error analysis and comparative study of numerical methods for the parabolic equation applied to tunnel propagation modeling parabolic equation pe methods have been widely applied to the modeling of wireless propagation in tunnel environments however the relevant literature does not include concrete guidelines for the choice of the parameters of these methods and the tradeoffs involved this paper provides a comprehensive analysis of the two sources of error that arise when pe methods are employed for the modeling of radio wave propagation scenarios the well known numerical dispersion error stemming from the finite difference solvers for pe and the approximation error stemming from the use of pe for the solution of wave propagation problems that are subject to maxwell s equations the analysis is performed for four methods three of which have been already used in pe based propagation studies namely the crank nicolson cn scheme the alternative direction implicit adi method and its locally one dimensional lod adi version the fourth method is the mitchell fairweather mf adi scheme that has been recently shown to be a promising alternative technique for tunnel propagation modeling the proposed method leads to robust criteria for the choice of spatial discretization in realistic propagation scenarios as shown via numerical examples

stability and convergence of the difference methods for the space time fractional advection diffusion equation abstract in this paper we consider a space time fractional advection dispersion equation stfade on a finite domain the stfade is obtained from the standard advection dispersion equation by replacing the first order time derivative by the caputo fractional derivative of order 0 1 and the first order and second order space derivatives by the riemman liouville fractional derivatives of order 0 1 and of order 1 2 respectively for the space fractional derivatives d x u x t and d x u x t we adopted the grunwald formula and the shift grunwald formula respectively we propose an implicit difference method idm and an explicit difference method edm to solve this equation stability and convergence of these methods are discussed using mathematical induction we prove that the idm is unconditionally stable and convergent but the edm is conditionally stable and convergent numerical results are in good agreement with theoretical analysis

a pieri type formula for isotropic flag manifolds we give the formula for multiplying a schubert class on an odd orthogonal or symplectic flag manifold by a special schubert class pulled back from the grassmannian of maximal isotropic subspaces this is also the formula for multiplying a type b respectively type c schubert polynomial by the schur p polynomial p m respectively the schur q polynomial q m geometric constructions and intermediate results allow us to ultimately deduce this formula from formulas for the classical flag manifold these intermediate results are concerned with the bruhat order of the infinite coxeter group b identities of the structure constants for the schubert basis of cohomology and intersections of schubert varieties we show that most of these identities follow from the pieri type formula and our analysis leads to a new partial order on the coxeter group b and formulas for many of these structure constants

an implicit fourth order compact finite difference scheme for one dimensional burgers equation abstract a fourth order compact finite difference method is proposed in this paper to solve one dimensional burgers equation the newly proposed method is based on the hopf cole transformation which transforms the original nonlinear burgers equation into a linear heat equation and transforms the dirichlet boundary condition into the robin boundary condition the linear heat equation is then solved by an implicit fourth order compact finite difference scheme a compact fourth order formula is also developed to approximate the robin boundary conditions while the initial condition for the heat equation is approximated using simpson s rule to maintain overall fourth order accuracy numerical experiments have been conducted to demonstrate the efficiency and high order accuracy of this method the numerical results also show that the method is unconditionally stable as there is no constraint on time step size

modeling radio transmission loss in curved branched and rough walled tunnels with the adi pe method we discuss the use of the parabolic equation pe along with the alternate direction implicit adi method in predicting the loss for three specialized tunnel cases curved tunnels branched tunnels and rough walled tunnels this paper builds on previous work which discusses the use of the adi pe in modeling transmission loss in smooth straight tunnels for each specialized tunnel case the adi pe formulation is presented along with necessary boundary conditions and tunnel geometry limitations to complete the study examples are presented where the adi pe numerical results for the curved and rough walled tunnel are compared to known analytical models and experimental data and the branched tunnel data is compared to the numerical solutions produced by hfss

games on large networks information and complexity in this work we study static and dynamic games on large networks of interacting agents assuming that the players have some statistical description of the interaction graph as well as some local information inspired by statistical physics we consider statistical ensembles of games and define a probabilistic approximate equilibrium notion for such ensembles a necessary information complexity notion is introduced to quantify the minimum amount of information needed for the existence of a probabilistic approximate equilibrium we then focus on some special classes of games for which it is possible to derive upper and or lower bounds for the complexity at first static and dynamic games on random graphs are studied and their complexity is determined as a function of the graph connectivity in the low complexity case we compute probabilistic approximate equilibrium strategies we then consider static games on lattices and derive upper and lower bounds for the complexity using contraction mapping ideas a lq game on a large ring is also studied numerically using a reduction technique approximate equilibrium strategies are computed and it turns out that the complexity is relatively low

on posets and hopf algebras abstract we generalize the notion of the rank generating function of a graded poset namely by enumerating different chains in a poset we can assign a quasi symmetric function to the poset this map is a hopf algebra homomorphism between the reduced incidence hopf algebra of posets and the hopf algebra of quasi symmetric functions this work implies that the zeta polynomial of a poset may be viewed in terms hopf algebras in the last sections of the paper we generalize the reduced incidence hopf algebra of posets to the hopf algebra of hierarchical simplicial complexes

error and extrapolation of a compact lod method for parabolic differential equations this paper is concerned with a compact locally one dimensional lod finite difference method for solving two dimensional nonhomogeneous parabolic differential equations an explicit error estimate for the finite difference solution is given in the discrete infinity norm it is shown that the method has the accuracy of the second order in time and the fourth order in space with respect to the discrete infinity norm a richardson extrapolation algorithm is developed to make the final computed solution fourth order accurate in both time and space when the time step equals the spatial mesh size numerical results demonstrate the accuracy and the high efficiency of the extrapolation algorithm

sixth order compact scheme combined with multigrid method and extrapolation technique for 2d poisson equation we develop a sixth order finite difference discretization strategy to solve the two dimensional poisson equation which is based on the fourth order compact discretization multigrid method richardson extrapolation technique and an operator based interpolation scheme we use multigrid v cycle procedure to build our multiscale multigrid algorithm which is similar to the full multigrid method fmg the multigrid computation yields fourth order accurate solution on both the fine grid and the coarse grid a sixth order accurate coarse grid solution is computed by using the richardson extrapolation technique then we apply our operator based interpolation scheme to compute sixth order accurate solution on the fine grid numerical experiments are conducted to show the solution accuracy and the computational efficiency of our new method compared to sun zhang s sixth order richardson extrapolation compact rec discretization strategy using alternating direction implicit adi method and the standard fourth order compact difference foc scheme using a multigrid method

a novel parallel parabolic equation method for electromagnetic scatterings the alternating group explicit age scheme together with the alternating direction implicit adi scheme is used for the parabolic equation pe method to analyze the electromagnetic scattering from electrically large conducting targets the calculation can be taken plane by plane by using the finite difference scheme along the paraxial direction furthermore each transverse plane is divided into two regions which are a small region around the target and an outer region combined with the air layer and a perfect matching layer the adi scheme is applied to the small region whereas the age scheme is introduced to the outer region in this manner only a few matrix equations are needed to be solved for the small region and the unknowns of the outer region can be directly obtained without solving any matrix equations therefore the cpu time can be reduced significantly by the proposed method and it can be easily adapted to the parallel technology the numerical results verify the accuracy and efficiency of the proposed method compared with the crank nicolson based pe method and the adi based pe method moreover the parallel efficiency is given as the last numerical example

on the linear quadratic gaussian nash game with one step delay observation sharing pattern this paper studies a two player nash dynamic discrete time linear quadratic game under the so called one step delay observation sharing pattern it is shown that under very weak assumptions the solution exists and is unique and linear in the information

a novel marching on in degree solver of time domain parabolic equation for transient em scattering analysis a novel marching on in degree mod solver of 3 d time domain parabolic equation is proposed to solve the transient electromagnetic scattering from electrically large perfect electric conductor pec targets the finite difference fd scheme is applied to the spatial discretization while the weighted laguerre polynomials are used as the temporal basis functions in this way a large number of computational resources can be saved by using the fd scheme along the paraxial direction and the late time stability can be guaranteed by the mod method both the fd schemes of crank nicolson and the alternating direction implicit types are discussed in this communication numerical results are given to demonstrate the accuracy and efficiency of the proposed method

a higher order compact lod method and its extrapolations for nonhomogeneous parabolic differential equations abstract a higher order compact locally one dimensional lod finite difference method for two dimensional nonhomogeneous parabolic differential equations is proposed the resulting scheme consists of two one dimensional tridiagonal systems and all computations are implemented completely in one spatial direction as for one dimensional problems the solvability and the stability of the scheme are proved almost unconditionally the error estimates are obtained in the discrete h 1 l 2 and l norms and show that the proposed compact lod method has the accuracy of the second order in time and the fourth order in space two richardson extrapolation algorithms are presented to increase the accuracy to the fourth order and the sixth order in both time and space when the time step is proportional to the spatial mesh size numerical results demonstrate the accuracy of the compact lod method and the high efficiency of its extrapolation algorithms

single cell discretizations of order two and four for biharmonic problems abstract new difference formulas are derived for solving the biharmonic problem in two dimensions over a rectangular domain these methods use only the nine grid points of a single mesh cell and do not require fictitious points in order to approximate the boundary conditions derivatives of the solution are obtained as a by product of the methods second order formulas are derived for both the first and second biharmonic problems in numerical experiments the new second order formulas compare favourably with the standard second order methods extensions to fourth order formulas are given the method of deriving these formulas can be used to derive similar formulas for arbitrarily shaped regions

non orthogonal domain parabolic equation and its tilted gaussian beam solutions a non orthogonal coordinate system which is a priori matched to localized initial field distributions for time harmonic wave propagation is presented applying in addition a rigorous paraxial asymptotic approximation results in a novel parabolic wave equation for beam type field propagation in 3d homogeneous media localized solutions to this equation that exactly match linearly phased gaussian aperture distributions are termed tilted gaussian beams these beams serve as the building blocks for various beam type expansion schemes application of the scalar waveobjects to electromagnetic field beam type expansion as well as reflection and transmission of these waveobjects by planar velocity dielectric discontinuity are presented a numerical example which demonstrates the enhanced accuracy of the tilted gaussian beams over the conventional ones concludes the paper

a random laplace transform method for solving random mixed parabolic differential problems this paper deals with the explicit solution of random mixed parabolic equations in unbounded domains by using the random laplace transform to second order stochastic processes the mean square random laplace operational calculus is stated and its application to the random parabolic equation together with previous results of the underlying random ordinary differential equations allow us to obtain an explicit solution of the problem a numerical example which includes simulations illustrates the developed method

an implicit high accuracy variable mesh scheme for 1 d non linear singular parabolic partial differential equations abstract in this paper we propose a new two level implicit difference scheme of o k 2 h l 1 kh l h l 3 for the solution of non linear parabolic equation eu xx x t u u x u t 0 x t 0 subject to appropriate initial and dirichlet boundary conditions prescribed where k 0 h l 0 are mesh sizes in t and x coordinates respectively and e 0 is a small parameter in addition we also discuss a new explicit variable mesh difference scheme of o kh l h l 3 for the estimates of u x in all cases we require only three spatial variable grid points the proposed schemes require less algebra and three evaluations of function a special technique is required to solve singular parabolic equations the proposed variable mesh scheme when applied to a linear diffusion equation is shown to be stable for all h l 0 and k 0 computational results are provided to support our derived schemes and analysis

stability of difference schemes for two dimensional parabolic equations with non local boundary conditions the stability of difference schemes for one dimensional and two dimensional parabolic equations subject to non local bitsadze samarskii type boundary conditions is dealt with to analyze the stability of difference schemes the structure of the spectrum of the matrix that defines the linear system of difference equations for a respective stationary problem is studied depending on the values of parameters in non local conditions this matrix can have one zero one negative or complex eigenvalues the stepwise stability is proved and the domain of stability of difference schemes is found

a fourth order accurate method for the incompressible navier stokes equations on overlapping grids a method is described to solve the time dependent incompressible navier stokes equations with finite differences on curvilinear overlapping grids in two or three space dimensions the scheme is fourth order accurate in space and uses the momentum equations for the velocity coupled to a poisson equation for the pressure the boundary condition for the pressure is taken as u 0 extra numerical boundary conditions are chosen to make the scheme accurate and stable the velocity is advanced explicitly in time any standard time stepping scheme such as runge kutta can be used the poisson equation is solved using direct or iterative sparse matrix solvers or by the multigrid algorithm computational results in two and three space dimensions are given

modeling radio wave propagation in tunnels with a vectorial parabolic equation to study radio wave propagation in tunnels we present a vectorial parabolic equation pe taking into account the cross section shape wall impedances slowly varying curvature and torsion of the tunnel axis for rectangular cross section two polarizations are decoupled and two families of adiabatic modes can be found explicitly giving a generalization of the known results for a uniform tunnel in the general case a boundary value problem arises to be solved by using finite difference finite element fd fe techniques numerical examples demonstrate the computational efficiency of the proposed method

radiowave propagation analysis with a bidirectional 3 d vector parabolic equation method a 3 d bidirectional solution to the parabolic approximation of the wave equation is investigated by using a vector field representation the backward propagating wave is integrated to the classical parabolic equation approach which represents the forward propagating wave propagation over flat terrain in the presence of knife edges is considered as well as over irregular terrain consisting of hills modeled by the succession of knife edges at each knife edge appropriate boundary conditions are enforced and the wave is partly reflected in the backward direction the wave is marched in both directions by using the split step algorithm different tests are conducted in order to analyze and validate the results obtained by the proposed algorithm comparisons with results from both 2 d parabolic equation based algorithm and 3 d finite difference time domain based algorithm are presented in this paper

unconditionally stable finite difference scheme and iterative solution of 2d microscale heat transport equation abstract a two dimensional time dependent heat transport equation at the microscale is derived a second order finite difference scheme in both time and space is introduced and the unconditional stability of the finite difference scheme is proved a computational procedure is designed to solve the discretized linear system at each time step by using a preconditioned conjugate gradient method numerical results are presented to validate the accuracy of the finite difference scheme and the efficiency of the proposed computational procedure

high order difference methods for heat equations in polar cylindrical coordinates abstract fourth order difference methods for the solution of poisson equations in cylindrical polar coordinates are proposed the same technique is then applied to obtain o k 2 h 4 two level unconditionally stable adi methods for the solution of the heat equation in two dimensional polar coordinates and three dimensional cylindrical coordinates numerical examples given here show that the methods developed here retain their order and accuracy everywhere including the region in the vicinity of the singularity r 0

high order difference schemes for the system of two space second order nonlinear hyperbolic equations with variable coefficients abstract in this paper we develop implicit difference schemes of o k4 k2h2 h4 where k 0 h 0 are grid sizes in time and space coordinates respectively for solving the system of two space dimensional second order nonlinear hyperbolic partial differential equations with variable coefficients having mixed derivatives subject to appropriate initial and boundary conditions the proposed difference method for the scalar equation is applied for the solution of wave equation in polar coordinates to obtain three level conditionally stable adi method of o k4 k2h2 h4 some physical nonlinear problems are provided to demonstrate the accuracy of the implementation

skew quasisymmetric schur functions and noncommutative schur functions abstract recently a new basis for the hopf algebra of quasisymmetric functions qsym called quasisymmetric schur functions has been introduced by haglund luoto mason van willigenburg in this paper we extend the definition of quasisymmetric schur functions to introduce skew quasisymmetric schur functions these functions include both classical skew schur functions and quasisymmetric schur functions as examples and give rise to a new poset l c that is analogous to young s lattice we also introduce a new basis for the hopf algebra of noncommutative symmetric functions nsym this basis of nsym is dual to the basis of quasisymmetric schur functions and its elements are the pre image of the schur functions under the forgetful map nsym sym we prove that the multiplicative structure constants of the noncommutative schur functions equivalently the coefficients of the skew quasisymmetric schur functions when expanded in the quasisymmetric schur basis are nonnegative integers satisfying a littlewood richardson rule analogue that reduces to the classical littlewood richardson rule under as an application we show that the morphism of algebras from the algebra of poirier reutenauer to sym factors through nsym we also extend the definition of schur functions in noncommuting variables of rosas sagan in the algebra ncsym to define quasisymmetric schur functions in the algebra ncqsym we prove these latter functions refine the former and their properties and project onto quasisymmetric schur functions under the forgetful map lastly we show that by suitably labeling l c skew quasisymmetric schur functions arise in the theory of pieri operators on posets

polynomial chaos for random fractional order differential equations fractional order models provide a powerful instrument for description of memory and hereditary properties of systems in comparison to integer order models where such effects are difficult to incorporate and often neglected also many physical real world problems that involve uncertainties and errors can be best modeled with random differential equations thus to be able to deal with many real life problems it is important to develop mathematical methodologies to solve systems that include both memory effects and uncertainty the aim of this paper is to study the application of the generalized polynomial chaos gpc to random fractional ordinary differential equations the method of polynomial chaos has played an increasingly important role when dealing with uncertainties the main idea of the method is the projection of the random parameters and stochastic processes in the system onto the space of polynomial chaoses however to apply polynomial chaos to random fractional differential equations requires careful attention due to memory effects and the increasing of the computation time in respect to the classic random differential equations in order to avoid more complex numerical computations and obtain accurate solutions we rely on richardson extrapolation it is shown that the application of generalized polynomial chaos method in conjunction with richardson extrapolation is a reliable and accurate method to numerically solve random fractional ordinary differential equations

a gaussian beam approximation approach for embedding antennas into vector parabolic equation based wireless channel propagation models vector parabolic equation vpe methods have been widely applied to the modeling of radio wave propagation in tunnel environments offering high computational efficiency and fidelity while the propagation environment can be discretized and represented in detail the representation of radiating sources such as transmitting antennas requires the calculation analytical if possible or numerical via another method such as ray tracing rt of the fields that the sources generate on the initial plane of the vpe model these initial conditions are necessary for subsequently applying vpe however the solutions offered so far compromise either the accuracy or the efficiency of vpe for example generating the initial conditions for vpe through rt adds significant computational overhead to the typically fast vpe solver to address this significant limitation of vpe methods we introduce a technique that allows one to directly embed antennas into a vpe mesh via a gaussian beam approximation of their radiated fields hence the initial conditions for vpe are generated for practical antenna patterns without invoking other techniques and with no compromise on the inherent efficiency of vpe concrete guidelines on how to choose parameters for gaussian beams are provided numerical results are compared to experimental measurements in various tunnel scenarios demonstrating the validity and usefulness of the technique

an adi pe approach for modeling radio transmission loss in tunnels alternate direction implicit adi method is used to study radio wave propagation in tunnels using the parabolic equation pe we formulate the adi technique for use in tunnels with rectangular circular and arched cross sections and with lossy walls the electrical parameters of the lossy walls are characterized by an equivalent surface impedance a vector pe is also formulated for use in tunnels with lossy walls it is shown that the adi is more computationally efficient than the crank nicolson method however boundary conditions become more difficult to model the boundary conditions of the adi intermediate planes are given the same boundary conditions as the physical plane and the overall accuracy is reduced also when implementing the adi in tunnels with circular cross sections the order at which the line by line decomposition occurs becomes important to validate the adi pe we show simulation results for tunnel test cases with known analytical solutions furthermore the adi pe is used to simulate real tunnels in order to compare with experimental data it is shown that the pe models the electric fields most accurately in real tunnels at large distances where the lower order modes dominate

a formally fourth order accurate compact scheme for 3d poisson equation in cylindrical coordinates in this paper we extend our previous work m c lai a simple compact fourth order poisson solver on polar geometry j comput phys 182 2002 337 345 to 3d cases more precisely we present a spectral finite difference scheme for poisson equation in cylindrical coordinates the scheme relies on the truncated fourier series expansion where the partial differential equations of fourier coefficients are solved by a formally fourth order accurate compact difference discretization here the formal fourth order accuracy means that the scheme is exactly fourth order accurate while the poles are excluded and is third order accurate otherwise despite the degradation of one order of accuracy due to the presence of poles the scheme handles the poles naturally thus no pole condition is needed the resulting linear system is then solved by the bi cgstab method with the preconditioner arising from the second order discretization which shows the scalability with the problem size

on the numerical solution of multi dimensional parabolic problem by the additive splitting up method several numerical methods arising from the difference methods for certain classes of two dimensional parabolic equation are based on an operator slitting from the theoretical point of view the success and the evaluation of the splitting approach is primarily determined by the accuracy and the stability constrained most of the splitting methods defined in the past decades were of multiplicative non parallel types with respect to the spatial variables in 1994 lui tai and neittaanmaki presented a parallelisable implicit splitting type of methods of different order of approximation and its of additive type with regard to the solution at the advanced time step in this paper the stability analysis will be presented for the implicit splitting methods by lu et al and also we presented a parallelisable explicit splitting algorithm several model problems are solved by the splitting up algorithms to enhance the theoretical results and the concluding remarks

a note on multigrid for the three dimensional poisson equation in cylindrical coordinates abstract following young and dauwalder 1965 iyengar and manohar 1988 derived high order difference methods for the solution of the three dimensional poisson equation and heat equation in polar cylindrical coordinates in this note we implement and compare s and v cycles in the multigrid context for the fourth order method derived by iyengar and manohar 1988 for the solution of the three dimensional poisson equation defect correction is also studied suitable restrictions and prolongations for the three dimensional problems are given it is seen that the s cycle is more efficient than the v cycle and defect correction is more economical than the direct implementation of the fourth order method

comparison of different numerical laplace inversion methods for engineering applications abstract laplace transform is a powerful method for enabling solving differential equation in engineering and science using the laplace transform for solving differential equations however sometimes leads to solutions in the laplace domain that are not readily invertible to the real domain by analytical means numerical inversion methods are then used to convert the obtained solution from the laplace domain into the real domain four inversion methods are evaluated in this paper several test functions which arise in engineering applications are used to evaluate the inversion methods we also show that each of the inversion methods is accurate for a particular case this study shows that among all these methods the fourier transform inversion technique is the most powerful but also the most computationally expensive stehfest s method which is used in many engineering applications is easy to implement and leads to accurate results for many problems including diffusion dominated ones and solutions that behave like e t type functions however this method fails to predict e t type functions or those with an oscillatory response such as sine and wave functions

a fourth order compact adi method for solving two dimensional unsteady convection diffusion problems n this article an exponential high order compact ehoc alternating direction implicit adi method in which the crank nicolson scheme is used for the time discretization and an exponential fourth order compact difference formula for the steady state 1d convection diffusion problem is used for the spatial discretization is presented for the solution of the unsteady 2d convection diffusion problems the method is temporally second order accurate and spatially fourth order accurate which requires only a regular five point 2d stencil similar to that in the standard second order methods the resulting ehoc adi scheme in each adi solution step corresponds to a strictly diagonally dominant tridiagonal matrix equation which can be inverted by simple tridiagonal gaussian decomposition and may also be solved by application of the one dimensional tridiagonal thomas algorithm with a considerable saving in computing time the unconditionally stable character of the method was verified by means of the discrete fourier or von neumann analysis numerical examples are given to demonstrate the performance of the method proposed and to compare mostly it with the high order adi method of karaa and zhang and the spatial third order compact scheme of note and tan

a finite difference scheme for the heat conduction equation abstract a symmetrical semi implicit ssi difference scheme is formulated for the heat conduction equation the scheme is easy to code fast and quite accurate the advantage of the scheme appears mainly when used for large complicated multidimensional grids and for nonlinear problems

high accuracy difference schemes for the system of two space nonlinear parabolic differential equations with mixed derivatives and variable coefficients abstract in this article two level compact implicit difference methods of o k2 kh2 h4 using 9 spatial grid points are proposed for the numerical solution of the system of two dimensional nonlinear parabolic equations with variable coefficients subject to the dirichlet boundary conditions where k 0 and h 0 are step lengths in time and space directions respectively the proposed difference method for scalar equation is applied for the solution of the heat conduction equation in polar coordinates to obtain the two level unconditionally stable adi method of o k2 h4 the method having two variables also has been successfully applied on two dimensional unsteady navier stokes model equations in polar coordinates some numerical examples are presented to demonstrate the accuracy of the implementation

on the informational properties of the nash solution of lqg dynamic games the m person n stage discrete time lqg nash game is considered the players use strategies that are linear functions of the current estimate of the state generated by a kalman filter we study the impact of improvements of the information on the costs of the players for certain classes of such problems we show that better information is beneficial to all the players if the number of stages or the number of players is larger than some bounds and which bounds are given explicitly in terms of the coefficient matrices related properties of the two person zero sum game are also investigated it is shown that under certain conditions better information is beneficial to the player who has better maneuverability while the saddle point cost is independent of the information if both players have the same maneuverability conditions guaranteeing the uniform boundedness of the solutions of the coupled riccati equations which arise in such games are also given

a transformation free hoc scheme for incompressible viscous flows past an impulsively started circular cylinder in this paper we present a higher order compact scheme for the unsteady two dimensional 2d navier stokes equations on nonuniform polar grids specifically designed for the incompressible viscous flows past a circular cylinder the scheme is second order accurate in time and at least third order accurate in space the scheme very efficiently computes both unsteady and time marching steady state flow for a wide range of reynolds numbers re ranging from 10 to 9500 for the impulsively started cylinder the robustness of the scheme is highlighted when it accurately captures the vortex shedding for moderate re represented by the von karman street and the so called and s phenomena for higher re comparisons are made with established numerical and experimental results and excellent agreement is found in all the cases both qualitatively and quantitatively

a full analysis of a new second order finite volume approximation based on a low order scheme using general admissible spatial meshes for the unsteady one dimensional heat equation abstract the present work is an extension of our previous works 3 4 5 which dealt with first order both in time and space and second order time accurate second order in time and first order in space implicit finite volume schemes for parabolic equations we aim in this work and some forthcoming studies at getting higher order both in time and space finite volume approximations for the exact solution of parabolic equations using the class of spatial generic meshes introduced recently in 13 we focus in the present contribution on the one dimensional heat equation and its implicit finite volume scheme described in 3 the implicit finite volume scheme approximating the one dimensional heat equation we consider hereafter referred to as the basic finite volume scheme yields linear systems to be solved successively the matrices involved in these linear systems are tridiagonal the finite volume approximate solution is of order h k where h resp k is the mesh size of the spatial resp time discretization we construct a new finite volume approximation of order h k 2 in several discrete norms which allows us to get approximations of order two for the exact solution and its first derivatives this new high order approximation can be computed using the same linear systems involved in the basic finite volume scheme while the right hand sides are corrected the construction of these right hand sides includes the approximations of the second third and fourth spatial derivatives of the exact solution the computation of the approximation of these high order derivatives can be performed using the same matrices stated above with another two tridiagonal matrices the manner by which this new high order approximation is constructed can be repeated to compute successively finite volume approximations of arbitrary order using the same matrices stated above these high order approximations can be obtained on any one dimensional admissible finite volume mesh in the sense of 12 without any restrictive condition on the spatial mesh a full analysis for the stated theoretical results as well as some numerical examples supporting the theory is presented the results obtained in the present study are based essentially on two facts the first fact is the use of the results provided in 3 which state the convergence order of the finite volume approximate solution in several norms the second fact is the comparison between the stated new higher order approximations and suitable auxiliary finite volume approximations

examination clarification and simplification of stability and dispersion analysis for adi fdtd and cnss fdtd schemes we describe a rigorous analysis of unconditional stability for the alternating direction implicit finite difference time domain adi fdtd and crank nicholson split step cnss fdtd schemes avoiding use of the von neumann spectral criterion the proof is performed in the spectral domain and uses skew hermitivity of matrix terms in the adi and cnss total amplification matrices a bound for the total adi amplification matrix is provided while the cn and cnss fdtd amplification matrices are unitary the bound on the adi fdtd depends on the courant number importantly we have found that the adi fdtd amplification matrix is not normal i e the unit spectral radius alone cannot be used to prove the adi fdtd unconditional stability the paper also shows that the adi fdtd and cnss fdtd schemes share the same dispersion equation by a similarity of their total amplification matrices and the two schemes have identical numerical dispersion in the frame of plane waves

a new adi technique for two dimensional parabolic equation with an integral condition abstract nonclassical parabolic initial boundary value problems arise in the study of several important physical phenomena this paper presents a new approach to treat complicated boundary conditions appearing in the parabolic partial differential equations with nonclassical boundary conditions a new fourth order finite difference technique based upon the noye and hayman n h alternating direction implicit adi scheme is used as the basis to solve the two dimensional time dependent diffusion equation with an integral condition replacing one boundary condition this scheme uses less central processor time cpu than a second order fully implicit scheme based on the classical backward time centered space btcs method for two dimensional diffusion it also has a larger range of stability than a second order fully explicit scheme based on the classical forward time centered space ftcs method the basis of the analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach developed from the 1974 work of warming and hyeet this allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference methods the results of numerical experiments for the new method are presented the central processor times needed are also reported error estimates derived in the maximum norm are tabulated

reproducing kernel method for numerical simulation of downhole temperature distribution abstract this paper research downhole temperature distribution in oil production and water injection using reproducing kernel hilbert space method rkhsm for the first time the aim of this paper is that using the highly accurate rkhsm can solve downhole temperature problems effectively according to 2 d mathematical models of downhole temperature distribution the analytical solution was given in a series expansion form and the approximate solution was expressed by n term summation of reproducing kernel functions which initial and boundary conditions were selected properly numerical results of downhole temperature distribution with multiple pay zones in which different radial distance and different injection production conditions such as injection rate injection temperature injection time oil layer thickness were carried out by mathematical 7 0 and numerical results correspond to general knowledge and show that use rkhsm to research downhole temperature distribution is feasible and effective

short note a high order pade adi method for unsteady convection diffusion equations a high order alternating direction implicit adi method for computations of unsteady convection diffusion equations is proposed by using fourth order pade schemes for spatial derivatives the present scheme is fourth order accurate in space and second order accurate in time the solution procedure consists of a number of tridiagonal matrix operations which make the computation cost effective the method is unconditionally stable and shows higher accuracy and better phase and amplitude error characteristics than the standard second order adi method d w peaceman h h rachford jr the numerical solution of parabolic and elliptic differential equations journal of the society of industrial and applied mathematics 3 1959 28 41 and the fourth order adi scheme of karaa and zhang high order adi method for solving unsteady convection diffusion problem journal of computational physics 198 2004 1 9

a new two level implicit discretization of o k 2 kh 2 h 4 for the solution of singularly perturbed two space dimensional non linear parabolic equations we propose a new two level implicit difference method of o k2 kh2 h4 for the solution of singularly perturbed non linear parabolic differential equation e uxx uyy f x y t u ux u y ut 0 0 subject to appropriate initial and dirichlet boundary conditions where k 0 and h 0 are grid sizes in time and space directions respectively and e 0 is a small parameter we also develop new methods of o kh2 h4 for the estimates of u x and u y in all cases we use 9 spatial grid points and a single computational cell the proposed methods are directly applicable to singular problems we do not require any special scheme to solve singular problems we also discuss alternating direction implicit adi method for solving diffusion equation in polar cylindrical coordinates this method permits multiple use of the one dimensional tri diagonal algorithm with a considerable saving in computing time and produces a very efficient solver it is shown that the adi method is unconditionally stable numerical experiments are conducted to test the high accuracy of the proposed methods and compared with the exact solutions

a difference scheme for burgers equation in an unbounded domain the numerical solution of the one dimensional burgers equation in an unbounded domain is considered two artificial boundaries are introduced to make the computational domain finite on both artificial boundaries two exact boundary conditions are proposed respectively to reduce the original problem to an initial boundary value problem in a finite computational domain a difference scheme is constructed by the method of reduction of order to solve the problem in the finite computational domain at each time level only a strictly diagonal dominated tridiagonal system of linear algebraic equations needs to be solved it is proved that the difference scheme is uniquely solvable and unconditional convergent with the convergence order 3 2 in time and order 2 in space in an energy norm a numerical example demonstrates the theoretical results

hopscotch method the numerical solution of the frank kamenetskii partial differential equation numerical solutions to the frank kamenetskii partial differential equation modelling a thermal explosion in a cylindrical vessel are obtained using the hopscotch scheme we observe that a nonlinear source term in the equation leads to numerical difficulty and hence adjust the scheme to accommodate such a term numerical solutions obtained via matlab mathematica and the crank nicolson implicit scheme are employed as a means of comparison to gain insight into the accuracy of the hopscotch scheme the solution is compared to a power series solution obtained via the lie group method the numerical solution is also observed to converge to a well known steady state solution a linear stability analysis is performed to validate the stability of the results obtained

a new high accuracy two level implicit off step discretization for the system of three space dimensional quasi linear parabolic partial differential equations we present a new two level compact implicit numerical scheme based on off step discretization for the solution of the system of three space dimensional quasi linear parabolic partial differential equations subject to suitable initial and boundary conditions further we derive the estimates of first order space derivatives of the solution the proposed methods are fourth order accurate in space and second order accurate in time and involve 19 spatial grid points of a single compact cell also we develop the alternating direction implicit adi scheme for a linear parabolic equation with variable coefficients which is shown to be unconditionally stable for the heat equation in polar coordinates the methods so proposed do not require any modification when applied to the singular problems at the point of singularity unlike the numerical scheme proposed earlier in mohanty and jain 1994 and mohanty 1997 2003 the proposed methods are directly applicable to parabolic equations with singular coefficients this is the main highlight of our work the method successfully works for the navier stokes equations of motion in polar coordinates many physical problems are solved to illustrate the accuracy of the proposed methods

high order difference methods for linear variable coefficient parabolic equation abstract in this paper we derive o k 2 h 4 two level three point finite difference methods for the solution of the general linear variable coefficient parabolic equation in one dimension u t a x u xx b x u x c x u 0 x 1 t 0 under suitable initial and boundary conditions the stability of the new schemes is examined using a linear stability analysis in particular we derive unconditionally stable o k 2 h 4 methods for the solution of the convection diffusion type equation in cylindrical and spherical coordinates viz u t u xx x u x cu 0

extrapolation algorithm of compact adi approximation for two dimensional parabolic equation abstract in this paper we propose a compact alternating direction implicit adi scheme for solving the two dimensional parabolic equation the maximum norm convergence is proven and the convergence rate is second order in time and fourth order in space we develop a richardson extrapolation algorithm to increase the accuracy to sixth order both in time and space numerical experiments show the effectiveness of the method

a brief survey on numerical methods for solving singularly perturbed problems in the present paper a brief survey on computational techniques for the different classes of singularly perturbed problems is given this survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000 2009 however some older important relevant papers are also included in this survey we also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper see appl math comput 30 1989 223 259 130 2002 457 510 134 2003 371 429 for details thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems

a high order hodie finite difference scheme for 1d parabolic singularly perturbed reaction diffusion problems this paper deals with the numerical approximation of the solution of 1d parabolic singularly perturbed problems of reaction diffusion type the numerical method combines the standard implicit euler method on a uniform mesh to discretize in time and a hodie compact fourth order finite difference scheme to discretize in space which is defined on a priori special meshes condensing the grid points in the boundary layer regions the method is uniformly convergent having first order in time and almost fourth order in space the analysis of the uniform convergence is made in two steps splitting the contribution to the error from the time and the space discretization although this idea has been previously used to prove the uniform convergence for parabolic singularly perturbed problems here the proof is based on a new study of the asymptotic behavior of the exact solution of the semidiscrete problems obtained after the time discretization by using the euler method some numerical results are given corroborating in practice the theoretical results

a simple compact fourth order poisson solver on polar geometry we present a simple and efficient compact fourth order poisson solver in polar coordinates this solver relies on the truncated fourier series expansion where the differential equations of the fourier coefficients are solved by the compact fourth order finite difference scheme by shifting a grid a half mesh away from the origin and incorporating the symmetry constraint of fourier coefficients we can easily handle coordinate singularities without pole conditions the numerical evidence confirms fourth order accuracy for the problem on an annulus and third order accuracy for the problem on a disk in addition a simple and comparably accurate approximation for the derivatives of the solution is also presented

high accuracy difference schemes for a class of three space dimensional singular parabolic equations with variable coefficients abstract in this article two level implicit difference methods of o k 2 h 4 using 19 spatial grid points for the solution of three space dimensional heat conduction equation and unsteady navier stokes equations in polar coordinates are proposed unconditionally stable adi methods for the solution of the heat conduction equation in polar coordinates are also discussed numerical examples given here show that the methods developed here retain their order and accuracy everywhere including the region in the vicinity of the singularity r 0

discretisation and multigrid solution of elliptic equations with mixed derivative terms and strongly discontinuous coefficients this paper introduces a new conservative cell centred finite volume scheme for the accurate solution of elliptic diffusion equations with strongly varying coefficients the discretisation is designed to model a wide range of heterogeneities and anisotropies and in particular the case where the diffusivity is represented by a nondiagonal matrix which may occur if the medium is anisotropic in a general direction such problems arise for example in oil reservoir simulation when renormalisation techniques are used to model the reservoir geology it is in this context that the method is described although it is applicable to a far wider class of problems in heat conduction electrostatics and potential theory the paper also describes the application of a mulitigrid scheme for efficient numerical solution of the algebraic equations derived using the new discretisation

a compact finite difference scheme for solving a one dimensional heat transport equation at the microscale 431 heat transport at the microscale is of vital importance in microtechnology applications the heat transport equation is different from the traditional heat diffusion equation since a second order derivative of temperature with respect to time and a third order mixed derivative of temperature with respect to space and time are introduced in this study we develop a high order compact finite difference scheme for the heat transport equation at the microscale it is shown by the discrete fourier analysis method that the scheme is unconditionally stable numerical results show that the solution is accurate

an alternating direction implicit scheme for parabolic equations with mixed derivative and convective terms an alternating direction implicit adi scheme is introduced which is capable of solving a general parabolic equation in two space dimensions with mixed derivative and convective terms in the case of constant coefficients the scheme is shown to be unconditionally stable the study was motivated by the investigation of flow in a dye laser cell a device used for the amplification of a laser beam a simple model for which involves laminar flow in a two dimensional symmetric channel subject to impulsive heating numerical results are presented for this problem and the qualitative behaviour of the temperature distribution within the channel for different peclet numbers is discussed

haar wavelet collocation method for three dimensional elliptic partial differential equations a new collocation method based on haar wavelet is presented for numerical solution of three dimensional elliptic partial differential equations with dirichlet boundary conditions an important advantage of the method is that it can be applied to both linear as well as nonlinear problems the algorithm based on this new method is simple and can be easily implemented in any programming language experimental rates of convergence of the proposed method are calculated which are in agreement with theoretical results the proposed method is applied to several benchmark problems from the literature including linear and nonlinear elliptic problems as well as systems of elliptic partial differential equations the numerical experiments confirm the accuracy and diverse applicability of the method

numerical investigation of the parabolic mixed derivative diffusion equation via alternating direction implicit methods the two dimensional analogue of a two parameter mixed derivative equation is considered via two numerical methods the adi and compact adi method these methods are compared and the influence of the kinematic viscosity and viscoelasticity is discussed in relation to the solutions obtained the two schemes are proven to be conditionally stable and the methods compared via an error analysis

analysis of stationary iterative methods for the discrete convection diffusion equation with a 9 point compact scheme we study the convergence of point and line stationary iterative methods for solving the linear system arising from a fourth order 9 point compact finite difference discretization of the two dimensional convection diffusion equation with constant coefficients we present new techniques to bound the spectral radii of iteration matrices in terms of the cell reynolds numbers we also derive analytic formulas for the spectral radii for special values of the cell reynolds numbers and study asymptotic behaviors of the analytic bounds the results provide rigorous justification for the numerical experiments conducted elsewhere which show good stability for the fourth order compact scheme in addition we compare the 9 point scheme with the traditional 5 point difference discretization schemes and conduct some numerical experiments to supplement our analyses

flow past an impulsively started circular cylinder using a higher order semicompact scheme in the present investigation an attempt has been made to solve the two dimensional incompressible viscous flow past an impulsively started circular cylinder for reynolds numbers ranging from 20 to 5000 using higher order semicompact scheme hosc unlike conventional higher order compact schemes the hosc scheme has been developed to handle the circular geometry of the chosen problem and the intensive algebraic manipulations have been reduced considerably by relaxing the compactness of the computational stencil for few terms but retained for most of the terms of the discretized equations for the flow past an impulsively started circular cylinder the results obtained at low and moderate reynolds numbers have been validated with the experimental and numerical observations available in the literature for high reynolds number flows the present scheme rightly captures the alpha phenomenon at re 1000 and both beta and alpha phenomena one after the other at re 5000

a two level linearized compact adi scheme for two dimensional nonlinear reaction diffusion equations abstract a novel two level linearized compact alternating direction implicit adi scheme is proposed for solving two dimensional nonlinear reaction diffusion equations the computational cost is reduced by use of the newton linearized method and the adi method the existence and uniqueness of the numerical solutions are proved moreover the error estimates in h 1 and l norms are presented numerical examples are given to illustrate our theoretical results

hopf algebras and edge labeled posets abstract given a finite graded poset with labeled hasse diagram we construct a quasi symmetric generating function for chains whose labels have fixed descents this is a common generalization of a generating function for the flag f vector defined by ehrenborg and of a symmetric function associated with certain edge labeled posets that arose in the theory of schubert polynomials we show that this construction gives a hopf morphism from an incidence hopf algebra of edge labeled posets to the hopf algebra of quasi symmetric functions

a new high accuracy two level implicit off step discretization for the system of two space dimensional quasi linear parabolic partial differential equations abstract in this article we propose a two level compact implicit off step discretization for the solution of the system of 2d quasi linear parabolic partial differential equations subject to suitable initial and boundary conditions new methods for the estimates of first order space derivatives of the solution are also derived these methods are fourth order accurate in space and second order accurate in time and involve only nine spatial grid points of a single compact cell we further develop the alternating direction implicit adi scheme for a general linear parabolic equation which is shown to be unconditionally stable for the heat equation in polar coordinates the proposed methods are directly applicable to singular problems without the need of any special technique which is the main advantage of this work the method is effectively applied to the time dependent navier stokes model equations in polar coordinates numerical examples are provided to illustrate the accuracy of the methods

numerical inversion of a laplace transform solution of a diffusion equation with a mixed derivative term we consider a diffusion equation with a mixed derivative term modelling the transient flow of a second grade fluid the same diffusion equation with a mixed derivative term also comes up in the modelling of heat conduction in which the history of the temperature gradient is included the equations differ in the sign of the coefficient of the mixed derivative term we show that the method of lines solution is stable for the transient flow of a second grade fluid and unstable for heat conduction with the temperature gradient history included a laplace transform solution for the heat conduction case is then investigated the laplace transform solution is inverted numerically using a fourier based method a stable solution is obtained the solutions are compared and discussed

a new fourth order discretization for singularly perturbed two dimensional non linear elliptic boundary value problems abstract in this article we derive a new difference method of o h4 so called arithmetic average discretization for the solution of two dimensional non linear singularly perturbed elliptic partial differential equation of the form e uxx uyy f x y u ux uy 0 0 is a small parameter we also derive new methods of o h4 for the estimates of u n which are quite often of interest in many physical problems in all cases we require only 9 grid points and a single computational cell the main advantage of the proposed methods is that the methods are directly applicable to singular problems we do not require any special technique or modification to solve singular problems numerical results are provided to demonstrate the usefulness of the methods discussed

an alternative direction decomposition scheme and error analysis for parabolic equation model alternate direction implicit adi method has been widely used to analyze the tunnel propagation or electromagnetic scattering problem for the finite difference method in this paper a novel alternate direction decomposition add scheme is proposed to simplify the split step fourier transform ssft algorithm in parabolic equation pe model the spatial matrix of electric field is decoupled into two subarrays by add ssft algorithm which is used to decrease the computational burden for 3 d pe problems in a large scale environment after the integration order exchanged also the phase error of this approximation is analyzed to show the algorithm performance along with several numerical results including the tunnel cases with traditional modal theory and they are provided to validate and calibrate the pe model and demonstrate the efficiency and accuracy of the proposed algorithm moreover the scattering computation is investigated and compared with multilevel fast multipole method to show that add ssft is appropriate for solving the scattering problems for the electrical large target

locally explicit schemes for three dimensional diffusion with a non local boundary specification two finite difference schemes based on the classical three point forward time centred space ftcs method and the five point ftcs formula are used to solve the three dimensional time dependent diffusion equation with a non local boundary specification in these methods locally one dimensional lod procedures are used to extend the one dimensional techniques to solve the three dimensional problem numerical integration procedures are also employed to overcome the non local boundary specification results from some numerical experiments are presented including error estimates derived in the maximum norm the results of numerical testing show that the lod techniques give better results than the fully explicit schemes the cpu times used for the procedures developed in this article are also reported