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

black box multigrid for nonsymmetric problems one major problem with the multigrid method has been that each new grid configuration has required a major programming effort to develop a code that specifically handles that grid configuration such a penalty is not required for methods like sor iccg etc in these methods one need only specify the matrix problem no matter what the grid configuration in previous work this defect was overcome and a code written in which one needed only to specify the logically rectangular positive definite matrix problem the code did everything else necessary to set up the auxiliary coarser problems to achieve a multigrid solution in this paper the previous work is extended to a single nonsymmetric equation

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

the frequency decomposition multi grid method a new variant of the multi grid algorithms is presented it uses multiple coarse grid corrections with particularly associated prolongations and restrictions in this paper the robustness with respect to anisotropic problems is considered

an excmg accelerated multiscale multigrid computation for 3d poisson equation abstract multiscale multigrid msmg method is an effective computational framework for efficiently computing high accuracy solutions for elliptic partial differential equations in the current msmg method compared to the cpu cost on computing sixth order solutions by applying extrapolation and other techniques on two fourth order solutions from different scales grids much more cpu time is spent on computing fourth order solutions themselves on coarse and fine grids particularly for high dimensional problems here we propose to embed extrapolation cascadic multigrid excmg method into the msmg framework to accelerate the whole process numerical results on 3d poisson equations show that the new excmg msmg method is more efficient than the existing msmg method and the excmg method for sixth order solution computation

an algorithm using the finite volume element method and its splitting extrapolation this paper is to present a new efficient algorithm by using the finite volume element method and its splitting extrapolation this method combines the local conservation property of the finite volume element method and the advantages of splitting extrapolation such as a high order of accuracy a high degree of parallelism less computational complexity and more flexibility than a richardson extrapolation because the splitting extrapolation formulas only require us to solve a set of smaller discrete subproblems on different coarser grids in parallel instead of on the globally fine grid a large scale multidimensional problem is turned into a set of smaller discrete subproblems additionally this method is efficient for solving interface problems if we regard the interfaces of the problems as the interfaces of the initial domain decomposition

on the net reproduction rate of continuous structured populations with distributed states at birth we consider a nonlinear structured population model with a distributed recruitment term the question of the existence of non trivial steady states can be treated at least in three different ways one approach is to study spectral properties of a parametrised family of unbounded operators the alternative approach which we develop here is based on the reformulation of the partial differential equation as an integral equation in this context we introduce a density dependent net reproduction rate and discuss its relationship to a biologically meaningful quantity finally we discuss a third approach which is based on a finite rank approximation of the recruitment operator

higher order adi method with completed richardson extrapolation for solving unsteady convection diffusion equations a higher order alternating direction implicit adi method with completed richardson extrapolation is proposed for solving unsteady two dimensional convection diffusion equations the method has sixth order accuracy in space and fourth order accuracy in time completed richardson extrapolation is used to improve the accuracy of the solution obtained from the high order adi method in spatial and temporal domains simultaneously stability analysis is given to show the influence of richardson extrapolation on the unconditionally stable adi method numerical experiments are conducted to test its high accuracy and to compare it with karaa zhang s high order adi method

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 new method to deduce high order compact difference schemes for two dimensional poisson equation abstract this paper introduces a novel method for deducing high order compact difference schemes for the two dimensional 2d poisson equation like finite volume method a dual partition is introduced combining simpson integral formula and parabolic interpolation a family of fourth order and sixth order compact difference schemes are obtained based on three different types of dual partitions moreover several new fourth order compact schemes are gained and numerical experiments are shown two of them are much better than almost any other fourth order schemes which have been presented in others work the outline for the nonlinear poisson equation is also given numerical experiments are presented to verify the feasibility of this new method and the high accuracy of these fourth order and sixth order compact difference schemes

petrov galerkin methods for linear volterra integro differential equations in this paper we study a class of petrov galerkin solutions that have global optimal convergence rates for linear volterra integro differential equations these solutions also possess certain local and global superconvergence asymptotic expansions of the errors in these solutions are established which can be used to form higher order approximations by richardson extrapolation and defect corrections several postprocessing techniques are introduced to enhance these solutions as by products these higher order numerical approximations can be used to generate a posteriori error estimators representative numerical results are also provided

one sided finite difference approximations suitable for use with richardson extrapolation new expressions for one sided finite difference approximations are proposed in these approximations the odd order error terms are eliminated while the even order terms are left to be taken care of by richardson extrapolation the effective local truncation error is shown to be less than for higher order one sided finite difference approximations but the solutions for a test problem are shown to have comparable accuracy for both approximations

stable high order methods for solving elliptic equations with large first order terms abstract stable high order methods are developed for solving elliptic partial differential equations with large first order terms in particular second order and fourth order methods are developed by stabilizing the central difference method the methods can solve second order linear elliptic partial differential equations and are found to be stable and accurate in all tested examples results are presented for the two dimensional convection diffusion equation involving problems with and without boundary layers the methods converged for all values of parameters attempted and the results compared favorably with other methods the methods developed in this paper are accurate stable easy to use and applicable to other problems

steady states of a selection mutation model for an age structured population abstract we introduce a selection mutation model for an age and age at maturity structured population the age at maturity is considered an evolutionary trait on which the vital rates depend subject to selection by competition and to mutation modelled by a birth term which takes the form of a convolution like operator we prove well posedness of the initial value problem and the existence of non trivial steady states under suitable hypotheses which yield uniqueness when the interaction variable is one dimensional

an algorithm for multistage dynamic networks with random arc capacities with an application to dynamic fleet management we consider the class of multistage dynamic networks with random arc capacities a framework that is well suited to model dynamic fleet management problems we propose a successive convex approximation approach that produces an approximation to the expected recourse function which captures the future effects of current decisions under uncertainty this method decomposes the network in each stage into tree subproblems whose expected recourse functions are easy to obtain we also compare this method with two alternative methods on a set of dynamic fleet management problems the numerical results show that this method is superior to the two alternative methods

high order accurate solution of the incompressible navier stokes equations high order methods are of great interest in the study of turbulent flows in complex geometries by means of direct simulation with this goal in mind the incompressible navier stokes equations are discretized in space by a compact fourth order finite difference method on a staggered grid the equations are integrated in time by a second order semi implicit method stable boundary conditions are implemented and the grid is allowed to be curvilinear in two space dimensions the method is extended to three dimensions by a fourier expansion in every time step a system of linear equations is solved for the velocity and the pressure by an outer and an inner iteration with preconditioning the convergence properties of the iterative method are analyzed the order of accuracy of the method is demonstrated in numerical experiments the method is used to compute the flow in a channel the driven cavity and a constricted channel

a tighter variant of jensen s lower bound for stochastic programs and separable approximations to recourse functions abstract in this paper we propose a new method to compute lower bounds on the optimal objective value of a stochastic program and show how this method can be used to construct separable approximations to the recourse functions we show that our method yields tighter lower bounds than jensen s lower bound and it requires a reasonable amount of computational effort even for large problems the fundamental idea behind our method is to relax certain constraints by associating dual multipliers with them this yields a smaller stochastic program that is easier to solve we particularly focus on the special case where we relax all but one of the constraints in this case the recourse functions of the smaller stochastic program are one dimensional functions we use these one dimensional recourse functions to construct separable approximations to the original recourse functions computational experiments indicate that our lower bounds can significantly improve jensen s lower bound and our recourse function approximations can provide good solutions

black box multigrid for systems this paper extends to systems the black box multigrid method in which one provides a logically rectangular matrix problem and in which the coarser problems necessary to achieve a multigrid solution are automatically derived the extension is straightforward if the components of the vector solution of the system are defined at the same grid locations otherwise as in the case of staggered grids the situation is less satisfactory

basic smoothing procedures for the multigrid treatment of elliptic 3d operators from 2d multigrid it is a fundamental insight that pointwise relaxation combined with standard coarsening gives no reasonable smoothing effect if the given elliptic operator shows appreciably anisotropic behavior corresponding questions and practical consequences are discussed in this paper for the 3d case in the general anisotropic 3d case even line relaxation is not sufficient if standard coarsening is maintained instead plane relaxation is necessary in certain cases if plane relaxation is applied correctly and performed by use of appropriate 2d multigrid the resulting 3d multigrid method has an asymptotic complexity of o n where n number of 3d grid points and is highly efficient even for moderate values of n in contrast to the common opinion plane relaxation turns out to be a simple and general smoothing concept for standard elliptic 3d problems

a non iterative algorithm based on richardson s extrapolation application to groundwater flow modelling in this report a promising third order algorithm based on richardson s extrapolation and crank nicholson s method is studied the algorithm having a wide scope for applications was initially intended to be used for time integration on non linear advection diffusion problems then stability low oscillations and low additional diffusion become important numerical properties it is shown that this method using three single steps for extrapolation is very close to be unconditionally stable under the previous requirements it behaves essentially as well as crank nicholson s scheme but with a better performance at least in relation to spurious oscillations and perhaps not so good for dispersion properties in addition the method is third order accurate and provides useful information related to time integration error to be used for time step control these features indicate that the algorithm is a good candidate to efficiently solve problems with significant variations in time additionally its non iterative version saves important amounts of cpu time specially for applied 3d modelling to illustrate the good performance of the algorithm we include several tests comparing it with crank nicholson s scheme and a 2d application to variably saturated groundwater flow modelling

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

skew hermitian based iterations for nine point approximations of convection diffusion problems we study skew hermitian based splitting methods for the iterative solution of nonsymmetric linear systems arising from high order compact hoc approximations of two dimensional convection diffusion problems such discretisations lead to nine point system matrices with block tridiagonal structures the nonsymmetric discretisation matrix for a constant coefficient grid aligned flow problem is shown to be positive definite and thus hermitian and skew hermitian splitting hss methods can be considered for the solution of the corresponding linear system for the solution of this linear system using the hss iteration we derive an analytical expression for the optimal value for the upper bound of the contraction factor comparison of the hss iteration with triangular and skew hermitian splitting tss methods and their corresponding block variants btss are carried out for various test problems cpu timings for the different splitting methods using experimentally determined optimal acceleration parameters are also given our results show that the combination of high order compact discretisations and skew hermitian based iterative methods for solving the corresponding linear systems provide efficient procedures for approximating the solutions of two dimensional convection diffusion equations

some optimal error estimates for piecewise linear finite element approximations it is shown that the ritz projection onto spaces of piecewise linear finite elements is bounded in the sobolev space wl for 2 p oc this implies that for functions in wf n w2 the error in approximation behaves like 0 h in wp for 2 p oo and like 0 h2 in lp for 2 p oo in all these cases the additional logarithmic factor previously included in error estimates for linear finite elements does not occur

high accuracy solutions of incompressible navier stokes equations abstract in recent years we have developed high accuracy finite difference approximations for partial differential equations of elliptic type with particular emphasis on the convection diffusion equation these approximations are of compact type have a local truncation error of fourth order and allow the use of standard iterative schemes to solve the resulting systems of algebraic equations in this paper we extend these high accuracy approximations to the solution of navier stokes equations solutions are obtained for the model problem of driven cavity and are compared with solutions obtained using other approximations and those obtained by other authors it is discovered that the high order approximations do indeed produce high accuracy solutions and have a potential for use in solving important problems of viscous fluid flows

comparison of second and fourth order discretizations for multigrid poisson solvers we combine a compact high order difference approximation with multigrid v cycle algorithm to solve the two dimensional poisson equation with dirichlet boundary conditions this scheme along with several different orderings of grid space and projection operators is compared with the five point formula to show the dramatic improvement in computed accuracy on serial and vector machines

shape a stochastic hybrid approximation procedure for two stage stochastic programs we consider the problem of approximating the expected recourse function for two stage stochastic programs our problem is motivated by applications that have special structure such as an underlying network that allows reasonable approximations to the expected recourse function to be developed in this paper we show how these approximations can be improved by combining them with sample gradient information from the true recourse function for the case of strictly convex nonlinear approximations we prove convergence for this hybrid approximation the method is attractive for practical reasons because it retains the structure of the approximation

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

fast and robust sixth order multigrid computation for the three dimensional convection diffusion equation we present a sixth order explicit compact finite difference scheme to solve the three dimensional 3d convection diffusion equation we first use a multiscale multigrid method to solve the linear systems arising from a 19 point fourth order discretization scheme to compute the fourth order solutions on both a coarse grid and a fine grid then an operator based interpolation scheme combined with an extrapolation technique is used to approximate the sixth order accurate solution on the fine grid since the multigrid method using a standard point relaxation smoother may fail to achieve the optimal grid independent convergence rate for solving convection diffusion equations with a high reynolds number we implement the plane relaxation smoother in the multigrid solver to achieve better grid independency supporting numerical results are presented to demonstrate the efficiency and accuracy of the sixth order compact soc scheme compared with the previously published fourth order compact foc scheme

spectral bound and reproduction number for infinite dimensional population structure and time heterogeneity spectral bounds of quasi positive matrices are crucial mathematical threshold parameters in population models that are formulated as systems of ordinary differential equations the sign of the spectral bound of the variational matrix at 0 decides whether at low density the population becomes extinct or grows another important threshold parameter is the reproduction number mathcal r which is the spectral radius of a related positive matrix as is well known the spectral bound and mathcal r 1 have the same sign provided that the matrices have a particular form the relation between spectral bound and reproduction number extends to models with infinite dimensional state space and then holds between the spectral bound of a resolvent positive closed linear operator and the spectral radius of a positive bounded linear operator we also extend an analogous relation between the spectral radii of two positive linear operators which is relevant for discrete time models we illustrate the general theory

single commodity network design with random edge capacities this paper examines the single commodity network design problem with stochastic edge capacities we characterize the structures of the optimal designs and compare with the deterministic counterparts we do this partly to understand what constitutes robust network designs but also to construct a heuristic for the stochastic problem leading to optimality gaps of about 10 in our view that is a rather good result for problems that otherwise cannot be solved at all

a gmres based plane smoother in multigrid to solve 3d anisotropic fluid flow problems for a discretization of the 3d steady incompressible navier stokes equations a solution method is presented for solving flow problems on stretched grids the discretization is a vertex centered finite volume discretization with a flux splitting approach for the convective terms second order accuracy is obtained with the well known defect correction technique b koren j comput phys 87 25 1990 the solution method used is multigrid for which a plane smoother is presented for obtaining good convergence in flow domains with severely stretched grids a matrix is set up in a plane which is solved iteratively with a preconditioned gmres method here a stop criterion for gmres is tested which reduces the number of inner iterations compared to an exact plane solver without affecting the multigrid convergence rates the performance of the solution method is shown for a poisson model problem and for 3d incompressible channel flow examples

stability and regularity results for a size structured population model in the present paper a nonlinear size structured population dynamical model with size and density dependent vital rate functions is considered the linearization about stationary solutions is analyzed by semigroup and spectral methods in particular the spectrally determined growth property of the linearized semigroup is derived from its long term regularity these analytical results make it possible to derive linear stability and instability results under biologically meaningful conditions on the vital rates the principal stability criteria are given in terms of a modified net reproduction rate

a comparison of uniformly convergent difference schemes for two dimensional convection diffusion problems galerkin and petrov galerkin finite element methods are used to obtain new finite difference schemes for the solution of linear two dimensional convection diffusion problems numerical estimates are made of the rates of convergence of these schemes uniformly with respect to the perturbation parameter and these uniform rates are shown to compare favourably with those of established methods

learning algorithms for separable approximations of discrete stochastic optimization problems we propose the use of sequences of separable piecewise linear approximations for solving nondifferentiable stochastic optimization problems the approximations are constructed adaptively using a combination of stochastic subgradient information and possibly sample information on the objective function itself we prove the convergence of several versions of such methods when the objective function is separable and has integer break points and we illustrate their behavior on numerical examples we then demonstrate the performance on nonseparable problems that arise in the context of two stage stochastic programming problems and demonstrate that these techniques provide near optimal solutions with a very fast rate of convergence compared with other solution techniques

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

an arc exchange decomposition method for multistage dynamic networks with random arc capacities abstract multistage dynamic networks with random arc capacities mdnrac have been successfully used for modeling various resource allocation problems in the transportation area however solving these problems is generally computationally intensive and there is still a need to develop more efficient solution approaches in this paper we propose a new heuristic approach that solves the mdnrac problem by decomposing the network at each stage into a series of subproblems with tree structures each subproblem can be solved efficiently the main advantage is that this approach provides an efficient computational device to handle the large scale problem instances with fairly good solution quality we show that the objective value obtained from this decomposition approach is an upper bound for that of the mdnrac problem numerical results demonstrate that our proposed approach works very well

convergence of a multigrid method for elliptic equations with highly oscillatory coefficients standard multigrid methods are not so effective for equations with highly oscillatory coefficients new coarse grid operators based on homogenized operators are introduced to restore the fast convergence rate of multigrid methods finite difference approximations are used for the discretization of the equations convergence analysis is based on the homogenization theory proofs are given for a two level multigrid method with the homogenized coarse grid operator for two classes of two dimensional elliptic equations with dirichlet boundary conditions

a three point combined compact difference scheme a new three point combined compact difference ccd scheme is developed for numerical models the major features of the ccd scheme are three point implicit sixth order accuracy and inclusion of boundary values due to its combination of the first and second derivatives the ccd scheme becomes more compact and more accurate than normal compact difference schemes the efficient twin tridiagonal for calculating derivatives and triple tridiagonal for solving partial difference equation with the ccd scheme methods are also presented besides the ccd scheme has sixth order accuracy at periodic boundaries and fifth order accuracy at nonperiodic boundaries the possibility of extending to a three point eighth order scheme is also included

asymptotic error expansion and richardson extrapolation for linear fine elements the elliptic ritz projection with linear finite elements is shown to admit asymptotic error expansions on certain uniform meshes this justifies the application of richardson extrapolation for increasing the accuracy

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

incremental unknowns for solving partial differential equations incremental unknowns have been proposed in t as a method to approximate fractal attractors by using finite difference approximations of evolution equations in the case of linear elliptic problems the utilization of incremental unknown methods provides a new way for solving such problems using several levels of discretization the method is similar but different from the classical multigrid method in this article we describe the application of incremental unknowns for solving laplace equations in dimensions one and two we provide theoretical results concerning two level approximations and we report on numerical tests done with multi level approximations

fast and high accuracy multiscale multigrid method with multiple coarse grid updating strategy for the 3d convection diffusion equation an improved multiscale multigrid method with multiple coarse grid updating strategy for a three dimensional convection diffusion equation is presented the novelty of the proposed method lies in a fine grid updating strategy arising from the idea of multiple coarse grid computation the new fine grid updating strategy is able to replace the iterative refinement procedure in the existing multiscale multigrid method wang and zhang 2010 16 to obtain high order solutions since the proposed method needs a fourth order compact scheme with unequal meshsize grids a 19 point fourth order compact difference scheme with unequal meshsize in different coordinate directions is also developed for the three dimensional convection diffusion equation numerical results are given to compare the computed accuracy and the computational efficiency of the multiscale multigrid method with the multiple coarse grid updating strategy against the multiscale multigrid method with the iterative refinement procedure

nonlinear singular perturbation problems and one sided difference schemes we consider the boundary value problem varepsilon y a y y b x y f x 1 leqq x leqq 1 for y 1 and y 1 given and b x 0 equiv 0 b y x y geqq 0 we construct a simple difference scheme approximating varepsilon u xx a u u x b x u f x u t for t geqq 0 1 leqq x leqq 1 with the same boundary conditions and an arbitrary initial guess as the numerical artificial time approaches infinity we prove convergence to a unique steady state solution as the mesh size delta x and the parameter varepsilon approach zero in any order we prove convergence to the solution of the corresponding o d e we use one sided differences to approximate the transport term the method applies equally well to multidimensional analogues we present numerical results verifying the theory

a finite element splitting extrapolation for second order hyperbolic equations splitting extrapolation is an efficient technique for solving large scale scientific and engineering problems in parallel this article discusses a finite element splitting extrapolation for second order hyperbolic equations with time dependent coefficients this method possesses a higher degree of parallelism less computational complexity and more flexibility than richardson extrapolation while achieving the same accuracy by means of domain decomposition and isoparametric mapping some grid parameters are chosen according to the problem the multiparameter asymptotic expansion of the d quadratic finite element error is also established the splitting extrapolation formulas are developed from this expansion an approximation with higher accuracy on a globally fine grid can be computed by solving a set of smaller discrete subproblems on different coarser grids in parallel some a posteriori error estimates are also provided numerical examples show that this method is efficient for solving discontinuous problems and nonlinear hyperbolic equations

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

a high order difference method for the steady state navier stokes equations a stable high order difference method is developed for solving the heated cavity and the driven cavity problems governed by the steady state navier stokes equations the numerical results compared favorably with other methods coordinate transformations were also used to reflect the boundary layer phenonmena without requiring many mesh points the methods used in this paper are accurate stable and applicable to other flow problems governed by the navier stokes equations

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

stability analysis of the gurtin maccamy model in this paper we propose a numerical scheme to investigate the stability of steady states of the nonlinear gurtin maccamy system which is a basic model in population dynamics in fact the analysis of stability is usually performed by the study of transcendental characteristic equations that are too difficult to approach by analytical methods the method is based on the discretization of the infinitesimal generator associated to the semigroup of the solution operator by using pseudospectral differencing techniques the method computes the rightmost characteristic roots and it is shown to converge with spectral accuracy behavior

asymptotic expansions and richardson extrapolation of approximate solutions for second order elliptic problems on rectangular domains by mixed finite element methods in this paper asymptotic error expansions for mixed finite element approximations of general second order elliptic problems are derived under rectangular meshes and the richardson extrapolation is applied to improve the accuracy of the approximations by two different schemes with the help of an interpolation postprocessing technique the results of this paper provide new asymptotic expansions and new approximate solutions which are one order and a half order higher in accuracy than those obtained in j wang math comp 56 1991 pp 477 503 and h chen r e ewing and r lazarov asymptotic error expansion for the lowest order raviart thomas rectangular mixed finite elements technical report isc 97 01 1997 respectively as a by product we illustrate that all the approximations of higher accuracy can be used to form a class of a posteriori error estimators for the mixed finite element method

a modified crank nicolson scheme with incremental unknowns for convection dominated diffusion equations a modified crank nicolson scheme based on one sided difference approximations is proposed for solving time dependent convection dominated diffusion equations in two dimensional space the modified scheme is consistent and unconditionally stable a priori l 2 error estimate for the fully discrete modified scheme is derived with the use of the incremental unknowns preconditioner at each time step a comparison among several classical numerical schemes has been made and numerical results confirm stability and efficiency of the modified crank nicolson scheme

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

a splitting extrapolation for solving nonlinear elliptic equations with d quadratic finite elements nonlinear elliptic partial differential equations are important to many large scale engineering and science problems for this kind of equations this article discusses a splitting extrapolation which possesses a high order of accuracy a high degree of parallelism less computational complexity and more flexibility than richardson extrapolation according to the problems some domain decompositions are constructed and some independent mesh parameters are designed multi parameter asymptotic expansions are proved for the errors of approximations based on the expansions splitting extrapolation formulas are developed to compute approximations with high order of accuracy on a globally fine grid because these formulas only require us to solve a set of smaller discrete subproblems on different coarser grids in parallel instead of on the globally fine grid a large scale multidimensional problem is turned into a set of smaller discrete subproblems additionally this method is efficient for solving interface problems

generation of finite difference formulas on arbitrarily spaced grids on etablit des recurrences simples pour le calcul des poids dans les formules aux differences finies compactes pour les derivees de tous ordres avec une precision d ordre arbitraire sur des grilles a une dimension d espacement arbitraire

spatial filter averaging approach of probabilistic method to linear second order partial differential equations of the parabolic type a backward in time probabilistic method with spatial filter averaging is presented to solve linear second order partial differential equations of the parabolic type an advantage of this methodology is that while forward methods are subject to region with loss of density of particles and hence loss of spatial resolution of the solution the solution given by backward methods is given on any desired grid however traditional backward time probabilistic method using monte carlo averaging are computationally expensive we prove a convergence result and present several examples the method leads to important improvement in computational efficiency and is expected to perform well to solve high dimensional problems where a solution is needed on a large grid

a high order finite difference method with richardsonextrapolation for 3d convection diffusion equation in this paper we extend the sun and zhang s 24 work on high order finite difference method which is based on the richardson extrapolation technique and an operator interpolation scheme for the one and two dimensional steady convection diffusion equations to the three dimensional case firstly we employ a fourth order compact difference scheme to get the fourth order accurate solution on the fine and the coarse grids then we use the richardson extrapolation technique by combining the two approximate solutions to get a sixth order accurate solution on coarse grid finally we apply an operator interpolation scheme to achieve the sixth order accurate solution on the fine grid during this process we use alternating direction implicit adi method to solve the resulting linear systems numerical experiments are conducted to verify the accuracy and effectiveness of the present method

a distance for multistage stochastic optimization models we describe multistage stochastic programs in a purely in distribution setting i e without any reference to a concrete probability space the concept is based on the notion of nested distributions which encompass in one mathematical object the scenario values as well as the information structure under which decisions have to be made the nested distance between these distributions is introduced and turns out to be a generalization of the wasserstein distance for stochastic two stage problems we give characterizations of this distance and show its usefulness in examples the main result states that the difference of the optimal values of two multistage stochastic programs which are lipschitz and differ only in the nested distribution of the stochastic parameters can be bounded by the nested distance of these distributions this theorem generalizes the well known kantorovich rubinstein theorem which is applicable only in two stage situations to multistage moreover a dual characterization for the nested distance is established the setup is applicable both for general stochastic processes and for finite scenario trees in particular the nested distance between general processes and scenario trees is well defined and becomes the important tool for judging the quality of the scenario tree generation minimizing at least heuristically this distance is what good scenario tree generation is all about

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

a dynamic network flow problem with uncertain arc capacities formulation and problem structure we consider a dynamic network flow problem where the arc capacities are random variables this gives a multistage stochastic linear program we describe the randomness using a multi scenario approach because of the multilayered structure there are several ways to decompose the linear program we classify different decomposition schemes and we develop a scheme calledcompath decomposition which is derived from path decomposition for network flows we give a polynomial time algorithm to find a cheapest compath that can solve the subproblems resulting from compath decomposition finally compath decomposition is extended to multicommodity flow problems

an eigenvalue problem related to cell growth abstract in this paper the eigenvalues of the operator corresponding to the partial differential equation which describes the evolution of a population reproducing by simple fission are investigated this is done by transforming the eigenvalue problem to an integral equation the theory concerning positive operators on a banach space appears to be very useful