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

cellular automata as an alternative to rather than an approximation of differential equations in modeling physics abstract cellular automata are models of distributed dynamical systems whose structure is particularly well suited to ultrafast exact numerical simulation on the other hand they constitute a radical departure from the traditional partial differential equation approach to distributed dynamics here we discuss the problem of encoding the state variables and evolution laws of a physical system into this new setting and of giving suitable correspondence rules for interpreting the model s behavior

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 cellular model of prisoners dilemma for prodding gratuity repeated prisoner s dilemma rpd is an evolutionary game extensively utilised for modelling socio economic dynamics such an instrument was used in order to investigate co operation in different interesting aspects and strategies e g gratuity bruni 2008 2 in terms of co operation that privileges a collective advantage this work explores rpd extension to a spatial context srpd by a stochastic distribution of agents with different strategies in the space of a cellular automaton ca according to the nowak approach nowak and may 1993 22 this approach coincides with spd extension to srpd in srpd each cell is occupied by one agent that relates with all other agents in neighbourhood cells in terms of rpd classic payoffs and symmetries results show that the srpd convergence to total co operation or defection is very rapid more than rpd but some values of payoff generate an interesting chaotic evolution of the system furthermore introduction of new behaviours of agents through altering payoffs and symmetry characterises this our model in such conditions unexpected emergence of co operative behaviours was detected for agents with complex strategies in spite of their reduced conditions of increasing own payoff

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

evolutionary prisoner s dilemma in random graphs abstract we study an evolutionary version of the spatial prisoner s dilemma game spd where the agents are placed in a random graph for graphs with fixed connectivity we show that for low values of the final density of cooperating agents c depends on the initial conditions however if the graphs have large connectivities c is independent of the initial conditions we characterize the phase diagram of the system using both extensive numerical simulations and analytical computations it is shown that two well defined behaviors are present a nash equilibrium where the final density of cooperating agents c is constant and a non stationary region where c fluctuates in time moreover we study the spd in poisson random graphs and find that the phase diagram previously developed looses its meaning in fact only one regime may be defined this regime is characterized by a non stationary final state where the density of cooperating agents varies in time

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

matrix decomposition algorithms for separable elliptic boundary value problems in two space dimensions abstract bialecki b and g fairweather matrix decomposition algorithms for separable elliptic boundary value problems in two space dimensions journal of computational and applied mathematics 46 1993 369 386 in recent years several matrix decomposition algorithms have been developed for the efficient solution of the linear algebraic systems arising when finite difference finite element galerkin and spectral methods are applied to separable elliptic boundary value problems in a rectangle the success of these methods depends on knowledge of the eigenvalues and eigenvectors of matrices arising in corresponding two point boundary value problems in one space dimension the primary purpose of this paper is to provide an overview of matrix decomposition algorithms and show how they can be expressed in terms of a unifying framework particular emphasis is placed on algorithms formulated recently by the authors for solving the linear systems arising in orthogonal spline collocation that is spline collocation at gauss points all of the methods discussed in this paper are modular and possess a great deal of natural parallelism

deciding game invariance abstract in a previous paper duchene and rigo introduced the notion of invariance for take away games on heaps roughly speaking these are games whose rulesets do not depend on the position given a sequence s of positive tuples of integers the question of whether there exists an invariant game having s as set of p positions is relevant in particular it was recently proved by larsson et al that if s is a pair of complementary beatty sequences then the answer to this question is always positive in this paper we show that for a fairly large set of sequences expressed by infinite words the answer to this question is decidable

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

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

on the sequentiality of the successor function abstract letube a strictly increasing sequence of integers by a greedy algorithm every nonnegative integer has a greedyu representation the successor function maps the greedyu representation ofnonto the greedyu representation ofn 1 we characterize the sequencesusuch that the successor function associated withuis a left resp a right sequential function we also show that the odometer associated touis continuous if and only if the successor function is right sequential

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

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

deterministic synchronization of automations with bounded delay resume nous montrons qu un automate a delai borne sous sequentiel i e deterministe sur les entrees est equivalent a un automate en ligne c est a dire lettre a lettre apres une periode transitoire fixe ou l automate lit le debut de l entree sans rien sortir et sous sequentiel les resultats de synchronisation etablis anterieurement pour les automates a delai borne ne preservaient pas la sous sequentialite

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

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

adi spectral collocation methods for parabolic problems we discuss the crank nicolson and laplace modified alternating direction implicit legendre and chebyshev spectral collocation methods for a linear variable coefficient parabolic initial boundary value problem on a rectangular domain with the solution subject to non zero dirichlet boundary conditions the discretization of the problems by the above methods yields matrices which possess banded structures this along with the use of fast fourier transforms makes the cost of one step of each of the chebyshev spectral collocation methods proportional except for a logarithmic term to the number of the unknowns we present the convergence analysis for the legendre spectral collocation methods in the special case of the heat equation using numerical tests we demonstrate the second order accuracy in time of the chebyshev spectral collocation methods for general linear variable coefficient parabolic problems

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

numeration systems linear recurrences and regular sets a numeration system based on a strictly increasing sequence of positive integers u0 1 u1u2 expresses a non negative integer n as a sum n j 0 i ajuj in this case we say the string a i a i 1 a1 a0 is a representation for n