symbolic computation of high order compact difference schemes for three dimensional linear elliptic partial differential equations with variable coefficients we present a symbolic computation procedure for deriving various high order compact difference approximation schemes for certain three dimensional linear elliptic partial differential equations with variable coefficients based on the maple software package we approximate the leading terms in the truncation error of the taylor series expansion of the governing equation and obtain a 19 point fourth order compact difference scheme for a general linear elliptic partial differential equation a test problem is solved numerically to validate the derived fourth order compact difference scheme this symbolic derivation method is simple and can be easily used to derive high order difference approximation schemes for other similar linear elliptic partial differential equations

an efficient sixth order solution for anisotropic poisson equation with completed richardson extrapolation and multiscale multigrid method we present an efficient numerical method for anisotropic poisson equations the sixth order accuracy is achieved through applying completed richardson extrapolation on two fourth order solutions computed from different scale grids with unequal mesh size discretization theoretical analysis is conducted to demonstrate that the richardson extrapolation is able to obtain a sixth order solution by removing the leading truncation error terms of the fourth order solution from grid with unequal mesh sizes the gain in efficiency is obtained through adopting partial semi coarsening multigrid method to solve the resulting linear systems and multiscale multigrid computation to speed up the whole solution numerical experiments are conducted to verify the accuracy and efficiency of the proposed method and the results are compared with the existing fourth order methods for solving 2d and 3d anisotropic poisson equations

high order compact scheme with multigrid local mesh refinement procedure for convection diffusion problems we derive a new fourth order compact finite difference scheme which allows different meshsize in different coordinate directions for the two dimensional convection diffusion equation a multilevel local mesh refinement strategy is used to deal with the local singularity problem a corresponding multilevel multigrid method is designed to solve the resulting sparse linear system numerical experiments are conducted to show that the local mesh refinement strategy works well with the high order compact discretization scheme to recover high order accuracy for the computed solution our solution method is also shown to be effective and robust with respect to the level of mesh refinement and the anisotropy of the problems

a numerical study of some radial basis function based solution methods for elliptic pdes abstract during the last decade three main variations have been proposed for solving elliptic pdes by means of collocation with radial basis functions rbfs in this study we have implemented them for infinitely smooth rbfs and then compared them across the full range of values for the shape parameter of the rbfs this was made possible by a recently discovered numerical procedure that bypasses the ill conditioning which has previously limited the range that could be used for this parameter we find that the best values for it often fall outside the range that was previously available we have also looked at piecewise smooth versus infinitely smooth rbfs and found that for pde applications with smooth solutions the infinitely smooth rbfs are preferable mainly because they lead to higher accuracy in a comparison of rbf based methods against two standard techniques a second order finite difference method and a pseudospectral method the former gave a much superior accuracy

improved multiquadric method for elliptic partial differential equations via pde collocation on the boundary the multiquadric radial basis function mq method is a recent meshless collocation method with global basis functions it was introduced for discretizing partial differential equations pdes by kansa in the early 1990s the mq method was originally used for interpolation of scattered data and it was shown to have exponential convergence for interpolation problems in 1 we have extended the kansa mq method to numerical solution and detection of bifurcations in 1d and 2d parameterized nonlinear elliptic pdes we have found there that the modest size nonlinear systems resulting from the mq discretization can be efficiently continued by a standard continuation software such as auto we have observed high accuracy with a small number of unknowns as compared with most known results from the literature in this paper we formulate an improved kansa mq method with pde collocation on the boundary mq pdecb we add an additional set of nodes which can lie inside or outside of the domain adjacent to the boundary and correspondingly add an additional set of collocation equations obtained via collocation of the pde on the boundary numerical results are given that show a considerable improvement in accuracy of the mq pdecb method over the kansa mq method with both methods having exponential convergence with essentially the same rates

a transformation free hoc scheme and multigrid method for solving the 3d poisson equation on nonuniform grids a high order compact hoc difference scheme is proposed to solve the three dimensional 3d poisson equation on nonuniform orthogonal cartesian grids involving no coordinate transformation from the physical space to the computational space theoretically the proposed scheme has third to fourth order accuracy its fourth order accuracy is achieved under uniform grid settings then a multigrid method is developed to solve the linear system arising from this hoc difference scheme and the corresponding multigrid restriction and interpolation operators are constructed using the volume law numerical experiments are conducted to show the computed accuracy of the hoc scheme and the computational efficiency of the multigrid method

truncation error and oscillation property of the combined compact difference scheme in this work we study a sixth order combined compact difference scheme for a one dimensional convection diffusion equation we derive the truncation error representation and analyze its oscillation property numerical experiments are performed to illustrate and support our analysis

radial basis function method for a multidimensional linear elliptic equation with nonlocal boundary conditions the development of numerical methods for the solution of partial differential equations pdes with nonlocal boundary conditions is important since such type of problems arise as mathematical models of various real world processes we use radial basis function rbf collocation technique for the solution of a multidimensional linear elliptic equation with classical dirichlet boundary condition and nonlocal integral conditions rbf based meshless methods are easily implemented and efficient especially for multidimensional problems formulated on complexly shaped domains in this paper properties of the method are investigated by studying two and three dimensional test examples with manufactured solutions we analyze the influence of the rbf shape parameter and the distribution of the nodes on the accuracy of the method as well as the influence of nonlocal conditions on the conditioning of the collocation matrix

some dynamically trivial mappings with applications to the improvement of simple iteration abstract we consider the iteration algorithm defined by x n k g x n x n 1 x n k 1 n 0 1 2 where is a k dimensional means we show that under certain conditions on the function g the algorithm will converge globally to a fixed point of g and that under less stringent conditions it will converge locally even when the corresponding algorithm for simple iteration x n 1 g x n n 0 1 2 does not a corollary of our main theorem answers a related question concerning the dynamic triviality of a class of mappings of r k into itself

minimal residual smoothing in multi level iterative method a minimal residual smoothing mrs technique is employed to accelerate the convergence of the multi level iterative method by smoothing the residuals of the original iterative sequence the sequence with smoothed residuals is re introduced into the multi level iterative process the new sequence generated by this acceleration procedure converges much faster than both the sequence generated by the original multi level method and the sequence generated by mrs technique the cost of this acceleration scheme is independent of the original operator and in many cases is negligible the emphasis of this paper is on the practical implementation of mrs acceleration techniques in the multi level method the discussions are focused on the two level method because the acceleration scheme is only applied on the finest level of the multi level method numerical experiments using the proposed mrs acceleration scheme to accelerate both the two level and multi level methods are conducted to show the efficiency and the cost effectiveness of this acceleration scheme

a modified diffusion coefficient technique for the convection diffusion equation a new modified diffusion coefficient mdc technique for solving convection diffusion equation is proposed the galerkin finite element discretization process is applied on the modified equation rather than the original one for a class of one dimensional convection diffusion equations we derive the modified diffusion coefficient analytically as a function of the equation coefficients and mesh size then prove that the discrete solution of this method coincides with the exact solution of the original equation for every mesh size and or equation coefficients the application of the derived analytic formula of mdc is extended for other classes of convection diffusion equations where the analytic formula is computed locally within each element according to the mesh size and the values of associated coefficients in each direction the numerical results of the proposed approach for two dimensional variable coefficients with boundary layers convection dominated problems show stable and accurate solutions even on coarse grids accordingly multigrid based solvers retain their efficient convergence rates

high order compact exponential finite difference methods for convection diffusion type problems a class of high order compact hoc exponential finite difference fd methods is proposed for solving one and two dimensional steady state convection diffusion problems the newly proposed hoc exponential fd schemes have nonoscillation property and yield high accuracy approximation solution as well as are suitable for convection dominated problems the o h 4 compact exponential fd schemes developed for the one dimensional 1d problems produce diagonally dominant tri diagonal system of equations which can be solved by applying the tridiagonal thomas algorithm for the two dimensional 2d problems o h 4 k 4 compact exponential fd schemes are formulated on the nine point 2d stencil and the line iterative approach with alternating direction implicit adi procedure enables us to deal with diagonally dominant tridiagonal matrix equations which can be solved by application of the one dimensional tridiagonal thomas algorithm with a considerable saving in computing time to validate the present hoc exponential fd methods three linear and nonlinear problems mostly with boundary or internal layers where sharp gradients may appear due to high peclet or reynolds numbers are numerically solved comparisons are made between analytical solutions and numerical results for the currently proposed hoc exponential fd methods and some previously published hoc methods the present hoc exponential fd methods produce excellent results for all test problems it is shown that besides including the excellent performances in computational accuracy efficiency and stability the present method has the advantage of better scale resolution the method developed in this article is easy to implement and has been applied to obtain the numerical solutions of the lid driven cavity flow problem governed by the 2d incompressible navier stokes equations using the stream function vorticity formulation

analysis of a fourth order compact scheme for convection diffusion in 1984 gupta et al introduced a compact fourth order finite difference convection diffusion operator with some very favorable properties in particular this scheme does not seem to suffer excessively from spurious oscillatory behavior and it converges with standard methods such as gauss seidel or sor hence multigrid regardless of the diffusion this scheme has been rederived developed including some variations and applied in both convection diffusion and navier stokes equations by several authors accurate solutions to high reynolds number flow problems at relatively coarse resolutions have been reported these solutions were often compared to those obtained by lower order discretizations such as second order central differences and first order upstream discretizations the latter it was stated achieved far less accurate results due to the artificial viscosity which the compact scheme did not include we show here that while the compact scheme indeed does not suffer from a cross stream artificial viscosity as does the first order upstream scheme when the characteristic direction is not aligned with the grid it does include a streamwise artificial viscosity that is inversely proportional to the natural viscosity this term is not always benign 7 refs 1 fig 1 tab

mathematical treatment of the magnetic confinement in a current carrying stellarator the model studied concerns the case of a stellarator machine and the magnetic confinement is modeled by using averaging methods and suitable vacuum coordinates this is shown to lead to a two dimensional grad shafranov type problem for the averaged poloidal flux function various problems are considered and it is pointed out that corresponding problems for models based on tokamak machines are essentially similar

finite difference methods for solution of nonlocal boundary value problems the article is devoted to the construction and investigation of various types of difference schemes relevant to nonlocal boundary value problems stated for multidimensional elliptic equations suitable tools for performing the investigation are developed including difference analogue of green s formula and inequalities corresponding to nonlocal boundary conditions

matrix dependent prolongations and restrictions in a blackbox multigrid solver multigrid methods are studied for the solution of linear systems resulting from the 9 point discretization of a general linear second order elliptic partial differential equation in two dimensions the rate of convergence of standard multigrid methods often deteriorates when the coefficients in the differential equation are discontinuous or when dominating first order terms are present these difficulties may be overcome by choosing the prolongation and restriction operators in a special way a novel way to do this is proposed as a result a blackbox solver written in standard fortran 77 has been developed numerical experiments for several hard test problems are described and comparison is made with other algorithms the standard mg method and a method introduced by kettler a significant improvement of robustness and efficiency is found

multigrid defect correction and fourth order compact scheme for poissons equation this paper presents an analysis of a multigrid defect correction to solve a fourth order compact scheme discretization of the poissons equation we focus on the formulation which arises in the velocity pressure decoupling methods encountered in computational fluid dynamics especially the poissons equation results of the divergence gradient formulation and neumann boundary conditions are prescribed the convergence rate of a multigrid defect correction is investigated by means of an eigenvalues analysis of the iteration matrix the stability and the mesh independency are demonstrated an improvement of the convergence rate is suggested by introducing the damped jacobi and incomplete lower upper smoothers based on an eigenvalues analysis the optimal damping parameter is proposed for each smoother numerical experiments confirm the findings of this analysis for periodic domain and uniform meshes which are the working assumptions further numerical investigations allow us to extend the results of the eigenvalues analysis to neumann boundary conditions and non uniform meshes the hodgehelmholtz decomposition of a vector field is carried out to illustrate the computational efficiency especially by making comparisons with a second order discretization of the poissons equation solved with a state of art of algebraic multigrid method

high accuracy iterative solution of convection diffusion equation with boundary layers on nonuniform grids abstract a fourth order compact finite difference scheme and a multigrid method are employed to solve the two dimensional convection diffusion equations with boundary layers the computational domain is first discretized on a nonuniform stretched grid to resolve the boundary layers a grid transformation technique is used to map the nonuniform grid to a uniform one the fourth order compact scheme is applied to the transformed uniform grid a multigrid method is used to solve the resulting linear system numerical experiments are used to show that a graded mesh and a grid transformation are necessary to compute high accuracy solutions for the convection diffusion problems with boundary layers and dicretized by the fourth order compact scheme

some new operational matrices and its application to fractional order poisson equations with integral type boundary constrains abstract enormous application of fractional order partial differential equations fpdes subjected to some constrains in the form of nonlocal boundary conditions motivated the interest of many scientists around the world the prime objective of this article is to find approximate solution of a general fpdes subject to nonlocal integral type boundary conditions on both ends of the domain the proposed method is based on spectral method we construct some new operational matrices which have the ability to handle integral type non local boundary constrains these operational matrices can be effectively applied to convert the fpdes together with its integral types boundary conditions to easily solvable matrix equation the accuracy and efficiency of proposed method are demonstrated by solving some bench mark problems the proposed method has the ability to solve non local fpdes with high accuracy and low computational cost different aspects of presented approach are compared with two other recently developed methods haar wavelets collocation method and a family of collocation methods which are based on radial base functions it is observed that the proposed method is highly accurate robust efficient and stable as compared to these methods

grid adaptive algorithms for the solution of the euler equations on irregular grids abstract the effect of grid irregularity on the accuracy of algorithms for one dimensional unsteady flow is investigated it is shown how standard second order accurate algorithms for equally spaced grids may be extended to act as first order accurate algorithms on an irregular grid the loss of accuracy on irregular grids is found to cause a significant reduction in the quality of the results to obtain solutions comparable in quality with second order accurate algorithms on equally spaced grids a class of second order accurate algorithms is derived which are conservative and compatible on randomly spaced grids they are shown to give solutions to the equations similar in quality to those obtainable on equally spaced grids

global attractivity in a nonlinear difference equation abstract we consider the nonlinear difference equation x n a k 1 m b k x n k n 0 1 2 where a and bk k 1 2 m are nonnegative numbers with b mk 1bk 0 and we are interested in whether all positive solutions are attracted by the positive equilibrium l a 2 a 2 2 b

radial basis collocation methods for elliptic boundary value problems this paper introduces radial basis functions rbf into the collocation methods and the combined methods for elliptic boundary value problems first the ritz galerkin method rgm is chosen using the rbf and the integration approximation leads to the collocation method of rbf for poisson s equation next the combinations of rbf with finite element method fem finite difference method fdm etc can be easily formulated by following li 1 and hu and li 2 3 but more analysis of inverse estimates is explored in this paper since the rbfs have the exponential convergence rates and since the collocation nodes may be scattered in rather arbitrary fashions in various applications the rbf may be competitive to orthogonal polynomials for smooth solutions moreover for singular solutions we may use some singular functions and rbfs together numerical examples for smooth and singularity problems are provided to display effectiveness of the methods proposed and to support the analysis made

on the recursive sequence xn axn 1 bxn 2c dxn 1xn 2 in this paper we study the invariant intervals the global attractivity of the equilibrium points and the asymptotic behavior of the solutions of the difference equationx n ax n 1 bx n 2c dx n 1x n 2 n 1 2 where a 0 b c d 0

global asymptotic stability in a rational recursive sequence in this paper we study the global stability of the difference equationx n a bx n 1 cx n 1 2d x n 2 n 1 2 where a b 0 and c d 0 we show that one nonnegative equilibrium point of the equation is a global attractor with a basin that is determined by the parameters and every positive solution of the equation in the basin exponentially converges to the attractor

imposing nonlocal boundary conditions in galerkin type methods based on non interpolatory functions abstract the imposition of inhomogeneous dirichlet essential boundary conditions is a fundamental challenge in the application of galerkin type methods based on non interpolatory functions i e functions which do not possess the kronecker delta property such functions typically are used in various meshfree methods as well as methods based on the isogeometric paradigm the present paper analyses a model problem consisting of the poisson equation subject to non standard boundary conditions namely instead of classical boundary conditions the model problem involves dirichlet and neumann type nonlocal boundary conditions variational formulations with strongly and weakly imposed inhomogeneous dirichlet type nonlocal conditions are derived and compared within an extensive numerical study in the isogeometric framework based on non uniform rational b splines nurbs the attention in the numerical study is paid mainly to the influence of the nonlocal boundary conditions on the properties of the considered discretisation methods

collocation methods for poisson s equation in this paper we provide an analysis on the collocation methods cm which uses a large scale of admissible functions such as orthogonal polynomials trigonometric functions radial basis functions and particular solutions etc the admissible functions can be chosen to be piecewise i e different functions are used in different subdomains the key idea is that the collocation method can be regarded as the least squares method involving integration approximation and optimal convergence rates can be easily achieved based on the traditional analysis of the finite element method the key analysis is to prove the uniformly vh elliptic inequality and some inverse inequalities used this paper explores the interesting fact that for the collocation methods given in this paper the integration rules only affect on the uniformly vh elliptic inequality but not on the solution accuracy the advantage of the cm is to formulate easily the associated algebraic equations which can be solved from the collocation equations directly by the least squares method thus to greatly reduce the condition number of the associated matrix moreover the new effective condition number is proposed to provide a better upper bound of condition number and to show a good stability for real problems solved by the collocation methods note that the boundary approximation method in li z c li combined methods for elliptic equations with singularities interfaces and infinities kluwer academic publishers boston london 1998 is a special case of the cm where the admissible functions satisfy the equations exactly numerical experiments are also carried for poisson s problem to support the analysis made

newton iteration with multiquadrics for the solution of nonlinear pdes abstract newton iteration is a standard tool for the numerical solution of nonlinear partial differential equations we show how globally supported multiquadric radial basis functions can be used for this task one of the insights gained is that the use of coarse meshes during the initial iterations along with a multiquadric parameter which is adjusted with the meshsize increases the efficiency and stability of the resulting algorithm some experiments with nash iteration are also included

a high order composite scheme for the second order elliptic problem with nonlocal boundary and its fast algorithm the elliptic problem with nonlocal boundary condition is widely applied in the field of science and engineering firstly we construct a linear finite element scheme for the nonlocal boundary problem and derive the optimal l 2 error estimate then based on the quadratic finite element and the extrapolation linear finite element methods we present a composite scheme and prove that it is convergent order three furthermore we design an upper triangular preconditioning algorithm for the linear finite element discrete system finally numerical results not only validate that the new algorithm is efficient but also show that the new scheme is convergent order three furthermore order four on uniform grids

a compact multigrid solver for convection diffusion equations we introduce a multigrid algorithm to solve the convection diffusion equations using a nine point compact difference scheme we test the efficiency of the algorithm with various smoothers and intergrid transfer operators the algorithm displays a grid independent convergence rate and produces solutions with high accuracy numerical results are presented to validate the conclusions

high accuracy multigrid solution of the 3d convection diffusion equation we present an explicit fourth order compact finite difference scheme for approximating the three dimensional 3d convection diffusion equation with variable coefficients this 19 point formula is defined on a uniform cubic grid fourier smoothing analysis is performed to show that the smoothing factor of certain relaxation techniques used with the scheme is smaller than 1 we design a parallelization oriented multigrid method for fast solution of the resulting linear system using a four color gauss seidel relaxation technique for robustness and efficiency and a scaled residual injection operator to reduce the cost of multigrid inter grid transfer operator numerical experiments on a 16 processor vector computer are used to test the high accuracy of the discretization scheme as well as the fast convergence and the parallelization or vectorization efficiency of the solution method several test problems are solved and highly accurate solutions of the 3d convection diffusion equations are obtained for small to medium values of the grid reynolds number effects of using different residual projection operators are compared on both vector and serial computers

a transient higher order compact scheme for incompressible viscous flows on geometries beyond rectangular in this paper we propose an implicit high order compact hoc finite difference scheme for solving the two dimensional 2d unsteady navier stokes n s equations on irregular geometries on orthogonal grids our scheme is second order accurate in time and fourth order accurate in space it is used to solve three pertinent fluid flow problems namely the flow decayed by viscosity the lid driven square cavity and the flow in a constricted channel it is seen to efficiently capture both transient and steady state solutions of the n s equations with dirichlet as well as neumann boundary conditions apart from including the good features of hoc schemes our formulation has the added advantage of capturing transient viscous flows involving free and wall bounded shear layers which invariably contain spatial scale variations detailed comparison data produced by the scheme for all the three test cases are provided and compared with analytical as well as established numerical results excellent comparison is obtained in all the cases

on the recursive sequence xn 1 α βxn γxn 1bxn cxn 1 we investigate the global character of solutions of the equation in the title with nonnegative parameters and positive initial conditions we give a detailed description of the semicycles of solutions and prove that the equilibrium of the equation is globally asymptotically stable

high order compact scheme for the steady stream function vorticity equations a higher order compact scheme that is o h4 on the nine point 2 d stencil is formulated for the steady stream function vorticity form of the navier stokes equations the resulting stencil expressions are presented and hence this new scheme can be easily incorporated into existing industrial software we also show that special treatment of the wall boundary conditions is required the method is tested on representative model problems and compares very favourably with other schemes in the literature

on the recursive sequence xn 1 α βxn k 1 γxn 2k 1bxn k 1 cxn 2k 1 abstract our aim in this paper is to investigate the global asymptotic stability of all positive solutions of the higher order nonlinear difference equation x n 1 x n k 1 x n 2 k 1 bx n k 1 cx n 2 k 1 n 0 1 2 where b c and are positive k 1 2 3 and the initial conditions x 2 k 1 x 1 x 0 are positive real numbers we show that the unique positive equilibrium of the equation is globally asymptotically stable and has some basins that depend on certain conditions posed on the coefficients our concentration is on invariant intervals the character of semicycles and the boundedness of the above mentioned equation our final comments are about informative examples

on rational recursive sequences our goal in this chapter is to gain some understanding of the dynamics of rational recursive sequences of the form x n 1 frac a sum limits i 0 k a i x n i b sum limits i 0 k b i x n i n 0 1 3 0 1 where k is a positive integer and the coefficients a b a 0 a k b 0 b k are nonnegative real numbers

multigrid method based on the transformation free hoc scheme on nonuniform grids for 2d convection diffusion problems in this paper a multigrid method based on the high order compact hoc difference scheme on nonuniform grids which has been proposed by kalita et al j c kalita a k dass d c dalal a transformation free hoc scheme for steady convection diffusion on non uniform grids int j numer methods fluids 44 2004 33 53 is proposed to solve the two dimensional 2d convection diffusion equation the hoc scheme is not involved in any grid transformation to map the nonuniform grids to uniform grids consequently the multigrid method is brand new for solving the discrete system arising from the difference equation on nonuniform grids the corresponding multigrid projection and interpolation operators are constructed by the area ratio some boundary layer and local singularity problems are used to demonstrate the superiority of the present method numerical results show that the multigrid method with the hoc scheme on nonuniform grids almost gets as equally efficient convergence rate as on uniform grids and the computed solution on nonuniform grids retains fourth order accuracy while on uniform grids just gets very poor solution for very steep boundary layer or high local singularity problems the present method is also applied to solve the 2d incompressible navier stokes equations using the stream function vorticity formulation and the numerical solutions of the lid driven cavity flow problem are obtained and compared with solutions available in the literature

residual scaling techniques in multigrid ii practical applications this paper focuses on the practical applications of the multigrid residual scaling techniques and is the continuation of a companion paper residual scaling techniques in multigrid i equivalence proof appl math comput 86 283 303 1997 we discuss the computational issues of some residual scaling techniques which have been proven mathematically equivalent a heuristic residual analysis technique based on the geometry of the grid points and the relaxation pattern is introduced to estimate the optimal residual scaling factor for a high order multigrid method we compare the performance of a typical pre optimization pre acceleration technique with a typical post optimization post acceleration technique and show that the pre optimization is preferable in both convergence and efficiency our numerical results support the theoretical conclusions made in the companion paper and demonstrate the full advantage of the pre optimization technique over the post optimization technique

preconditioned iterative methods and finite difference schemes for convection diffusion we conduct experimental study on numerical solution of the two dimensional convection diffusion equation discretized by three finite difference schemes the traditional central difference scheme the standard upwind scheme and the fourth order compact scheme we study the computed accuracy achievable by each scheme the algebraic properties of the coefficient matrices arising from different schemes and the performance of the gauss seidel iterative method the preconditioned gmres iterative method and the multigrid method for solving linear systems arising from these schemes

miscellaneous error bounds for multiquadric and related interpolators we establish several types of a a priori error bounds for multiquadric and related interpolators the results are stated and proven in the general multivariate case these estimates show for example that in many cases such interpolators converge very quickly and can be used in the recovery of band limited functions from discrete data we also include numerical experiments which illustrate the theoretical results

on the recursive sequencexn 1 α xn 1 xn abstract we study the global stability the boundedness character and the periodic nature of the positive solutions of the difference equationxn 1 xn 1 xn where 0 and where the initial conditionsx 1andx0are arbitrary positive real numbers

derivation of high order compact finite difference schemes for non uniform grid using polynomial interpolation in this paper simple polynomial interpolation is used to derive arbitrarily high order compact schemes for the first derivative and tridiagonal compact schemes for the second derivative consisting of three second derivative nodes in the interior and two on the boundary on non uniform grids boundary and near boundary schemes of the same order as the interior are also developed using polynomial interpolation and for a general compact scheme on a non uniform grid it is shown that polynomial interpolation is more efficient than the conventional method of undetermined coefficients for finding coefficients of the scheme the high order non uniform schemes along with boundary closure of up to 14th order thus obtained are shown to be stable on a non uniform grid with appropriate stretching so that more grid points are clustered near the boundary the stability and resolution properties of the high order non uniform grid schemes are studied and the results of three numerical tests on stability and accuracy properties are also presented

combined compact difference scheme for linear second order partial differential equations with mixed derivative a combined compact difference scheme is proposed for linear second order partial differential equations with mixed derivative the scheme is based on a nine point stencil at the interior with sixth order accurate local truncation error fourier analysis is used to analyze the spectral resolution of the proposed scheme numerical tests demonstrate at least sixth order convergence rate with dirichlet boundary condition and fifth order with robin boundary condition a bonus is that high reynolds numbers do not interfere with the order of accuracy

application of the multiquadric method for numerical solution of elliptic partial differential equations abstract we have used the multiquadric mq approximation scheme for the solution of elliptic partial differential equations with dirichlet and or neumann boundary conditions the scheme has the advantage of using the data points in arbitrary locations with an arbitrary ordering two dimensional laplace poisson and biharmonic equations describing the various physical processes have been taken as the test examples the agreement is found to be very good between the computed and exact solutions the method also provides an excellent approximation with a curved boundary

the parameter r2 in multiquadric interpolation the multiquadric mq method is an effective bivariate interpolant to three dimensional data xi yi zi where the xi yi are arbitrarily located in the plane the accuracy of the mq method is dependent on a user defined parameter r2 and most practitioners select r2 based upon the number of data points and the locations of the xi yi in the plane we observe that the optimal value of r2 is a strong function of the zi and that it is essentially independent of both the number and the locationsof the xi yi points contrary to some opinions we observe that the mq method can effectively interpolate to track data that is data sampled densely along tracks in the plane together with several other observations we present an algorithm that generally yields an effective value for r2

on the solution of a nonlocal problem in a rectangular domain we consider the bitsadze samarskii nonlocal boundary value problem for the two dimensional poisson equation the solution of this problem is defined as a solution of the local dirichlet boundary value problem by constructing a special method to find a function as the boundary value on the side of the rectangle where the nonlocal condition was given further the five point approximation of the laplace operator is used for the realization of the proposed method numerical experiments are illustrated in the last section to support the analysis made

radial basis function interpolation in the quantum trajectory method optimization of the multi quadric shape parameter this paper investigates multi quadric radial basis function rbf interpolation and its application in the quantum trajectory method qtm for wave packet propagation in the multi quadric r r2 2 1 2 r is the radial distance from the observation point to the origin of the basis function and is known as the shape parameter due to its affect on the functional form of the basis function the quality of any rbf interpolation scheme is dictated by the choice of this parameter many recent studies have investigated a suitable means for obtaining an optimized time independent parameter the purpose of this study however is to not only to find this optimized shape parameter but also to analyze its time dependence in four different dynamical models the anisotropic harmonic oscillator the downhill ramp the uphill ramp and a harmonic oscillator coupled with a downhill ramp to obtain the optimized shape parameter at each time step an algorithm similar to the leave one out method of cross validation is utilized the results for each of the four models are presented and the feasibility and necessity of employing a shape parameter optimization algorithm for each of the models is discussed

residual scaling techniques in multigrid i equivalence proof this paper addresses the residual scaling techniques coarse grid correction optimization techniques in multigrid methods we surveyed recent developments in this area and prove the equivalence of the overweighted residual technique and the overcorrection technique this leads to the proof of mathematical equivalence of the prescaling and postscaling acceleration techniques two theorems have been proved to unify the concept of the residual scaling techniques these theoretical results clear the way for developing efficient prescaling acceleration techniques for practical applications those practical prescaling acceleration techniques are discussed in a companion paper residual scaling techniques ii practical applications

compact h4 finite difference approximations to operators of navier stokes type a method of obtaining compact finite difference approximations of h4 accuracy to operators of navier stokes type is considered the basic procedure is developed for operators in one space dimension and subsequently applied to problems in more space dimensions and in time four illustrative numerical examples are given which indicate clearly in various cases that excellent accuracy may be obtained using the methods comparisons with previous results and with the results of h2 accurate computations are made

the role of the multiquadric shape parameters in solving elliptic partial differential equations this study examines the generalized multiquadrics mq f j x x x j 2 c j 2 b in the numerical solutions of elliptic two dimensional partial differential equations pdes with dirichlet boundary conditions the exponent b as well as c j 2 can be classified as shape parameters since these affect the shape of the mq basis function we examined variations of b as well as c j 2 where c j 2 can be different over the interior and on the boundary the results show that increasing b has the most important effect on convergence followed next by distinct sets of c j 2 w w c j 2 w additional convergence accelerations were obtained by permitting both c j 2 w w and c j 2 w to oscillate about its mean value with amplitude of approximately 1 2 for odd and even values of the indices our results show high orders of accuracy as the number of data centers increases with some simple heuristics

global attractivity in a recursive sequence in this paper we study the invariant intervals the globally attractivity of the two equilibrium points and the oscillatory behavior of the solutions of the difference equationx n ax n 1 bx n 2c x n 2 n 1 2 where a b c 0