the hall effect dynamo exhibiting considerable nonlinearity the algorithm exhibits reliable numerical accuracy and stability using both the analytical and the numerical techniques of this paper the adjoint dynamo system can be solved directly with the same order of computational complexity as that required to solve the forward problem these numerical techniques form a foundation for ultimate application to observations of the geomagnetic field over the time scale of centuries

a fourier legendre spectral element method in polar coordinates in this paper a new fourier legendre spectral element method based on the galerkin formulation is proposed to solve the poisson type equations in polar coordinates the 1 r singularity at r 0 is avoided by using gauss radau type quadrature points in order to break the time step restriction in the time dependent problems the clustering of collocation points near the pole is prevented through the technique of domain decomposition in the radial direction a number of poisson type equations subject to the dirichlet or neumann boundary condition are computed and compared with the results in literature which reveals a desirable result

a fully spectral methodology for magnetohydrodynamic calculations in a whole sphere abstract we present a fully spectral methodology for magnetohydrodynamic mhd calculations in a whole sphere the use of jones worland polynomials for the radial expansion guarantees that the physical variables remain infinitely differentiable throughout the spherical volume furthermore we present a mathematically motivated and systematic strategy to relax the very stringent time step constraint that is present close to the origin when a spherical harmonic expansion is used for the angular direction the new constraint allows for significant savings even on relatively simple solutions as demonstrated on the so called full sphere benchmark a specific problem with a very accurately known solution the numerical implementation uses a 2d data decomposition which allows it to scale to thousands of cores on present day high performance computing systems in addition to validation results we also present three new whole sphere dynamo solutions that present a relatively simple structure

convergence analysis of spectral collocation methods for a singular differential equation solutions of partial differential equations with coordinate singularities often have special behavior near the singularities which forces them to be smooth special treatment for these coordinate singularities is necessary in spectral approximations in order to avoid degradation of accuracy and efficiency it has been observed numerically in the past that for a scheme to attain high accuracy it is unnecessary to impose all the pole conditions the constraints representing the special solution behavior near singularities in this paper we provide a theoretical justification for this observation specifically we consider an existing approach which uses a pole condition as the boundary condition at a singularity and solves the reformulated boundary value problem with a commonly used gauss lobatto collocation scheme spectral convergence of the legendre and chebyshev collocation methods is obtained for a singular differential equation arising from polar and cylindrical geometries

axisymmetric pulse train solutions in narrow gap spherical couette flow abstract we numerically compute the flow induced in a spherical shell by fixing the outer sphere and rotating the inner one the aspect ratio r o r i r i is set at 0 04 and 0 02 and in each case the reynolds number measuring the inner sphere s rotation rate is increased to 10 beyond the first bifurcation from the basic state flow for 0 04 the initial bifurcations are the same as in previous numerical work at 0 154 and result in steady one and two vortex states further bifurcations yield travelling wave solutions similar to previous analytic results valid in the 0 limit for 0 02 the steady one vortex state no longer exists and the first bifurcation is directly to these travelling wave solutions consisting of pulse trains of taylor vortices travelling toward the equator from both hemispheres and annihilating there in distinct phase slip events we explore these time dependent solutions in detail and find that they can be both equatorially symmetric and asymmetric as well as periodic or quasi periodic in time

spectral approximation on the unit ball spectral approximation by polynomials on the unit ball is studied in the frame of the sobolev spaces w s p mathbb b d 1 p infty the main results give sharp estimates on the order of approximation by polynomials in the sobolev spaces and explicit construction of approximating polynomials one major effort lies in understanding the structure of orthogonal polynomials with respect to an inner product of the sobolev space w 2 s mathbb b d as an application a direct and efficient spectral galerkin method based on our orthogonal polynomials is proposed for the second and fourth order elliptic equations on the unit ball its optimal error estimates are explicitly derived for both procedures in the sobolev spaces and finally numerical examples are presented to illustrate the theoretic results

on local borg marchenko uniqueness results we provide a new short proof of the following fact first proved by one of us in 1998 if two weyl titchmarsh m functions m j z of two schrodinger operators h j d 2 dx 2 q j j 1 2 in l 2 o r o r are exponentially close that is m 1 z m 2 z z o e 2 im z 1 2 a o r then q 1 q 2 a e on o the result applies to any boundary conditions at x o and x r and should be considered a local version of the celebrated borg marchenko uniqueness result which is quickly recovered as a corollary to our proof moreover we extend the local uniqueness result to matrix valued schrodinger operators

spectral radial basis functions for full sphere computations the singularity of cylindrical or spherical coordinate systems at the origin imposes certain regularity conditions on the spectral expansion of any infinitely differentiable function there are two efficient choices of a set of radial basis functions suitable for discretising the solution of a partial differential equation posed in either such geometry one choice is methods based on standard chebyshev polynomials although these may be efficiently computed using fast transforms differentiability to all orders of the obtained solution at the origin is not guaranteed the second is the so called one sided jacobi polynomials that explicitly satisfy the required behavioural conditions in this paper we compare these two approaches in their accuracy differentiability and computational speed we find that the most accurate and concise representation is in terms of one sided jacobi polynomials however due to the lack of a competitive fast transform chebyshev methods may be a better choice for some computationally intensive timestepping problems and indeed will yield sufficiently although not infinitely differentiable solutions provided they are adequately converged

expansion functions for two dimensional incompressible fluid flow in arbitrary domains expansion functions are presented for two dimensional incompressible fluid flow in arbitrary domains that optimally conserve the 2d structure of vortex dynamics this is obtained by conformal mapping of the domain onto a circle and by constructing orthogonal radial polynomials and angular harmonics on the new domain such that the kinetic energy is diagonal and the separate components satisfy all of the required physical boundary conditions

on inverse spectral problem for non selfadjoint sturm liouville operator on a finite interval an inverse spectral problem is studied for a non selfadjoint sturm liouville operator on a finite interval with an arbitrary behavior of the spectrum the spectral data introduced generalize the classical discrete spectral data corresponding to the specification of the spectral function in the selfadjoint case the connection with other types of spectral characteristics is investigated and a uniqueness theorem is proved a constructive procedure for solving the inverse problem is given

modeling extreme waves based on equations of potential flow with a free surface a method for numerical investigation of nonlinear wave dynamics based on direct hydrodynamical modeling of 1 d potential periodic surface waves is created the model is a part of an interactive wind wave model using a non stationary conformal mapping the principal equations are rewritten in a surface following coordinate system and reduced to two simple evolutionary equations for the elevation and the velocity potential of the surface fourier expansion is used to approximate these equations high accuracy was confirmed by validation of the non stationary model against known solutions and by comparison between the results obtained with different resolution in the horizontal the method developed is applied to the simulation of waves evolution with different initial conditions numerical experiments with initially monochromatic waves with different steepness show that the model is able to simulate breaking conditions when the surface becomes a multi valued function of the horizontal coordinate an estimate of the critical initial wave height that divides between non breaking and eventually breaking waves is obtained simulations of nonlinear evolution of a wave field is represented initially by two modes with close wave numbers amplitude modulation and a wave field with a phase modulation both runs result in the appearance of large and very steep waves these also break if the initial amplitudes are sufficiently large

a terrain following boussinesq system a long wave model is derived asymptotically from the nonlinear potential theory equations the flow regime of interest is incompressible irrotational and inviscid asymptotic analysis leads to a weakly nonlinear weakly dispersive variable coefficient boussinesq system valid for a wide class of topographies the mild slope hypothesis is not required and rapidly varying topographies are also considered in analogy with atmospheric models we use a terrain following coordinate system the novelty is that this coordinate system naturally suggests the weighted averaging of terrain following velocity components as opposed to the depth average of horizontal velocity components found in standard shallow water formulations furthermore a schwarz christoffel toolbox is used to provide additional insight on these new results regarding applications the proposed model can be used for studying solitary waves interacting with fine scale inhomogeneities a theme of great interest the terrain following model als

an application of one sided jacobi polynomials for spectral modeling of vector fields in polar coordinates a spectral tau method is proposed for solving vector field equations defined in polar coordinates the method employs one sided jacobi polynomials as radial expansion functions and fourier exponentials as azimuthal expansion functions all the regularity requirements of the vector field at the origin and the physical boundary conditions at a circumferential boundary are exactly satisfied by adjusting the additional tau coefficients of the radial expansion polynomials of the highest order the proposed method is applied to linear and nonlinear dispersive time evolution equations of hyperbolic type describing internal kelvin and poincare waves in a shallow stratified lake on a rotating plane

a spectral method for polar coordinates a new set of polynomial functions that can be used in spectral expansions of c functions in polar coordinates r is defined by a singular sturm liouville equation with the use of the basis functions the spectral representations remain analytic at the pole despite the coordinate singularity because the pole condition is exactly satisfied at the origin for ag azimuthal modes not just a few of the gravest modes which is the usual case based on recurrence relations fast and stable numerical operators for 1 r r d dr the laplacian and helmholtz operators and their inverses are developed although the spacings in the azimuthal direction of the collocation points near the origin are small i e 1 m2 where m is the number of radial modes the explicit numerical method for euler s equation is not stiff at the origin namely the cfl number is o 1 where the grid size in is defined as m i e the maximum allowable timestep is proportional to 1 m not 1 m2

an efficient spectral projection method for the navier stokes equations in cylindrical geometries an efficient and accurate numerical scheme is presented for the three dimensional navier stokes equations in primitive variables in a cylinder the scheme is based on a spectral galerkin approximation for the space variables and a second order projection scheme for time the new spectral projection scheme is implemented to simulate unsteady incompressible flows in a cylinder

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 direct spectral collocation poisson solver in polar and cylindrical coordinates in this paper we present a direct spectral collocation method for the solution of the poisson equation in polar and cylindrical coordinates the solver is applied to the poisson equations for several different domains including a part of a disk an annulus a unit disk and a cylinder unlike other poisson solvers for geometries such as unit disks and cylinders no pole condition is involved for the present solver the method is easy to implement fast and gives spectral accuracy we also use the weighted interpolation technique and nonclassical collocation points to improve the convergence

on magnetic energy instability in spherical stationary flows eigenmode or linear analysis of the induction equation yields magnetic field structures which grow or decay exponentially in time under the influence of a prescribed flow geometry however due to

radial collocation methods for the onset of convection in rotating spheres the viability of using collocation methods in radius and spherical harmonics in the angular variables to calculate convective flows in full spherical geometry is examined as a test problem the stability of the conductive state of a self gravitating fluid sphere subject to rotation and internal heating is considered a study of the behavior of different radial meshes previously used by several authors in polar coordinates including or not the origin is first performed the presence of spurious modes due to the treatment of the singularity at the origin to the spherical harmonics truncation and to the initialization of the eigenvalue solver is shown and ways to eliminate them are presented finally to show the usefulness of the method the neutral stability curves at very high taylor and moderate and small prandtl numbers are calculated and shown several radial collocation meshes have been tested to solve convective flows in full spherical geometry the inclusion of the origin in the mesh together with a single regularity condition is enough to obtain good results for eigenvalue problems several sources of spurious modes have been detected and methods to eliminate them are proposed the onset of convection in an internally heated sphere has been studied and asymptotic laws for high taylor and moderate prandtl numbers have been found

spectral collocation methods and polar coordinate singularities this paper considers the numerical solution of elliptic differential equations on the unit disk using polar coordinates the disk is mapped onto a rectangle the resulting transformed problem is solved by a method related to collocation since the origin is a coordinate singularity some natural trial functions are singular there and a special technique is applied to use zero as a collocation point for poisson and helmholtz equations a fast algorithm with an operation count of o n2 log n is presented numerical results show the different stability and convergence properties of the algorithms

a local borg marchenko theorem for complex potentials the borg marchenko theorem states that the weyl titchmarsh m function of the differential expression d2 dx2 q with a real valued potential q determines this potential uniquely we investigate the validity of the borg marchenko theorem and its local version for complex valued potentials

a spectral b spline method for the navier stokes equations in unbounded domains the numerical method presented in this paper aims at solving the incompressible navier stokes equations in unbounded domains the problem is formulated in cylindrical coordinates and the method is based on a galerkin approximation scheme that makes use of vector expansions that exactly satisfy the continuity constraint more specifically the divergence free basis vector functions are constructed with fourier expansions in the and z directions while mapped b splines are used in the semi infinite radial direction special care has been taken to account for the particular analytical behaviors at both end points r 0 and r a modal reduction algorithm has also been implemented in the azimuthal direction allowing for a relaxation of the cfl constraint on the timestep size and a possibly significant reduction of the number of dof the time marching is carried out using a mixed quasi third order scheme besides the advantages of a divergence free formulation and a quasi spectral convergence the local character of the b splines allows for a great flexibility in node positioning while keeping narrow bandwidth matrices numerical tests show that the present method compares advantageously with other similar methodologies using purely global expansions

poloidal toroidal decomposition in a finite cylinder i influence matrices for the magnetohydrodynamic equations the navier stokes equations and magnetohydrodynamics equations are written in terms of poloidal and toroidal potentials in a finite cylinder this formulation insures that the velocity and magnetic fields are divergence free by construction but leads to systems of partial differential equations of higher order whose boundary conditions are coupled the influence matrix technique is used to transform these systems into decoupled parabolic and elliptic problems the magnetic field in the induction equation is matched to that in an exterior vacuum by means of the dirichlet to neumann mapping thus eliminating the need to discretize the exterior the influence matrix is scaled in order to attain an acceptable condition number

a borg levinson theorem for trees we prove that the dirichelet to neumann map for a schrodinger operator on a finite simply connected tree determines uniquely the potential on that tree

galerkin orthogonal polynomials abstract the galerkin method offers a powerful tool in the solution of differential equations and function approximation on the real interval 1 1 by expanding the unknown function in appropriately chosen global basis functions each of which explicitly satisfies the given boundary conditions in general this scheme converges exponentially fast and almost always supplies the most terse representation of a smooth solution to date typical schemes have been defined in terms of a linear combination of two jacobi polynomials however the resulting functions do not inherit the expedient properties of the jacobi polynomials themselves and the basis set will not only be non orthogonal but may in fact be poorly conditioned using a gram schmidt procedure it is possible to construct in an incremental fashion polynomial basis sets that not only satisfy any linear homogeneous boundary conditions but are also orthogonal with respect to the general weighting function 1 x 1 x however as it stands this method is not only cumbersome but does not provide the structure for general index n of the functions and obscures their dependence on the parameters in this paper it is shown that each of these galerkin basis functions as calculated by the gram schmidt procedure may be written as a linear combination of a small number of jacobi polynomials with coefficients that can be determined moreover this terse analytic representation reveals that for large index the basis functions behave asymptotically like the single jacobi polynomial p n x this new result shows that such galerkin bases not only retain exponential convergence but expedient function fitting properties too in much the same way as the jacobi polynomials themselves this powerful methodology of constructing galerkin basis sets is illustrated by many examples and it is shown how the results extend to polar geometries in exploring more generalised definitions of orthogonality involving derivatives we discuss how a large class of differential operators may be discretised by galerkin schemes and represented in a sparse fashion by the inverse of band limited matrices

modified jacobi bernstein basis transformation and its application to multi degree reduction of bezier curves this paper reports new modified jacobi polynomials mjps we derive the basis transformation between mjps and bernstein polynomials and vice versa this transformation is merging the perfect least square performance of the new polynomials together with the geometrical insight of bernstein polynomials the mjps with indexes corresponding to the number of endpoints constraints are the natural basis functions for least square approximation of bezier curves using mjps leads us to deal with the constrained jacobi polynomials and the unconstrained jacobi polynomials as orthogonal polynomials the mjps are automatically satisfying the homogeneous boundary conditions thereby the main advantage of using mjps in multi degree reduction of bezier curves on computer aided geometric design cagd is that the constraints in cagd are also satisfied and that decreases the steps of multi degree reduction algorithm several numerical results for the multi degree reduction of bezier curves on cagd are given we establish and prove the basis transformation between the modified jacobi polynomials mjps and bernstein polynomials and vice versa this transformation is merging the perfect least square performance of the mjps with the geometrical insight of the bernstein form the mjps with indexes corresponding to the number of endpoint constraints are the natural basis functions for least square approximation of bezier curves several numerical results are considered

navier stokes spectral solver in a sphere the paper presents the first implementation of a primitive variable spectral method for calculating viscous flows inside a sphere a variational formulation of the navier stokes equations is adopted using a fractional step time discretization with the classical second order backward difference scheme combined with explicit extrapolation of the nonlinear term the resulting scalar and vector elliptic equations are solved by means of the direct spectral solvers developed recently by the authors the spectral matrices for radial operators are characterized by a minimal sparsity diagonal stiffness and tridiagonal mass matrix closed form expressions of their nonzero elements are provided here for the first time showing that the condition number of the relevant matrices grows as the second power of the truncation order a new spectral elliptic solver for the velocity unknown in spherical coordinates is also described that includes implicitly the coriolis force in a rotating frame but requires a minimal coupling between the modal velocity components in the fourier space the numerical tests confirm that the proposed method achieves spectral accuracy and ensures infinite differentiability to all orders of the numerical solution by construction these results indicate that the new primitive variable spectral solver is an effective alternative to the spectral method recently proposed by kida and nakayama where the velocity field is represented in terms of poloidal and toroidal functions

a 3d pseudospectral method for cylindrical coordinates application to the simulations of rotating cavity flows the present work proposes a collocation spectral method for solving the three dimensional navier stokes equations using cylindrical coordinates the whole diameter r

finite differences for coarse azimuthal discretization and for reduction of effective resolution near origin of cylindrical flow equations in this paper the errors generated by the computation of derivatives in the azimuthal direction when flow equations are solved in cylindrical coordinates using finite differences are investigated they might be large for coarse discretizations even using high order schemes which led us to design explicit finite differences specially for 8 16 32 and 64 points per circle these schemes are shown to improve accuracy with respect to standard finite differences and to provide solutions for a two dimensional propagation problem similar to those obtained using fourier spectral methods in the direction a method is then presented to alleviate the time step limitation resulting from explicit time integration near cylindrical origin when finite differences are used it consists in calculating azimuthal derivatives at coarser resolutions than permitted by the grid in the same way as usually done using spectral methods in practice a series of doublings of the effective discretization in is implemented thus simulations can for instance be performed on a grid containing n 256 points with a time step 32 times larger with an accuracy comparable to that achieved in corresponding simulations involving fourier spectral methods in the direction

poloidal toroidal decomposition in a finite cylinder the navier stokes equations in a finite cylinder are written in terms of poloidal and toroidal potentials in order to impose incompressibility regularity of the solutions is ensured in several ways first the potentials are represented using a spectral basis which is analytic at the cylindrical axis second the non physical discontinuous boundary conditions at the cylindrical corners are smoothed using a polynomial approximation to a steep exponential profile third the nonlinear term is evaluated in such a way as to eliminate singularities the resulting pseudo spectral code is tested using exact polynomial solutions and the spectral convergence of the coefficients is demonstrated our solutions are shown to agree with exact polynomial solutions and with previous calculations of axisymmetric vortex breakdown and of onset of non axisymmetric helical spirals parallelization by azimuthal wavenumber is shown to be highly effective

pulse train solutions of a spatially heterogeneous amplitude equation arising in the subcritical instability of narrow gap spherical couette flow we investigate some complex solutions a x t of the heterogeneous complex ginzburg landau equation a t x ix a 2 a 2 a x 2

global bifurcation to travelling waves with application to narrow gap spherical couette flow abstract in a previous paper physica d 137 2000 260 an inhomogeneous complex landau equation was derived in the context of the amplitude modulation of taylor vortices between two rapidly rotating concentric spheres which bound a narrow gap and almost co rotate about a common axis of symmetry in this weakly nonlinear regime the latitudinal vortex width is comparable to the gap between the shells the vortices are located close to the equator and are modulated on a latitudinal length scale large compared to the gap width but small compared to the shell radius the system is characterised by two parameters which is proportional to the taylor number and which provides a measure of phase mixing only when the inner and outer spheres almost co rotate is of order unity otherwise is large in physica d 137 2000 260 it was shown that there is a finite amplitude steady solution branch at fixed that connects the first two bifurcation points 0 and 1 for sufficiently large the branch lies on 0 1 but for smaller it extends beyond 1 there on 1 n a large and small amplitude solution co exist and coalesce at the nose of the branch n in this paper we investigate both analytically and numerically the stability of the steady solutions and their subsequent evolution two types of modes exist one sp preserves the reflectional symmetry of the steady solutions with respect to the equatorial plane while the other sb breaks it three sp global bifurcation scenarios are identified each lead to limit cycles which correspond to vortices drifting towards the equator from both sides for small a heteroclinic connection is made between the steady nose solution and its reversed flow state opposite sign the same occurs for moderate except that two oppositely signed small amplitude steady solutions are connected for large a homoclinic cycle forms joining the undisturbed state to itself and this leads to a gluing bifurcation this homoclinic cycle evolves from the vacillating wave limit cycle shed by an sp hopf bifurcation of the large amplitude solution an sb pitchfork bifurcation of the steady solutions leads to asymmetric drifting phase solutions travelling waves time stepping reveals that they are generally unstable and evolve into larger amplitude periodic solutions which for small and moderate are sp states for large the sb drifting phase solutions are strongly subcritical the realised larger amplitude periodic state to which they evolve may be either of sb or sp type depending on the value of these complicated solutions consist of trains of stationary pulses each modulating travelling waves with distinct frequencies the asymmetric sb waves correspond to vortices drifting across the equator yet far from it where these vortices are very weak they drift towards the equator as in the case of the sp waves

an exact equation for the free surface of a fluid in a porous medium we study the problem of the evolution of the free surface of a fluid in a saturated porous medium bounded from below by a flat impermeable bottom and described by the laplace equation with moving boundary conditions by making use of a convenient conformal transformation we show that the solution to this problem is equivalent to the solution of the laplace equation on a fixed domain with new variable coefficients the boundary conditions we use a kernel of the laplace equation which allows us to write the dirichlet to neumann operator and in this way we are able to find an exact differential integral equation for the evolution of the free surface in one space dimension although not amenable to direct analytical solutions this equation turns out to allow an easy numerical implementation we give an explicit illustrative case at the end of the article

efficient spectral galerkin method ii direct solvers of second and fourth order equations using chebyshev polynomials efficient direct solvers based on the chebyshev galerkin methods for second and fourth order equations are presented they are based on appropriate base functions for the galerkin formulation that lead to discrete systems with special structured matrices that can be efficiently inverted numerical results indicate that the direct solvers presented in this paper are significantly more accurate and efficient than those based on the chebyshev tau method

pole condition for singular problems the pseudospectral approximation abstract this paper deals with the pseudospectral solution of differential equations with coordinate singularities such as those which describe situations in spherical or cylindrical geometries we use the differential equation together with a smoothness assumption on the solution to construct pole conditions the pole conditions which are straightforward and easily implemented serve as numerical boundary conditions at the coordinate singularity standard pseudospectral methods including fast transformation techniques can then be applied to the singular problem the method is illustrated using the eigenvalue problem of bessel s equation and a poisson equation on the unit disk numerical results show that spectral convergence is achieved

numerical simulation of gravity waves abstract we present a new spectral method to simulate numerically the waterwave problem in a channel for a fluid of finite or infinite depth it is formulated in terms of the free surface elevation and the velocity potential the numerical method is based on the reduction of this problem to a lower dimensional computation involving surface variables alone to accomplish this we describe the taylor expansion of the dirichlet neumann operator in homogeneous powers of the surface elevation each term is a concatenation of fourier multipliers with powers of and its derivatives and is valid uniformly in wavenumber these are easily calculated using the fast fourier transform the method is illustrated by computing the long time evolution of modulated wave packets and of approximations to the stokes steady wave train by imposing a surface pressure we observe surface steepening in large amplitude evolution and wake and bow wave development for flows with a close to critical froude number finally we give an example of nonlinear evolution of the distribution of energy among normal modes

on classification of second order differential equations with complex coefficients this paper is concerned with the deficiency index problem of second order differential equations with complex coefficients it is known that this class of equations is classified into cases i ii and iii according to the number of linearly independent solutions in suitable weighted square integrable spaces in this study the original equation is reformulated into a new formally self adjoint differential system by introducing a new spectral parameter and the relationship between the classifications of the equation and the system is obtained moreover the exact dependence of cases ii and iii on the corresponding half planes is given and some criteria of the three cases are established

a spectral model for two dimensional incompressible fluid flow in a circular basin in the accompanying paper part i w t m verkley 1997 j comput phys 100 114136 a spectral numerical scheme is developed for two dimensional incompressible fluid flow in a circular basin the model is formulated in terms of basis functions that are products of jacobi polynomials and complex exponentials the jacobi polynomials are used for the radial dependence of the fields and the complex exponentials for the angular dependence the basis functions are orthogonal with respect to the natural inner product for a circular domain the nonlinear advection term is calculated without aliasing using the transform method based on a grid of which the radii are gaussian and the angles are equidistant in the present paper we discuss the performance of the model by showing examples of time integrations the differences between these examples concern the spatial structure of the planetary vorticity plane s plane f plane the temporal and spatial resolution of the model and the form strength and type of the forcing and dissipation

a spectral method for unbounded domains a spectral method for an unbounded domain is presented rational basis functions which are algebraically mapped legendre functions are used for expansion in the radial direction of polar coordinates r or r z they satisfy the pole condition exactly at the coordinate singularity and their behavior asr is suitable for expanding smooth functions which decay algebraically or exponentially asr the method is not stiff when it is applied to initial value problems despite the presence of the coordinate singularity solenoidal vector fields are treated efficiently by the toroidal and poloidal decomposition which reduces the number of dependent variables from 3 to 2 examples include the computation of vortex dynamics in two and three dimensions

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

a sparse mesh for compact finite difference fourier solvers with radius dependent spectral resolution in circular domains this paper presents a new method for the resolution of elliptic and parabolic equations in circular domains it can be trivially extended to cylindrical domains the algorithm uses a mixed fourier compact finite difference method the main advantage of the method is achieved by a new concept of mesh the topology of the new grid keeps constant the aspect ratio of the cells avoiding the typical clustering for radial structured meshes at the center the reduction of the number of nodes has as a consequence the reduction in memory consumption in the case of fluid mechanics problems this technique also increases the time step for a constant courant number several examples are given in the paper which show the potential of the method

spectral collocation schemes on the unit disc two spectral collocation schemes on the unit disc are presented the first one is based on the mapping of gordon and hall here the unit square is directly mapped onto the unit disc by means of an interpolation technique unlike other poisson solvers on the unit disc no polar coordinates are involved hence the usual problems with the singularity of polar coordinates are avoided this is also shown for more complex geometries the second method is based on a diameter approach where the collocation nodes are no more clustering in the center numerical results are presented which demonstrate the high accuracy of our new spectral collocation schemes

an efficient spectral projection method for the navier stokes equations in cylindrical geometries i axisymmetric cases abstract an efficient and accurate numerical scheme is presented for the axisymmetric navier stokes equations in primitive variables in a cylinder the scheme is based on a new spectral galerkin approximation for the space variables and a second order projection scheme for the time variable the new spectral projection scheme is implemented to simulate the unsteady incompressible axisymmetric flow with a singular boundary condition which is approximated to within a desired accuracy by using a smooth boundary condition a sensible comparison is made with a standard second order in time and space finite difference scheme based on a stream function vorticity formulation and with available experimental data the numerical results indicate that both schemes produce very reliable results and that despite the singular boundary condition the spectral projection scheme is still more accurate in terms of a fixed number of unknowns and more efficient in terms of cpu time required for resolving the flow at a fixed reynolds number to within a prescribed accuracy than the finite difference scheme more importantly the spectral projection scheme can be readily extended to three dimensional nonaxisymmetric cases