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

parallel spectral element solution of the stokes problem abstract in this paper we present a high efficiency medium grained parallel spectral element method for numerical solution of the stokes problem in general domains the method is based upon naturally concurrent uzawa and jacobi preconditioned conjugate gradient iterative methods geometry based data parallel distribution of work amongst processors nearest neighbor sparsity and high order substructuring for minimum communication general locally structured globally unstructured parallel constructs and efficient embedding of vector reduction operations for inner product and norm calculation an analysis is given for the computational complexity of the algorithm on a native medium grained parallel processor and the potential communication superiority of high order discretizations is described lastly the method is implemented on the fast intel vector hypercube and the performance of this algorithm architecture coupling is evaluated in a technical and economic framework that reflects the true advantages of parallel solution of partial differential equations

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

openmx 2 0 extended structural equation and statistical modeling the new software package openmx 2 0 for structural equation and other statistical modeling is introduced and its features are described openmx is evolving in a modular direction and now allows a mix and match computational approach that separates model expectations from fit functions and optimizers major backend architectural improvements include a move to swappable open source optimizers such as the newly written csolnp entire new methodologies such as item factor analysis and state space modeling have been implemented new model expectation functions including support for the expression of models in lisrel syntax and a simplified multigroup expectation function are available ease of use improvements include helper functions to standardize model parameters and compute their jacobian based standard errors access to model components through standard r mechanisms and improved tab completion from within the r graphical user interface

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

the fractional diffusion equation in one space and in one time dimension a diffusion equation is solved where the first time derivative is replaced by the fractional time derivative 0 1 the solution is given in closed form in terms of fox functions

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

a real distinct poles rational approximation of generalized mittag leffler functions and their inverses applications to fractional calculus abstract the generalized mittag leffler function and its inverse is very useful in solving fractional differential equations and structural derivatives respectively however their computational complexities have made them difficult to deal with numerically we propose a real distinct pole rational approximation of the generalized mittag leffler function under some mild conditions this approximation is proven and empirically shown to be l acceptable due to the complete monotonicity property of the mittag leffler function we derive a rational approximation for the inverse generalized mittag leffler function these approximations are especially useful in developing efficient and accurate numerical schemes for partial differential equations of fractional order several applications are presented such as complementary error function solution of fractional differential equations and the ultraslow diffusion model using the structural derivative

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

a spectral element semi lagrangian sesl method for the spherical shallow water equations a spectral element semi lagrangian sesl method for the shallow water equations on the sphere is presented the sphere is discretized using a hexahedral grid although any grid imaginable can be used as long as it is comprised of quadrilaterals the equations are written in cartesian coordinates to eliminate the pole singularity which plagues the equations in spherical coordinates in a previous paper int j numer methods fluids 35 2001 869 we showed how to construct an explicit eulerian spectral element se model on the sphere we now extend this work to a semi lagrangian formulation the novelty of the lagrangian formulation presented is that the high order se basis functions are used as the interpolation functions for evaluating the values at the lagrangian departure points this makes the method not only high order accurate but quite general and thus applicable to unstructured grids and portable to distributed memory computers the equations are discretized fully implicitly in time in order to avoid having to interpolate derivatives at departure points by incorporating the coriolis terms into the lagrangian derivative the block lu decomposition of the equations results in a symmetric positive definite pseudo helmholtz operator which we solve using the generalized minimum residual method gmres with a fast projection method comput methods appl mech eng 163 1998 193 results for eight test cases are presented to confirm the accuracy and stability of the method these results show that sesl yields the same accuracy as an eulerian spectral element semi implicit sesi while allowing for time steps 10 times as large and being up to 70 more efficient

asymptotic green s functions for time fractional diffusion equation and their application for anomalous diffusion problem abstract asymptotic green s functions for short and long times for time fractional diffusion equation derived by simple and heuristic method are provided in case if fractional derivative is presented in caputo sense the applicability of the asymptotic green s functions for solving the anomalous diffusion problem on a semi infinite rod is demonstrated the initial value problem for longtime solution of the time fractional diffusion equation by green s function approach is resolved

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

ulam hyers stability of fractional langevin equations in this paper we discuss ulam hyers stability of nonlinear fractional langevin equations by using the boundedness monotonicity and nonnegative properties of classical and generalized mittag leffler functions further the same problem for the associated impulsive systems is investigated finally an example is given to illustrate our theoretical results

high order splitting methods for the incompressible navier stokes equations abstract a new pressure formulation for splitting methods is developed that results in high order time accurate schemes for the solution of the incompressible navier stokes equations in particular improved pressure boundary conditions of high order in time are introduced that minimize the effect of erroneous numerical boundary layers induced by splitting methods a new family of stiffly stable schemes is employed in mixed explicit implicit time intgration rules these schemes exhibit much broader stability regions as compared to adams family schemes typically used in splitting methods their stability properties remain almost constant as the accuracy of the integration increases so that robust third or higher order time accurate schemes can readily be constructed that remain stable at relatively large cfl number the new schemes are implemented within the framework of spectral element discretizations in space so that flexibility and accuracy is guaranteed in the numerical experimentation a model stokes problem is studied in detail and several examples of navier stokes solutions of flows in complex geometries are reported comparison is made with the previously used first order in time spectral element splitting and non splitting e g uzawa schemes high order splitting spectral element methods combine accuracy in space and time and flexibility in geometry and thus can be very efficient in direct simulations of turbulent flows in complex geometries

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

some recent advances in theory and simulation of fractional diffusion processes to offer an insight into the rapidly developing theory of fractional diffusion processes we describe in some detail three topics of current interest i the well scaled passage to the limit from continuous time random walk under power law assumptions to space time fractional diffusion ii the asymptotic universality of the mittag leffler waiting time law in time fractional processes iii our method of parametric subordination for generating particle trajectories

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

rational solutions for the time fractional diffusion equation a uniform rational approximation of the mittag leffler function is derived which serves as a global approximant accounting for both the taylor series for small arguments and asymptotic series for

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

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

spectral element methods for axisymmetric stokes problems abstract the approximation of the stokes problem in axisymmetric geometries using the spectral element method is considered the presence of the volume element r dr dz in the weak formulation of the problem is shown to be a potential source of difficulty the discrete equations associated with nodes on the axis of symmetry can lead to a degeneracy in the global system of equations this difficulty is resolved by incorporating the factor r into the weight function for spectral elements adjacent to the axis of symmetry and using appropriate basis functions in these elements in the radial direction properties of the jacobi polynomials are used to construct the elements of the modified method numerical results are presented demonstrating some of the features of the proposed approach

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

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

a note on the usefulness of constrained fourth order latent differential equation models in the case of small t constrained fourth order latent differential equation folde models have been proposed e g boker et al 2020 as alternative to second order latent differential equation solde models to estimate second order linear differential equation systems such as the damped linear oscillator model when however only a relatively small number of measurement occasions t are available i e t 50 the recommendation of which model to use is not clear boker et al 2020 based on a data set which consists of t 56 observations of daily stress for n 44 individuals we illustrate that folde can help to choose an embedding dimension even in the case of a small t this is of great importance as parameter estimates depend on the embedding dimension as well as on the latent differential equations model consequently the wavelength as quantity of potential substantive interest may vary considerably we extend the modeling approaches used in past research by including multiple subjects by accounting for individual differences in equilibrium and by including multiple instead of one single observed indicator

time delay embedding increases estimation precision of models of intraindividual variability this paper investigates the precision of parameters estimated from local samples of time dependent functions we find that time delay embedding i e structuring data prior to analysis by constructing a data matrix of overlapping samples increases the precision of parameter estimates and in turn statistical power compared to standard independent rows of panel data we show that the reason for this effect is that the sign of estimation bias depends on the position of a misplaced data point if there is no a priori knowledge about initial conditions of the time dependent function hence we reason that the advantage of time delayed embedding is likely to hold true for a wide variety of functions we support these conclusions both by mathematical analysis and two simulations

a spectral element method for fluid dynamics laminar flow in a channel expansion a spectral element method that combines the generality of the finite element method with the accuracy of spectral techniques is proposed for the numerical solution of the incompressible navier stokes equations in the spectral element discretization the computational domain is broken into a series of elements and the velocity in each element is represented as a high order lagrangian interpolant through chebyshev collocation points the hyperbolic piece of the governing equations is then treated with an explicit collocation scheme while the pressure and viscous contributions are treated implicitly with a projection operator derived from a variational principle the implementation of the technique is demonstrated on a one dimensional inflow outflow advection diffusion equation and the method is then applied to laminar two dimensional separated flow in a channel expansion comparisons are made with experiment and previous numerical work

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

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