shape and morse theory of attractors we study the shape in the sense of borsuk of attractors of continuous semi dynamical systems on general metric spaces we show in particular that the natural inclusion of the global attractor into the state space is a shape equivalence this and other results of the paper are used to develop an elementary morse theory of an attractor 2000 john wiley sons inc

a split step numerical method for the time dependent dirac equation in 3 d axisymmetric geometry abstract a numerical method is developed to solve the time dependent dirac equation in cylindrical coordinates for 3 d axisymmetric systems the time evolution is treated by a splitting scheme in coordinate space using alternate direction iteration while the wave function is discretized spatially on a uniform grid the longitudinal coordinate evolution is performed exactly by the method of characteristics while the radial coordinates evolution uses poisson s integral solution which allows to implement the radial symmetry of the wave function the latter is evaluated on a time staggered mesh by using hermite polynomial interpolation and by performing the integration analytically the cylindrical coordinate singularity problem at r 0 is circumvented by this method as the integral is well defined at the origin the resulting scheme is reminiscent of non standard finite differences in the last step of the splitting the remaining equation has a solution in terms of a time ordered exponential which is approximated to a higher order than the time evolution scheme we study the time evolution of gaussian wave packets and we evaluate the eigenstates of hydrogen like systems by using a spectral method we compare the numerical results to analytical solutions to validate the method in addition we present three dimensional simulations of relativistic laser matter interactions using the dirac equation

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

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

cech cohomology of attractors of discrete dynamical systems abstract let f r n r n be a homeomorphism and k an asymptotically stable attractor for f the aim of this paper is to study when the inclusion of k in its basin of attraction a k induces isomorphisms in cech cohomology we show that i this is true if coefficients are taken in q or z p p prime and ii it is true for integral cohomology if and only if the cech cohomology of k or a k is finitely generated we compute the cech cohomology of periodic point free attractors of volume contracting r 3 homeomorphisms and present applications to quite general models in population dynamics

a fast iterative solver for the variable coefficient diffusion equation on a disk we present an efficient iterative method for solving the variable coefficient diffusion equation on a unit disk the equation is written in polar coordinates and is discretized by the standard centered difference approximation under the grid arrangement by shifting half radial mesh away from the origin so that the coordinate singularity can be handled naturally without pole conditions the resultant matrix is symmetric positive definite so the preconditioned conjugate gradient pcg method can be applied different preconditioners have been tested for comparison in particular a fast direct solver derived from the equation and the semi coarsening multigrid are shown to be almost scalable with the problem size and outperform other preconditioners significantly the present elliptic solver has been applied to study the vortex dynamics of the ginzburg landau equation with a variable diffusion coefficient

a class of projection and contraction methods for asymmetric linear variational inequalities and their relations to fukushima s descent method for solving asymmetric linear variational inequalities we present a class of projection and contraction methods under the general g norm the search direction of our methods is just a convex combination of two descent directions of fukushima s merit function however we use the direction to reduce the distance function 12 u u 2g where is a solution point of the problem finally we report some numerical results for spatial price equilibrium problems by using the presented methods

a formulation of navier stokes problem in cylindrical coordinates applied to the 3d entry jet in a duct a pseudospectral formulation of the three dimensional navier stokes equations in the cylindrical system of co ordinates is presented which automatically includes the regularity conditions at the polar axis for the fourier harmonics the mathematical system of equations is numerically implemented using standard methods for spatial discretisation and time advancement the method presented here for incompressible flows can be easily extended to different set of equations and to different numerical schemes of various levels of accuracy the formulation is applied to the solution of incompressible three dimensional entry jets in a circular duct

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

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

carrying simplices in nonautonomous and random competitive kolmogorov systems abstract the purpose of this paper is to investigate the asymptotic behavior of positive solutions of nonautonomous and random competitive kolmogorov systems via the skew product flows approach it is shown that there exists an unordered carrying simplex which attracts all nontrivial positive orbits of the skew product flow associated with a nonautonomous random competitive kolmogorov system

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

spectral chebyshev fourier collocation for the helmholtz and variable coefficient equations in a disk the paper is concerned with the spectral collocation solution of the helmholtz equation in a disk in the polar coordinates r and q we use spectral chebyshev collocation in r spectral fourier collocation in q and a simple integral condition to specify the value of the approximate solution at the center of the disk the scheme is solved at a quasi optimal cost using the idea of superposition a matrix decomposition algorithm and fast fourier transforms both the dirichlet and neumann boundary conditions are considered and extensions to equations with variable coefficients are discussed numerical results confirm the spectral convergence of the method

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

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

infinite series in cohomology attractors and conley index in this paper we study the cohomological conley index of arbitrary isolated invariant continua for continuous maps f u subseteq mathbb r d rightarrow mathbb r d by analyzing the topological structure of their unstable manifold we provide a simple dynamical interpretation for the first cohomological conley index describing it completely and relate it to the cohomological conley index in higher degrees a number of consequences are derived including new computations of the fixed point indices of isolated invariant continua in dimensions 2 and 3 our approach exploits a certain attractor repeller decomposition of the unstable manifold reducing the study of the cohomological conley index to the relation between the cohomology of an attractor and its basin of attraction this is a classical problem that in the present case is particularly difficult because the dynamics is discrete and the topology of the unstable manifold can be very complicated to address it we develop a new method that may be of independent interest and involves the summation of power series in cohomology if z is a metric space and k subseteq z is a compact global attractor for a continuous map g z rightarrow z we show how to interpret series of the form sum j ge 0 a j g j as endomorphisms of the cohomology group of the pair z k

efficient spectral galerkin methods iii polar and cylindrical geometries we present in this paper several extremely efficient and accurate spectral galerkin methods for second and fourth order equations in polar and cylindrical geometries these methods are based on appropriate variational formulations which incorporate naturally the pole condition s in particular the computational complexities of the chebyshev galerkin method in a disk and the chebyshev legendre galerkin method in a disk or a cylinder are quasi optimal optimal up to a logarithmic term as an indication of efficiency the cpu time for the poisson solver on a disk by our chebyshev galerkin method is only about 70 of the corresponding finite difference code in fishpack

shape of global attractors in topological spaces in this paper we apply the notion and properties of compactly generated shape to study attractors in topological spaces

a highly accurate technique for the treatment of flow equations at the polar axis in cylindrical coordinates using series expansions numerical methods for solving the flow equations in cylindrical or spherical coordinates should be able to capture the behavior of the exact solution near the regions where the particular form of the governing equations is singular in this work we focus on the treatment of these numerical singularities for finite difference methods by reinterpreting the regularity conditions developed in the context of pseudospectral methods a generally applicable numerical method for treating the singularities present at the polar axis when nonaxisymmetric flows are solved in cylindrical coordinates using highly accurate finite difference schemes e g pade schemes on nonstaggered grids is presented governing equations for the flow at the polar axis are derived using series expansions near r 0 the only information needed to calculate the coefficients in these equations are the values of the flow variables and their radial derivatives at the previous iteration or time level these derivatives which are multivalued at the polar axis are calculated without dropping the accuracy of the numerical method using a mapping of the flow domain from 0 r 0 2 to r r 0 where r is the radius of the computational domain this allows the radial derivatives to be evaluated using high order differencing schemes e g compact schemes at points located on the polar axis the accuracy of the method is checked by comparison with the theoretical solution corresponding to a circular compressible forced jet in the regime of linear growth the proposed technique is illustrated by results from simulations of laminar forced jets and turbulent compressible jets using large eddy simulation les methods in terms of the general robustness of the numerical method and smoothness of the solution close to the polar axis the present results compare very favorably to similar calculations in which the equations are solved in cartesian coordinates at the polar axis or in which the singularity is removed by employing a staggered mesh in the radial direction without a mesh point at r 0 following the method proposed recently by mohseni and colonius 1 extension of the method described here for incompressible flows or for any other set of equations that is solved on a nonstaggered mesh in cylindrical or spherical coordinates with finite difference schemes of various levels of accuracy is immediate

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

smoothness of the carrying simplex for discrete time competitive dynamical systems a characterization of neat embedding abstract the paper is concerned with the question of smoothness of the carrying simplex s for a discrete time dissipative competitive dynamical system we give a necessary and sufficient criterion for s being a c 1 submanifold with corners neatly embedded in the nonnegative orthant formulated in terms of inequalities between lyapunov exponents for ergodic measures supported on the boundary of the orthant this completes one thread of investigation occasioned by a question posed by m w hirsch in 1988 besides amenable conditions are presented to guarantee the c r r 1 smoothness of s in the time periodic competitive kolmogorov systems of odes examples are also presented one in which s is of class c 1 but not neatly embedded the other in which s is not of class c 1

uniqueness and attractivity of the carrying simplex for discrete time competitive dynamical systems abstract for the discrete time dissipative competitive dynamical system on r n it is proved that generically there is a unique n 1 dim invariant carrying simplex attracting all nontrivial orbits which solved the conjecture raised by h l smith in 1986

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

finite difference method for incompressible navier stokes equations in arbitrary orthogonal curvilinear coordinates a finite difference method for solving three dimensional time dependent incompressible navier stokes equations in arbitrary curvilinear orthogonal coordinates is presented the method is oriented on turbulent flow simulations and consists of a second order central difference approximation in space and a third order semi implicit runge kutta scheme for time advancement spatial discretization retains some important properties of the navier stokes equations including energy conservation by the nonlinear and pressure gradient terms numerical tests cover cartesian cylindrical polar spherical cylindrical elliptic and cylindrical bipolar coordinate systems both laminar and turbulent flows are considered demonstrating reasonable accuracy and stability of the method

a rational spectral collocation method for third order singularly perturbed problems a new numerical method is developed for solving a class of third order singularly perturbed boundary value problems first of all the given problem is transformed into a system of two ordinary differential equations odes subject to suitable initial and boundary conditions then the rational spectral collocation method in barycentric form with sinh transformation is applied to solve the system of odes according to the asymptotic analysis the location and width of boundary layer of the given problem which are chosen as parameters in the sinh transformation can be determined ample numerical experiments are presented to illustrate the computational efficiency and accuracy of the our method

splitting based schemes for numerical solution of nonlinear diffusion equations on a sphere we provide an advanced study of our recently developed method for the numerical solution of nonlinear diffusion equations on a sphere in particular we analyse the method in detail when applied to solving diverse diffusion phenomena with specific conditions on the smoothness of the solution the degree of nonlinearity and the initial data and sources the main idea of the method consists in splitting the original differential operator by coordinates and subsequent constructing finite difference schemes for the split one dimensional problems using different coordinate maps for the sphere at the two split time intervals the essential advantage of this technique is that each split 1d equation can be equipped with a periodic boundary condition despite the sphere being not a doubly periodic domain therefore unlike the existing methods this one does not require applying special numerical procedures for careful computing the solution near the poles which is always a challenge each split 1d equation is approximated by a second or a fourth order finite difference scheme that keeps all the substantial properties of the differential problem it is balanced and dissipative since the spatial finite difference operator is negative definite the developed algorithm is cheap to implement from the computational point of view the theoretical results are confirmed numerically by simulating various nonlinear diffusion processes a numerical example in particular shows that the competition of the three basic mechanisms the nonlinear interaction forcing and dissipation can generate wave solutions whose spatial structures on the sphere are subjected to the alternating influence of the processes of self organisation and self destruction the accuracy of the method is evaluated by comparing the numerical solutions versus the analytical ones obtained with specially chosen forcings the convergence of the numerical solution to the analytics is verified by refining spatial grids

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