high order difference schemes for unsteady one dimensional diffusion convection problems abstract for unsteady 1d diffusion convection problems this paper develops an extensive analysis of two level three point finite difference schemes of order 2 in time and 4 in space this general class of fds includes several schemes independently proposed by different authors one main objective is the identification of difference schemes yielding satisfactory numerical results for strongly convective problems i e when the cell reynolds number b h 2 is greater than unity the stability and the oscillatory behaviour of the schemes are carefully studied and the analyses are completed by some numerical experiments we outline some key points i the great difficulty to obtain accurate numerical results for large values of ii the possibility of virtually optimum schemes is essentially theoretical and requires in practice careful experiments iii for strongly convective problems some second order explicit schemes are almost as efficient and less costly than implicit fourth order schemes

stability and error estimates of a new high order compact adi method for the unsteady 3d convection diffusion equation abstract in this paper a new high order compact adi method for the unsteady convection diffusion equation in three dimension 3d is considered collecting the truncation error of the finite difference operator by the recursion method we derive a new high order compact finite difference scheme for the unsteady 1d convection diffusion equation firstly then based on the adi method applying a correction technique to reduce the error of the splitting term a high order compact adi method for the unsteady 3d convection diffusion equation is proposed in which we solve a series of 1d problems with strictly diagonal dominant tri diagonal structures instead of the high dimensional ones the scheme is proved to be unconditional stable moreover the regularity and error estimate of the numerical solution are derived finally some numerical examples are performed which confirm the theoretical prediction and show that much better computational accuracy results can be got by applying the new scheme

a new family of 5 5 cc 4oc schemes applicable for unsteady navier stokes equations in this paper a new family of implicit compact finite difference schemes for computation of unsteady convection diffusion equation with variable convection coefficient is proposed the schemes which are fourth order accurate in space and second or lower order accurate in time depending on the choice of weighted time average parameter are then applied to unsteady navier stokes system the proposed schemes where transport variable and its first derivatives are carried as the unknowns combine virtues of compact discretization and pade scheme for spatial derivative these schemes which are based on a five point stencil with constant coefficients named as 5 5 constant coefficient 4th order compact 5 5 cc 4oc give rise to a diagonally dominant system of equations and shows higher accuracy and better phase and amplitude error characteristics than some of the standard methods these schemes are capable of using a grid aspect ratio other than unity and are unconditionally stable they efficiently capture both transient and steady solutions of linear and nonlinear convection diffusion equations with dirichlet as well as neumann boundary conditions subsequently the proposed schemes are applied to problems governed by the incompressible navier stokes equations the results obtained are in excellent agreement with analytical and available numerical results in all cases establishing efficiency and accuracy of the proposed schemes

cross sectional dependence robust block bootstrap panel unit root tests in this paper we consider the issue of unit root testing in cross sectionally dependent panels we consider panels that may be characterized by various forms of cross sectional dependence including but not exclusive to the popular common factor framework we consider block bootstrap versions of the group mean im et al 2003 and the pooled levin et al 2002 unit root coefficient df tests for panel data originally proposed for a setting of no cross sectional dependence beyond a common time effect the tests suited for testing for unit roots in the observed data can be easily implemented as no specification or estimation of the dependence structure is required asymptotic properties of the tests are derived for t going to infinity and n finite asymptotic validity of the bootstrap tests is established in very general settings including the presence of common factors and cointegration across units properties under the alternative hypothesis are also considered in a monte carlo simulation the bootstrap tests are found to have rejection frequencies that are much closer to nominal size than the rejection frequencies for the corresponding asymptotic tests the power properties of the bootstrap tests appear to be similar to those of the asymptotic tests

positive solutions to fourth order singular boundary value problems with integral boundary conditions in abstract spaces abstract under various weaker conditions we establish various results on the existence of positive solutions to fourth order singular boundary value problems with integral boundary conditions in abstract spaces the arguments are based upon a specially constructed cone and the fixed point theory in cone for a strict set contraction operator our results improve and extend that obtained in x zhang m feng w ge existence results for nonlinear boundary value problems with integral boundary conditions in banach spaces nonlinear anal 69 2008 3310 3321 y liu multiple positive solutions to fourth order singular boundary value problems in abstract spaces electron j diff eqns 120 2004 1 13 even in the scalar spaces

solvability of some fourth and higher order singular boundary value problems abstract existence results for a variety of singular fourth order boundary value problems of the form yiv f t y y y are given here our nonlinear term f may be singular at t 0 t 1 y 0 and or y 0 for example some singularities of the type y a and y b are included also we discuss and treat the extension of these results to nth order boundary value problems

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

the method of lower and upper solutions for fourth order two point boundary value problems abstract in this paper we develop the monotone method in the presence of lower and upper solutions for the problem u iv f x u x u x u 0 u 1 u 0 u 1 0 where f 0 1 r r r is continuous

multiple positive solutions for singular nth order nonlocal boundary value problems in banach spaces in this paper we consider a class of singular nth order nonlocal boundary value problems in banach spaces the existence of multiple positive solutions for the problem is obtained by using the fixed point index theory of strict set contraction operators to demonstrate the applications of our results two examples are also given in the paper

determining the number of factors in approximate factor models in this paper we develop some econometric theory for factor models of large dimensions the focus is the determination of the number of factors r which is an unresolved issue in the rapidly growing literature on multifactor models we first establish the convergence rate for the factor estimates that will allow for consistent estimation of r we then propose some panel criteria and show that the number of factors can be consistently estimated using the criteria the theory is developed under the framework of large cross sections n and large time dimensions t no restriction is imposed on the relation between n and t simulations show that the proposed criteria have good finite sample properties in many configurations of the panel data encountered in practice

nonparametric rank tests for non stationary panels we develop a set of nonparametric rank tests for non stationary panels based on multivariate variance ratios which use untruncated kernels as such the tests do not require the choice of tuning parameters associated with bandwidth or lag length and also do not require choices with respect to numbers of common factors the tests allow for unrestricted cross sectional dependence and dynamic heterogeneity among the units of the panel provided simply that a joint functional central limit theorem holds for the panel of differenced series we provide a discussion of the relationships between our setting and the settings for which first and second generation panel unit root tests are designed in monte carlo simulations we illustrate the small sample performance of our tests when they are used as panel unit root tests under the more restrictive dgps for which panel unit root tests are typically designed and for more general dgps we also compare the small sample performance of our nonparametric tests to parametric rank tests finally we provide an empirical illustration by testing for income convergence among countries

a perturbational h 4 exponential finite difference scheme for the convective diffusion equation a perturbational h4 compact exponential finite difference scheme with diagonally dominant coefficient matrix and upwind effect is developed for the convective diffusion equation perturbations of second order are exerted on the convective coefficients and source term of an h2 exponential finite difference scheme proposed in this paper based on a transformation to eliminate the upwind effect of the convective diffusion equation four numerical examples including one to three dimensional model equations of fluid flow and a problem of natural convective heat transfer are given to illustrate the excellent behavior of the present exponential schemes the h4 accuracy of the perturbational scheme is verified using double precision arithmetic

unit root tests in arma models with data dependent methods for the selection of the truncation lag abstract we analyze the choice of the truncation lag in the context of the said dickey test for the presence of a unit root in a general autoregressive moving average model it is shown that a deterministic relationship between the truncation lag and the sample size is dominated by data dependent rules that take sample information into account in particular we study data dependent rules that are not constrained to satisfy the lower bound condition imposed by said dickey akaike s information criterion falls into this category the analytical properties of the truncation lag selected according to a class of information criteria are compared to those based on sequential testing for the significance of coefficients on additional lags the asymptotic properties of the unit root test under various methods for selecting the truncation lag are analyzed and simulations are used to show their distinctive behavior in finite samples our results favor methods based on sequential tests over those based on informat

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

trend function hypothesis testing in the presence of serial correlation in this paper test statistics are proposed that can be used to test hypotheses about the parameters of the deterministic trend function of a univariate time series the tests are valid in the presence of general forms of serial correlation in the errors and can be used without having to estimate the serial correlation parameters either parametrically or nonparametrically the tests are valid for i 0 and i 1 errors trend functions that are permitted include general linear polynomial trend functions that may have breaks at either known or unknown locations asymptotic distributions are derived and consistency of the tests is established the general results are applied to a model with a simple linear trend a local asymptotic analysis is used to compute asymptotic size and power of the tests for this example size is well controlled and is relatively unaffected by the variance of the initial condition asymptotic power curves are computed for the simple linear trend model and are compared to existing tests it is shown that the new tests have nontrivial asymptotic power a simulation study shows that the asymptotic approximations are adequate for sample sizes typically used in economics the tests are used to construct confidence intervals for average gnp growth rates for eight industrialized countries using post war data

symmetric positive solutions of boundary value problems with integral boundary conditions under conditions weaker than those used by ma we establish various results on the existence and nonexistence of symmetric positive solutions to fourth order boundary value problems with integral boundary conditions the arguments are based upon a specially constructed cone and the fixed point theory in a cone our results improve and extend that obtained in h ma symmetric positive solutions for nonlocal boundary value problems of fourth order nonlinear anal 68 2008 645 651 we illustrate our results by one example which cannot be handled using the existing results

uniform high order difference schemes for a singularly perturbed two point boundary value problem on construit une famille de schemas aux differences finies uniformement precis pour le probleme du type eu a x u b x u f x on donne une analyse complete de l erreur de discretisation basee sur les resultats de stabilite on presente des essais numeriques

simple robust testing of regression hypotheses in this paper we consider the problem of hypothesis testing in models with errors that have serial correlation or heteroskedasticity of unknown form this situation is often encountered in regression models applied to economic time series data it is a classic textbook result that while ordinary least squares ols estimates of regression parameters remain consistent and asymptotically normal when errors are heteroskedastic or autocorrelated provided usual regularity conditions hold and no lagged dependent variables are in the model standard tests are no longer valid if the true form of serial correlation heteroskedasticity were known then generalized least squares gls provides efficient estimates and standard inference can be conducted on the gls transformed model but in practice the form of serial correlation heteroskedasticity is often unknown and this has led to the development of techniques that permit valid asymptotic inference without having to specify a model of the serial correlation or heteroskedasticity the most common approach is to estimate the variance covariance matrix of the ols estimates nonparametrically using spectral methods heteroskedasticity and autocorrelation consistent hac estimators and construct standard tests using the asymptotic normality of the ols estimates hac estimators have been extensively analyzed in the econometrics literature and important contributions are given by andrews 1991 andrews and monahan 1992 gallant 1987 hansen 1992 newey and west 1987 robinson 1991 1998 and white 1984 among others the benefit of hac estimator tests is asymptotically valid inference that is robust to general forms of serial correlation heteroskedasticity in the errors we propose an alternative method of constructing robust test statistics we apply a nonsingular data dependent stochastic transformation to the ols estimates the asymptotic distribution of the transformed estimates does not depend on nuisance parameters then test statistics that are asymptotically invariant to nuisance parameters asymptotic pivotal statistics are constructed the resulting test statistics have nonstandard asymptotic distributions that only depend on the number of restrictions being tested and critical values are easy to simulate using standard techniques the main advantage of our approach compared to standard approaches is that estimates of the variance covariance matrix are not explicitly required to construct the tests this is potentially important for two reasons first with the exception of the estimator proposed by robinson 1998 2 consistent nonparametric estimates of variance covariance matrices in models with serial

linear regression limit theory for nonstationary panel data this paper develops a regression limit theory for nonstationary panel data with large numbers of cross section n and time series t observations the limit theory allows for both sequential limits wherein t followed by n and joint limits where t n simultaneously and the relationship between these multidimensional limits is explored the panel structures considered allow for no time series cointegration heterogeneous cointegration homogeneous cointegration and near homogeneous cointegration the paper explores the existence of long run average relations between integrated panel vectors when there is no individual time series cointegration and when there is heterogeneous cointegration these relations are parameterized in terms of the matrix regression coefficient of the long run average covariance matrix in the case of homogeneous and near homogeneous cointegrating panels a panel fully modified regression estimator is developed and studied the limit theory enables us to test hypotheses about the long run average parameters both within and between subgroups of the full population

multigrid method and fourth order compact scheme for 2d poisson equation with unequal mesh size discretization abstract a fourth order compact difference scheme with unequal mesh sizes in different coordinate directions is employed to discretize a two dimensional poisson equation in a rectangular domain multigrid methods using a partial semicoarsening strategy and line gauss seidel relaxation are designed to solve the resulting sparse linear systems numerical experiments are conducted to test the accuracy of the fourth order compact difference scheme and to compare it with the standard second order difference scheme convergence behavior of the partial semicoarsening and line gauss seidel relaxation multigrid methods is examined experimentally

extreme canonical correlations and high dimensional cointegration analysis abstract we prove that the extreme squared sample canonical correlations between a random walk and its own innovations almost surely converge to the upper and lower boundaries of the support of the wachter distribution when the sample size and the dimensionality go to infinity proportionally this result is used to derive previously unknown analytic expressions for the bartlett type correction coefficients for johansen s trace and maximum eigenvalue tests in a high dimensional var 1 an analysis of cointegration among a large number of log exchange rates illustrates the usefulness of our theoretical results

multiple solutions of two point boundary value problems of ordinary differential equations in banach spaces soit p un cone normal de l espace de banach reel e on note par n la constante normale de p on considere le probleme aux deux points limites x f t x t i x 0 x 1 ou i 0 1 et f une application continue de i p dans p on etudie l existence de solutions positives multiples

testing for unit roots in heterogeneous panels abstract this paper proposes unit root tests for dynamic heterogeneous panels based on the mean of individual unit root statistics in particular it proposes a standardized t bar test statistic based on the augmented dickey fuller statistics averaged across the groups under a general setting this statistic is shown to converge in probability to a standard normal variate sequentially with t the time series dimension followed by n the cross sectional dimension a diagonal convergence result with t and n while n t k k being a finite non negative constant is also conjectured in the special case where errors in individual dickey fuller df regressions are serially uncorrelated a modified version of the standardized t bar statistic is shown to be distributed as standard normal as n for a fixed t so long as t 5 in the case of df regressions with intercepts and t 6 in the case of df regressions with intercepts and linear time trends an exact fixed n and t test is also developed using the simple average of the df statistics monte carlo results show that if a large enough lag order is selected for the underlying adf regressions then the small sample performances of the t bar test is reasonably satisfactory and generally better than the test proposed by levin and lin unpublished manuscript university of california san diego 1993

testing for a unit root in a random coefficient panel data model this paper proposes new unit root tests in the context of a random autoregressive coefficient panel data model in which the null of a unit root corresponds to the joint restriction that the autoregressive coefficient has unit mean and zero variance the asymptotic distributions of the test statistics are derived and simulation results are provided to suggest that they perform very well in small samples

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

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

positive solution for m point boundary value problems of fourth order in this paper we consider the existence of positive solutions to the fourth order boundary value problem u u u f t u 0 t 1 u 0 i 1m 2aiu i u 1 i 1m 2biu i u 0 i 1m 2aiu i u 1 i 1m 2biu i where r i 0 1 ai bi 0 for i 1 2 m 2 are given constants satisfying some suitable conditions the proofs are based on the fixed point index theorem in cones

heteroskedasticity autocorrelation robust standard errors using the bartlett kernel without truncation in this paper we analyze heteroskedasticity autocorrelation hac robust tests constructed using the bartlett kernel without truncation we show that while such an hac estimator is not consistent asymptotically valid testing is still possible we show that tests using the bartlett kernel without truncation are exactly equivalent to recent hac robust tests proposed by kiefer vogelsang and bunzel 2000 econometrica 68 pp 695 714

bootstrap unit root tests in panels with cross sectional dependency we apply bootstrap methodology to unit root tests for dependent panels with n cross sectional units and t time series observations more specifically we let each panel be driven by a general linear process which may be different across cross sectional units and approximate it by a finite order autoregressive integrated process of order increasing with t as we allow the dependency among the innovations generating the individual series we construct our unit root tests from the estimation of the system of the entire n cross sectional units the limit distributions of the tests are derived by passing t to infinity with n fixed we then apply bootstrap method to the approximated autoregressions to obtain critical values for the panel unit root tests and establish the asymptotic validity of such bootstrap panel unit root tests under general conditions the proposed bootstrap tests are indeed quite general covering a wide class of panel models they in particular allow for very general dynamic structures which may vary across individual units and more importantly for the presence of arbitrary cross sectional dependency the finite sample performance of the bootstrap tests is examined via simulations and compared to that of commonly used panel unit root tests we find that our bootstrap tests perform relatively well especially when n is small

existence results for nonlinear boundary value problems with integral boundary conditions in banach spaces abstract this paper investigates the multiplicity of positive solutions for a class of nonlinear boundary value problems of fourth order differential equations with integral boundary conditions in ordered banach spaces the arguments are based upon a specially constructed cone and the fixed point index theory in cone for strict set contraction operator the nonexistence of positive solution is also studied meanwhile an example is worked out to demonstrate the main results

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