exponentially fitted spline in compression for the numerical solution of singular perturbation problems abstract some exponentially fitted difference schemes for singularly perturbed two point boundary value problems are derived using spline in compression these schemes are second order accurate numerical examples are given in support of the theoretical results

numerical integration of non linear two point boundary value problems using iterated deferred corrections i abstract in several recent papers the idea has been proposed of using implicit runge kutta formulae for the approximate integration of non linear two point boundary value problems these formulae often take a very special form so that computational efficiencies are obtained in certain well defined circumstances in this paper we examine the relative merits of these different classes of formulae and discuss their efficient implementation in a deferred correction framework a theoretical comparison of these formulae is important firstly because they call for different computational tasks to be performed and secondly because they are not always applicable to exactly the same classes of problems the most efficient of these formulae are then compared with methods based on finite differences and with methods based on spline collocation for some simple smooth test functions on the basis of operation counts and numerical experimentation it is concluded that runge kutta methods can be generally competitive with the other methods considered

on the solution of block tridiagonal systems of linear algebraic equations having a special structure recently the present author has developed some high order finite difference formulae for the approximate numerical integration of general two point boundary value problems for ordinary differential equations the algebraic equations arising from using these formulae in conjunction with a modified newton iteration scheme are block tridiagonal with an additional special structure efficient algorithms for solving these equations are given and these appear to make high order finite difference schemes particularly attractive for the numerical solution of general two point boundary value problems

analysis of the parareal time parallel time integration method the parareal algorithm is a method to solve time dependent problems parallel in time it approximates parts of the solution later in time simultaneously to parts of the solution earlier in time in this paper the relation of the parareal algorithm to space time multigrid and multiple shooting methods is first briefly discussed the focus of the paper is on new convergence results that show superlinear convergence of the algorithm when used on bounded time intervals and linear convergence for unbounded intervals

the numerical analysis of ordinary differential equations runge kutta and general linear methods mathematical and computational introduction the euler method and its generalizations analysis of runge kutta methods general linear methods bibliography

adaptive runge kutta methods for nonlinear two point boundary value problems with mild boundary layers abstract a variable mesh deferred correction algorithm based on implicit runge kutta formulae is described for the approximate numerical solution of nonlinear two point boundary value problems a strategy for automatically choosing the variable mesh spacing is described and this seeks to equidistribute an approximation to the global truncation error of the runge kutta formula the facility of being able to use a variable mesh extends considerably the range of applicability of runge kutta methods and in particular allows the possibility of solving rather more difficult problems such as those with mild boundary layers or turning points in an efficient manner an extensive set of numerical results is given to illustrate the various algorithms described and a comparison is made with the deferred correction code of lentini and pereyra appearing in the nag library

a highly stable deferred correction scheme with interpolant for systems of nonlinear two point boundary value problems iterated deferred correction is a very popular approach to the numerical solution of general first order systems of nonlinear two point boundary value problems typically the order of accuracy of the solution is increased by 2 each time that a deferred correction is applied and this allows inexpensive local error estimates to be computed using embedding recent research has shown how deferred correction schemes having the very desirable property of superconvergence can be derived this marks a significant step forward in the design of deferred correction algorithms although the price to be paid for superconvergence is that it is difficult to derive realistic embedded local error estimates in the present paper we show that richardson extrapolation is a viable alternative for error estimation and we consider in some detail the relative merits of embedding and extrapolation guided by this analysis we are able to derive for the first time a lobatto deferred correction code which is very efficient for the solution of stiff problems particularly in cases where the mirk code twpbvp f is unstable furthermore we are able to derive efficient interpolants for this lobatto code which are also applicable to mirk formulae and we consider the problem of estimating the error in these interpolants this completely solves the interpolation problem for our deferred correction codes apart perhaps for deriving interpolants for extremely stiff problems this new lobatto code is available on the web page of one of the present authors and fills a large gap in currently available deferred correction software

an adaptive boundary value runge kutta solver for first order boundary value problems the use of finite difference methods which combine features of both runge kutta processes and gap schemes is proposed for developing adaptive codes for the solution of first order differential equations with two point boundary conditions the order conditions for the coefficients of these processes are given a set of conditions by which these order conditions can be reduced in number are developed an eighth order a stable method which has second fourth and sixth order a stable methods embedded in it is given as an example a variable order variable step finite difference solver using embedded methods of this kind is described

adjoint and defect error bounding and correction for functional estimates we present two error estimation approaches for bounding or correcting the error in functional estimates such as lift or drag adjoint methods quantify the error in a particular output functional that results from residual errors in approximating the solution to the partial differential equation defect methods can be used to bound or reduce the error in the entire solution with corresponding improvements to functional estimates both approaches rely on smooth solution reconstructions and may be used separately or in combination to obtain highly accurate solutions with asymptotically sharp error bounds the adjoint theory is presented for both smooth and shocked problems numerical experiments confirm fourth order error estimates for a pressure integral of shocked quasi id euler flow by employing defect and adjoint methods together and accounting for errors in approximating the geometry it is possible to obtain functional estimates that exceed the order of accuracy of the discretization process and the reconstruction approach superconvergent drag estimates are obtained for subsonic euler flow over a lifting airfoil

random initial imperfections of structures abstract this paper is a theoretical study on random initial imperfections of structures the explicit form of probability density function of the load bearing capacity critical load of structures is derived for random initial imperfections based on a decomposition of the space of imperfection vectors into two orthogonal subspaces the subspace that asymptotically affects the load bearing capacity and the other that does not tight bounds on the range of load bearing capacity are presented for various types of simple critical points by means of the asymptotic theory of statistics we show the inefficiency of a conventional random method that approximates the minimum loadhearing capacity by the minimum load for a number of random initial imperfections the theoretical and empirical probability distribution functions for simple truss structures are compared to show the validity and effectiveness of the present method

a comparison of some global methods for solving two point boundary value problems a numerical comparison is made between three codes which implement global methods for solving two point boundary value problems the codes compared are a collocation method colsys a deferred correction method pasva 3 and a deferred correction method based on mono implicit runge kutta formulae hagron although the difficulty of conducting a fair comparison of complex codes is widely recognised it is nevertheless of great practical importance to try to gauge the relative performance of different codes and the three codes considered in this paper are particularly appropriate for numerical comparison by choosing a test set which includes both smooth and increasingly stiff problems some interesting characteristics of the codes are noted and some recommendations regarding their use are made

experiments with a local adaptive grid h refinement for the finite difference solution of bvps in singularly perturbed second order odes abstract despite many years of the development of adaptive grid methods there is still a need for re evaluation tuning and comparisons of the various approaches to grid adaptation in this study we seek to identify the most reliable and computationally inexpensive variant of the iterative grid refinement de refinement algorithm for the finite difference solution of singularly perturbed bvps in second order odes that is also easily extendable to evolutionary pdes conventional three point discretisations of the first and second derivatives are used errors of the discrete solution and its first derivative are simultaneously controlled the grid refinement is accomplished by adding nodes at inter nodal positions and de refinement by removing previously added nodes in reverse order computational experiments are performed for 15 examples of odes assuming several alternative choices of the a posteriori error estimators grid refinement indicators and regridding strategies the most satisfactory results are obtained by combining deferred approach to error estimation based on higher order reference derivative approximations for truncation error evaluation interpolation based indicator for grid refinement and regridding strategy that uses the mean indicator value as the indicator threshold for refinement

on random imperfections for structures of regular polygonal symmetry a theoretical study is made on the effect of random initial imperfections on the load bearing capacities of structures of regular polygonal symmetry the group theoretic method is employed to implement the symmetry in the formulation in view of the concept of critical initial imperfection the explicit form of probability density function of the load bearing capacity of structures is derived for random initial imperfections in particular tight bounds on the range of load bearing capacity are presented for various types of simple and double critical points the theoretical and empirical probability distribution functions are compared for simple truss structures to show the validity and effectiveness of the present method

high orderp stable formulae for the numerical integration of periodic initial value problems recently there has been considerable interest in the approximate numerical integration of the special initial value problemy f x y for cases where it is known in advance that the required solution is periodic the well known class of stormer cowell methods with stepnumber greater than 2 exhibit orbital instability and so are often unsuitable for the integration of such problems an appropriate stability requirement for the numerical integration of periodic problems is that ofp stability however lambert and watson have shown that ap stable linear multistep method cannot have an order of accuracy greater than 2 in the present paper a class of 2 step methods of runge kutta type is discussed for the numerical solution of periodic initial value problems p stable formulae with orders up to 6 are derived and these are shown to compare favourably with existing methods

probabilistic eigenvalue buckling analysis solved through the ratio of polynomial response surface an efficient procedure for the reliability analysis of frame structures with respect to the buckling limit state is proposed under the assumption that no imperfections are present and that the elastic parameters are uncertain and modeled as random variables the approach allows a deeper investigation of structures which are not sensitive to imperfections the procedure relies on a response surface method adopting simple ratio of polynomials without cross terms as performance function such a relationship approximates analytically the dependence between the buckling load and the basic variables furnishing a limit state equation which is very close to the exact one when a proper experimental design is adopted in this way a monte carlo simulation applied to the response surface leads to a good approximation with low computational effort several numerical examples show the accuracy and effectiveness of the method varying structural complexity correlation between basic variables and their distribution

on improving the absolute stability of local extrapolation a widely used technique for improving the accuracy of solutions of initial value problems in ordinary differential equations is local extrapolation it is well known however that when using methods appropriate for solving stiff systems of odes the stability of the method can be seriously degraded if local extrapolation is employed this is due to the fact that performing local extrapolation on a low order method is equivalent to using a higher order formula and this high order formula may not be suitable for solving stiff systems in the present paper a general approach is proposed whereby the correction term added on in the process of local extrapolation is in a sense a rational rather than a polynomial function this approach allows high order formulae with bounded growth functions to be developed as an example we derive ana stable rational correction algorithm based on the trapezoidal rule this new algorithm is found to be efficient when low accuracy is requested say a relative accuracy of about 1 and its performance is compared with that of the more familiar richardson extrapolation method on a large set of stiff test problems

a new mesh selection algorithm based on conditioning for two point boundary value codes we present a hybrid mesh selection strategy for use in codes for the numerical solution of two point boundary value problems this new mesh strategy is based on the estimation of two parameters which characterise the conditioning of the continuous problem as well as on a standard estimate of the local discretisation error we have implemented this algorithm in the well known code twpbvp and have found that the modified code is often considerably more efficient than the original another strong advantage of using the new mesh selection algorithm is that it automatically provides an estimate of the conditioning of the discrete problem this is very valuable arguably indispensible for use either in an a posteriori error estimate or in situations where the conditioning constants are large as a warning that the accuracy obtained in the solution may be worse than anticipated

statistics of normally distributed initial imperfections abstract this paper offers a theoretical study on the probabilistic nature of critical loads buckling loads of structures subject to normally distributed initial imperfections explicit form of probability density function of critical loads are derived for various types of critical points double bifurcation points of structures with regular polygonal symmetry are dealt with by means of the group theoretic bifurcation theory the distribution of minimum values of the critical loads is investigated to present a statistical design index the theoretical and empirical probability density functions for simple structures are compared to show the validity and effectiveness of this method the method is quite efficient when it is directly applicable otherwise the explicit forms at least can greatly supplement the inefficiency of the conventional random method

on spline basis selection for solving differential equations the suitability of b splines as a basis for piecewise polynomial solution representation for solving differential equations is challenged two alternative local solution representations are considered in the context of collocating ordinary differential equations hermite type and lmonomial both are much easier and shorter to implement and somewhat more efficient than b splines a new condition number estimate for the b splines and hermite type representations is presented one choice of the hermite type representation is experimentally determined to produce roundoff errors at most as large as those for b splines the monomial representation is shown to have a much smaller condition number than the other ones and correspondingly produces smaller roundoff errors especially for extremely nonuniform meshes the operation counts for the two local representations considered are about the same the hermite type representation being slightly cheaper it is concluded that both representations are preferable

the order of accuracy for deferred corrections using uncentered end formulas a result of christiansen and russell siam j numer math 35 1980 pp 21 33 on the order of accuracy of an iterated deferred corrections algorithm of lentini and pereyra siam j numer math 14 1977 pp 91 111 is improved at the same time providing a realistic example for a theoretical framework developed by the author for proving accuracy results for deferred corrections

a variable order deferred correction algorithm for the numerical solution of nonlinear two point boundary value problems abstract we develop two variable order deferred correction algorithms based on finite difference methods for the approximate numerical solution of nonlinear two point boundary value problems the deferred corrections provide a powerful local error estimator and this estimate can be used to refine the mesh if need be in order to obtain a prescribed tolerance mesh refinement is particularly straightforward since our algorithms are truly one step in nature the approach described is particularly appropriate for problems where only low or engineering precision is required and for very large systems of equations where storage space is an extremely important consideration some numerical results are given and a comparison with the method developed by lentini and pereyra and with richardson extrapolation is made

adaptive time stepping for vesicle suspensions we present an adaptive high order accurate time stepping numerical scheme for the flow of vesicles suspended in stokesian fluids our scheme can be summarized as an approximate implicit spectral deferred correction sdc method applying a fully implicit sdc scheme to vesicle flows is prohibitively expensive for this reason we introduce several approximations our scheme is based on a semi implicit linearized low order time stepping method our discretization is spectrally accurate in space we expect that the accuracy can be arbitrary order but our examples suffer from order reduction which limits the observed accuracy to third order we also use invariant properties of vesicle flows constant area and boundary length in two dimensions to reduce the computational cost of error estimation for adaptive time stepping we present results in two dimensions for single vesicle flows constricted geometry flows converging flows and flows in a couette apparatus we experimentally demonstrate that the proposed scheme enables automatic selection of the time step size and high order accuracy

stability of collocation at gaussian points symmetric runge kutta schemes are particularly useful for solving stiff two point boundary value problems such a stable schemes perform well in many cases but it is demonstrated that in some instances the stronger property of algebraic stability is required a characterization of symmetric algebraically stable runge kutta schemes is given the class of schemes thus defined turns out not to be very rich the only collocation schemes in it are those based on gauss points and other schemes in the class do not seem to offer any advantage over collocation at gaussian points

diagonally implicit runge kutta formulae for the numerical integration of nonlinear two point boundary value problems abstract fully implicit runge kutta formulae based on interpolatory quadrature schemes for the approximate numerical solution of nonlinear two point boundary value problems have been investigated by weiss such formulae are not used very often in practice however because they generally require such a large computational effort as to make them uncompetitive with for example integration schemes based on the trapezoidal or implicit mid point rules in this paper we consider an alternative class of runge kutta formulae namely diagonally implicit runge kutta dirk formulae which can be implemented more efficiently than the fully implicit formulae considered by weiss we also consider how these dirk formulae can be implemented in a defect or deferred correction framework and we give some numerical results to illustrate the algorithms derived one particular formula belonging to the dirk class is the implicit mid point rule in this paper we derive an efficient implementation of this formula which is applicable when the given boundary conditions are non separated

a theoretical framework for proving accuracy results for deferred corrections general techniques are described for proving accuracy results for deferred correction solutions to differential equations these techniques apply also to computational estimates of the local discretization error the proofs avoid the necessity of demonstrating the existence of asymptotic expansions of the global error in powers of some meshsize parameter

numerical steady state and hopf bifurcation analysis on the diffusive nicholson s blowflies equation1research partially supported by natural sciences and engineering research council of canada 1 for the dirichlet boundary value problem of the diffusive nicholson s blowflies equation it was shown in ref 17 that in a certain range of the parameter space there is a unique positive steady state solution in this paper we propose a scheme to compute this steady state numerically in addition we describe an iterative procedure to locate the critical values of the delay where a hopf bifurcation of time periodic solutions takes place near the steady state some numerical simulations of both schemes are given

systematic description of imperfect bifurcation behavior of symmetric systems abstract a systematic method is presented for describing experimental curves of force vs strain of a system with regular polygonal dihedral group symmetry subject to bifurcation behavior with an aim toward overcoming the following problems 1 it is difficult to judge whether the system is undergoing bifurcation or not 2 the perfect behavior of the system cannot be known due to the presence of initial imperfections 3 those curves are often qualitatively different from bifurcation diagrams predicted by mathematics the tools employed are the asymptotic theory for imperfect bifurcation such as the koiter law and the stochastic theory of initial imperfections the former theory is extended in this paper to the system with regular polygonal symmetry to present asymptotic laws for recovering perfect curves with reference to the experimental ones these laws are formulated for physically observable displacements instead of the variables in the mathematical bifurcation diagrams in order to make them readily applicable to the experimental curves the stochastic theory is combined with an asymptotic law to develop a means to identify the multiplicity of the bifurcation point the systematic method for describing the experimental curves developed in this manner is applied to the bifurcation analysis of regular polygonal truss domes to testify its validity furthermore this method is applied to the shear behavior of cylindrical sand specimens to show that they in fact are undergoing bifurcation and in turn to demonstrate the importance of a viewpoint of bifurcation in the study of shear behavior of materials the need of a dual viewpoint of bifurcation and plasticity in the study of constitutive relationship of materials is emphasized to conclude the paper

order barriers and characterizations for continuous mono implicit runge kutta schemes the mono implicit runge kutta mirk schemes a subset of the family of implicit runge kutta irk schemes were originally proposed for the numerical solution of initial value odes more than fifteen years ago dur ing the last decade a considerable amount of attention has been given to the use of these schemes in the numerical solution of boundary value ode prob lems where their efficient implementation suggests that they may provide a worthwhile alternative to the widely used collocation schemes recent work in this area has seen the development of some software packages for bound ary value odes based on these schemes unfortunately these schemes lead to algorithms which provide only a discrete solution approximation at a set of mesh points over the problem interval while the collocation schemes provide a natural continuous solution approximation the availability of a continuous solution is important not only to the user of the software but also within the code itself for example in estimation of errors defect control mesh selection and the provision of initial solution estimates for new meshes an approach for the construction of a continuous solution approximation based on the mirk schemes is suggested by recent work in the area of continuous extensions for explicit runge kutta schemes for initial value odes in this paper we describe our work in the investigation of continuous versions of the mirk schemes i we give some lower bounds relating the stage order to the minimal number of stages for general continuous irk schemes ii we establish lower bounds on the number of stages needed to derive continuous mirk schemes of orders 1 through 6 and iii we provide characterizations of these schemes having a minimal number of stages for each of these orders

capillary rise between planar surfaces minimization of free energy is used to calculate the equilibrium vertical rise and meniscus shape of a liquid column between two closely spaced parallel planar surfaces that are inert and immobile states of minimum free energy are found using standard variational principles which lead not only to an euler lagrange differential equation for the meniscus shape and elevation but also to the boundary conditions at the three phase junction where the liquid meniscus intersects the solid walls the analysis shows that the classical young dupr e equation for the thermodynamic contact angle is valid at the three phase junction as already shown for sessile drops with or without the influence of a gravitational field integration of the euler lagrange equation shows that a generalized laplace young ly equation first proposed by o brien craig and peyton j colloid interface sci 26 500 1968 gives an exact prediction of the mean elevation of the meniscus at any wall separation whereas the classical ly equation for the elevation of the midpoint of the meniscus is accurate only when the separation approaches zero or infinity when both walls are identical the meniscus is symmetric about the midpoint and the midpoint elevation is a more traditional and convenient measure of capillary rise than the mean elevation therefore for this symmetric system a different equation is fitted to numerical predictions of the midpoint elevation and is shown to give excellent agreement for contact angles between 15 ifmmode circ else textdegree fi and 160 ifmmode circ else textdegree fi and wall separations up to 30 phantom rule 0 3em 0ex mathrm mm when the walls have dissimilar surface properties the meniscus generally assumes an asymmetric shape and significant elevation of the liquid column can occur even when one of the walls has a contact angle significantly greater than 90 ifmmode circ else textdegree fi the height of the capillary rise depends on the spacing between the walls and also on the difference in contact angles at the two surfaces when the contact angle at one wall is greater than 90 ifmmode circ else textdegree fi but the contact angle at the other wall is less than 90 ifmmode circ else textdegree fi the meniscus can have an inflection point separating a region of positive curvature from a region of negative curvature the inflection point being pinned at zero height however this condition arises only when the spacing between the walls exceeds a threshold value that depends on the difference in contact angles

gaussian collocation via defect correction for the numerical integration of boundary value problems for first order ordinary differential systems collocation on gaussian points is known to provide a powerful method in this paper we introduce a defect correction method for the iterative solution of such high order collocation equations the method uses the trapezoidal scheme as the basic discretization and an adapted form of the collocation equations for defect evaluation the error analysis is based on estimates of the contractive power of the defect correction iteration it is shown that the iteration produceso h 2 convergence rates for smooth starting vectors a new result is that the iteration damps all kind of errors so that it can also handle non smooth starting vectors successfully

bayes prediction in regressions with elliptical errors abstract in this paper the prediction problem is considered for linear regression models with elliptical errors when the bayes prior is non informative we show that the bayes prediction density under the elliptical errors assumption is exactly the same as that obtained with normally distributed errors thus assuming that the errors have a normal distribution when the true distribution is elliptical will not lead to incorrect predictive inferences if the error variance structure is correctly specified this extends the results of zellner 1976 finally based on monte carlo numerical integration procedures computations are provided in a model with multiplicative heteroscedasticity

mono implicit runge kutta formulae for the numerical solution of second order nonlinear two point boundary value problems mono implicit runge kutta mirk formulae are widely used for the numerical solution of first order systems of nonlinear two point boundary value problems in order to avoid costly matrix multiplications mirk formulae are usually implemented in a deferred correction framework and this is the basis of the well known boundary value code twpbvp however many two point boundary value problems occur naturally as second or higher order equations or systems and for such problems there are significant savings in computational effort to be made if the mirk methods are tailored for these higher order forms in this paper we describe mirk algorithms for second order equations and report numerical results that illustrate the substantial savings that are possible particularly for second order systems of equations where the first derivative is absent

a variable step runge kutta nystrom integrator for reversible systems of second order initial value problems a class of implicit runge kutta nystrom rkn integration formulae is developed for the numerical integration of reversible systems of second order ordinary differential equations of the form y f x y the formulae derived have high stage order provide a continuous solution are symmetric and have asymptotically correct local error estimates available at each step point the availability of these error estimates coupled with the symmetry of the algorithm allows the formulae to be implemented in a variable stepsize mode using the ideas of stoffer computing 55 1995 pp 1 22 numerical results presented for both fixed and variable stepsizes of integration show the power of the new methods

block runge kutta methods for the numerical integration of initial value problems in ordinary differential equations i the nonstiff case block runge kutta formulae suitable for the approximate numerical integration of initial value problems for first order systems of ordinary differential equations are derived considered in detail are the problems of varying both order and stepsize automatically this leads to a class of variable order block explicit runge kutta formulae for the integration of nonstiff problems and a class of variable order block implicit formulae suitable for stiff problems the central idea is similar to one due to c w gear in developing runge kutta starters for linear multistep methods some numerical results are given to illustrate the algorithms developed for both the stiff and nonstiff cases and comparisons with standard runge kutta methods are made

a stability criterion for explicit boundary value runge kutta methods abstract a function characterizing the stability of explicit boundary value runge kutta methods for the solution of ordinary differential equations is introduced it can be used to design methods with better stability and also in the selection of mesh points it is based on a generalization of the algebraic stability criterion numerical examples demonstrating the use of the proposed function are given

accelerating the convergence of spectral deferred correction methods in the recent paper by dutt greengard and rokhlin a variant of deferred or defect correction methods is presented which couples gaussian quadrature with the picard integral equation formulation of the initial value ordinary differential equation the resulting spectral deferred correction sdc methods have been shown to possess favorable accuracy and stability properties even for versions with very high order of accuracy in this paper we show that for linear problems the iterations in the sdc algorithm are equivalent to constructing a preconditioned neumann series expansion for the solution of the standard collocation discretization of the ode this observation is used to accelerate the convergence of sdc using the gmres krylov subspace method for nonlinear problems the gmres acceleration is coupled with a linear implicit approach stability and accuracy analyses show the accelerated scheme provides an improvement in the accuracy efficiency and stability of the original sdc approach furthermore preliminary numerical experiments show that accelerating the convergence of sdc methods can effectively eliminate the order reduction previously observed for stiff ode systems

high order convergent deferred correction schemes based on parametrerized runge kutta nystrom methods for second order boundary value problems abstract iterated deferred correction is a widely used approach to the numerical solution of first order systems of nonlinear two point boundary value problems normally the orders of accuracy of the various methods used in a deferred correction scheme differ by 2 and as a direct result each time deferred correction is used the order of the overall scheme is increased by a maximum of 2 in 16 however it has been shown that there exist schemes based on parameterized runge kutta methods which allow a higher increase of the overall order a first example of such a high order convergent scheme which allows an increase of 4 orders per deferred correction was based on two mono implicit runge kutta methods in the present paper we will investigate the possibility for high order convergence of schemes for the numerical solution of second order nonlinear two point boundary value problems not containing the first derivative two examples of such high order convergent schemes based on parameterized runge kutta nystrom methods of orders 4 and 8 are analysed and discussed

exponentially fitted gauss laguerre quadrature rule for integrals over an unbounded interval new quadrature formulae are introduced for the computation of integrals over the whole positive semiaxis when the integrand has an oscillatory behaviour with decaying envelope the new formulae are derived by exponential fitting and they represent a generalization of the usual gauss laguerre formulae their weights and nodes depend on the frequency of oscillation in the integrand and thus the accuracy is massively increased rules with one up to six nodes are treated with details numerical illustrations are also presented

order results for mono implicit runge kutta methods the mono implicit runge kutta methods are a subclass of the well known implicit runge kutta methods and have application in the efficient numerical solution of systems of initial and boundary value ordinary differential equations although the efficiency and stability properties of this class of methods have been studied in a number of papers the specific question of determining the maximum order of an s stage mono implicit runge kutta method has not been dealt with in addition to the complete characterization of some subclasses of these methods having a number of stages s leq 5 a main result of this paper is a proof that the order of an s stage mono implicit runge kutta method is at most s 1

a spline based method of solution of nonlinear two point boundary value problems abstract a method is presented that is meant to solve nonlinear two point boundary value problems the method is based on a spline interpolation of the function of interest with undetermined supporting points these are determined from the differential equation written at the supporting points which leads together with the boundary conditions to a nonlinear algebraic system this is next solved resorting to a newton type method with continuation to ensure convergence the procedure is illustrated with examples arising from computer aided design and robotics applications

defect correction from a galerkin viewpoint we consider the numerical solution of systems of nonlinear two point boundary value problems by galerkin s method an initial solution is computed with piecewise linear approximating functions and this is then improved by using higher order piecewise polynomials to compute defect corrections this technique including numerical integration is justified by typical galerkin arguments and properties of piecewise polynomials rather than the traditional asymptotic error expansions of finite difference methods

a second order decoupled penalty projection method based on deferred correction for mhd in elsasser variable abstract we study the deferred correction method for the magnetohydrodynamics mhd system written in elsasser variables the proposed algorithm is based on the penalty projection with grad div stabilized taylor hood solutions of the elsasser formulation in this way second order accuracy of the method in time is obtained through the deferred correction method with excellent mass conservation properties for the proposed method stability is rigorously proven and numerical experiments are presented to verify the proposed scheme and theory

deferred correction with mono implicit runge kutta methods for first order ivps abstract to reach a high order of accuracy for numerical solutions of ivps with mono implicit runge kutta mirk methods the technique of deferred correction is used special attention is paid to the possible increase of the order and the stability of such schemes several schemes are given

an automatic continuation strategy for the solution of singularly perturbed linear two point boundary value problems an automatic continuation alogorithm for the solution of linear singular perturbation problems is developed and incorporated into two codes which implement global methods for solving two point boundary value problems specifically the algorithm will be based upon error estimates formulated in the collocation code colsys and the deferred correction code hagron the benefits of using continuation are clearly demonstrated for both codes for a large class of problems

on the numerical integration of nonlinear two point boundary value problems using iterated deferred corrections part 2 the development and analysis of highly stable deferred correction formulae iterated deferred correction methods have been very widely used for the numerical solution of general nonlinear two point boundary value problems in ordinary differential equations however there

superconvergent deferred correction methods for first order systems of nonlinear two point boundary value problems iterated deferred correction is a widely used approach to the numerical solution of first order systems of nonlinear two point boundary value problems normally the orders of accuracy of the various methods used in a deferred correction scheme differ by 2 and as a direct result each time a deferred correction is applied the order of the overall scheme is increased by a maximum of 2 in this paper we consider the construction of mono implicit runge kutta mirk methods where an increase of four orders of accuracy is obtained for each deferred correction we develop a very powerful yet rather straightforward theory which allows us to identify the appropriate runge kutta formulae for inclusion in such schemes in particular we will focus on the construction of pairs of mirk formulae of order 4 and 8 which will allow this superconvergence to be realized we will further show that it is possible to derive formulae of this type for which high order interpolants and accurate error estimates are readily available

a survey of numerical techniques for solving singularly perturbed ordinary differential equations this survey paper contains a surprisingly large amount of material and indeed can serve as an introduction to some of the ideas and methods of singular perturbation theory starting from prandtl s work a large amount of work has been done in the area of singular perturbations this paper limits its coverage to some standard singular perturbation models considered by various workers and the numerical methods developed by numerous researchers after 1984 2000 the work done in this area during the period 1905 1984 has already been surveyed by the first author of this paper see appl math comput 30 1989 223 for details due to the space constraints we have covered only singularly perturbed one dimensional problems

on the robustness of lm lr and w tests in regression models in this paper we show that in the small sample case while the lr test is robust the lm and w tests are not robust when the errors are student t for the large sample however all three tests are found to be robust a variation of the w test is also considered and it turns out to be nonrobust the implications of assuming the errors to be distributed as multivariate student t rather than multivariate normal are also discussed it is found that not all of evans and savin s 4 results carry through when the errors are student t

discrete newton methods and iterated defect corrections in this paper we present a general theory for discrete newton methods iterated defect corrections via neighbouring problems and deferred corrections based on asymptotic expansions of the discretization error

an enhanced parareal algorithm based on the deferred correction methods for a stiff system in this study we consider a variant of the hybrid parareal algorithm based on deferred correction techniques in order to increase the convergence order even for the stiff system a hybrid parareal scheme introduced by minion 2011 20 improves the efficiency of the original parareal by utilizing a spectral deferred correction sdc strategy for a fine propagator within the parareal iterations in this paper we use krylov deferred correction kdc for a fine propagator to solve the stiff system and differential algebraic equations daes stably also we employ a deferred correction technique based on the backward euler method for a coarse propagator in order to make the global order of accuracy reasonably high while limiting the cost of sequential steps as small as possible numerical experiments on the efficiency of our method are promising

asymptotic expansions for the discretization error of least squares solutions of linear boundary value problems on presente des developpements asymptotiques pour l erreur de discretisation des solutions des moindres carres des problemes aux limites lineaires

testing conditional independence via rosenblatt transforms this paper proposes new tests of conditional independence of two random variables given a single index involving an unknown finite dimensional parameter the tests employ rosenblatt transforms and are shown to be distribution free while retaining computational convenience some results from monte carlo simulations are presented and discussed

collocation for singular perturbation problems i first order systems with constant coefficients the application of collocation methods for the numerical solution of singularly perturbed ordinary differential equations is investigated collocation at gauss radau and lobatto points is considered for both initial and boundary value problems for first order systems with constant coefficients particular attention is paid to symmetric schemes for boundary value problems these problems may have boundary layers at both interval ends br our analysis shows that certain collocation schemes in particular those based on gauss or lobatto points do perform very well on such problems provided that a fine mesh with steps proportional to the layers width is used in the layers only and a coarse mesh just fine enough to resolve the solution of the reduced problem is used in between ways to construct appropriate layer meshes are proposed of all methods considered the lobatto schemes appear to be the most promising class of methods as they essentially retain their usual superconvergence power for the smooth reduced solution whereas gauss legendre schemes do not br we also investigate the conditioning of the linear systems of equations arizing in the discretization of the boundary value problem for a row equilibrated version of the discretized system we obtain a pleasantly small bound on the maximum norm condition number which indicates that these systems can be solved safely by gaussian elimination with scaled partial pivoting

echelon modes in uniform materials abstract the mechanism of the echelon mode a series of parallel short wrinkles of soils that looks like a flight of stairs or wild geese arranged in formation observed in the triaxial compression test is revealed by means of the group theoretic bifurcation theory the study of the direct bifurcation of uniform materials in cylindrical domains with periodic boundaries has revealed the existence of various kinds of deformation patterns in particular a classical diamond checkerboard pattern solution and a pair of oblique stripe pattern solutions simultaneously branch at a bifurcation point of multiplicity four with the use of the periodic boundaries these oblique stripe solutions which cannot be directly obtained by the customary bifurcation analysis with non periodic boundaries have thus been made accessible the secondary bifurcation from these stripe solutions realizes the echelon mode the triaxial compression tests on soil specimens have demonstrated that the pattern of wrinkles shear bands and echelon modes of the specimens do indeed follow the present theory and can be understood as a natural consequence of pattern selection or in other words symmetry breaking bifurcation

testing for stochastic monotonicity we propose a test of the hypothesis of stochastic monotonicity this hypothesis is of interest in many applications in economics our test is based on the supremum of a rescaled u statistic we show that its asymptotic distribution is gumbel the proof is difficult because the approximating gaussian stochastic process contains both a stationary and a nonstationary part and so we have to extend existing results that only apply to either one or the other case we also propose a refinement to the asymptotic approximation that we show works much better in finite samples we apply our test to the study of intergenerational income mobility

conditional and unconditional statistical independence abstract conditional independence almost everywhere in the space of the conditioning variates does not imply unconditional independence although it may well imply unconditional independence of certain functions of the variables an example that is important in linear regression theory is discussed in detail this involves orthogonal projections on random linear manifolds which are conditionally independent but not unconditionally independent under normality necessary and sufficient conditions are obtained under which conditional independence does imply unconditional independence

on the method of modified equations vi asymptotic analysis of and asymptotic successive corrections techniques for two point boundary value problems in ode s the modified equation technique is extended to two point boundary value problems and a second order accurate implicit centered finite difference scheme for non homogeneous second order ordinary differential equations with linear boundary conditions is analyzed the first second and third modified equations or equivalent second equivalent and simply modified equations respectively for this scheme and its boundary conditions are presented it is shown that the three kinds of modified equations are asymptotically equivalent when the equivalent equation is used for the boundary conditions since an asymptotic analysis of these equations with the grid size as small parameter yields exactly the same results for a linear problem multiple scales and summed up asymptotic techniques are used and the resulting uniform asymptotic expansions are shown to be equivalent to the solution of the original finite difference scheme asymptotic successive corrections techniques are also applied to the three kinds of modified equations to obtain higher order schemes higher order boundary conditions are easily treated in the asymptotic successive corrections technique although these boundary conditions must be obtained by using the equivalent equation in order to obtain a correct estimate of the global error near the domain boundaries the methods introduced in this paper are applied to homogeneous and non homogeneous second order linear and non linear ordinary differential equations and yield very accurate results

a picture of the global bifurcation diagram in ecological interacting and diffusing systems abstract this paper concerns global phenomena of pattern formation in stationary reaction diffusion equations possessing turing s diffusion induced instability which appear typically in mathematical biology global bifurcation diagrams with respect to two diffusion parameters are presented by integrating two complementary approaches analytical and numerical simple and double bifurcation analysis using group theoretic methods for the compact lie group d x results from singular perturbations when one diffusion constant is sufficiently small and a global existence theorem on primary bifurcated branches when the other diffusion is sufficiently large are the main analytical results a new numerical method is used to trace all bifurcating branches we examine interrelations between those local and semi global results under the light of global pictures obtained by numerical studies a variety of interesting and new diagrams near double eigenvalues are observed coexistence of multiple stable stationary states and global extension and deformation of local double structures are one of the main conclusions

defect correction for two point boundary value problems on nonequidistant meshes new finite difference formulae of arbitrary order are derived for the speciai class of second order two point boundary value problems y f x y x a x b variable mesh spacing is possible and the required accuracy is achieved under a very mild mesh condition a naturai defect correction framework is set up to compute the higher order approximations

critical initial imperfection of structures abstract a method is introduced for determining the critical initial imperfection of discretized structures that decreases the load bearing capacity most rapidly the effects of imperfections on simple critical points such as limit points of loads and simple bifurcation points are theoretically investigated based on the idea of the lyapunov schmidt decomposition developed in bifurcation theory imperfection sensitivity varies with the types of points nonetheless critical imperfection pattern is expressed in the same formula regardless of the types among various imperfections the most influential can be found in a quantitative manner the validity and the usability of the proposed method are illustrated through its application to simple example structures

critical imperfection of symmetric structures a method is presented to determine the critical most unfavorable initial imperfection of structures of regular polygonal symmetry denoted by dihedral groups a critical point of such structures is either simple or forced to be double by the symmetry and the group representation theory is employed to deal with the degeneracy due to symmetry the method is developed further to incorporate the symmetry of imperfections in many cases of practical interest the critical imperfection is given explicitly as the product of imperfection sensitivity matrix and the critical eigenvector it is shown that the symmetry in the critical eigenvectors of the tangent stiffness matrix is inherited to the symmetry in the critical imperfection mode

superconvergent interpolants for the collocation solution of boundary value ordinary differential equations a long standing open question associated with the use of collocation methods for boundary value ordinary differential equations is concerned with the development of a high order continuous solution approximation to augment the high order discrete solution approximation obtained at the mesh points which subdivide the problem interval it is well known that the use of collocation at gauss points leads to solution approximations at the mesh points for which the global error is o h2k where k is the number of collocation points used per subinterval and h is the subinterval size this discrete solution is said to be superconvergent the collocation solution also yields a c0 continuous solution approximation that has a global error of o hk 1 in this paper we show how to efficiently augment the superconvergent discrete collocation solution to obtain c1 continuous superconvergent interpolants whose global errors are o h2k the key ideas are to use the theoretical framework of continuous runge kutta schemes and to augment the collocation solution with inexpensive monoimplicit runge kutta stages specific schemes are derived for k 1 2 3 and 4 numerical results are provided to support the theoretical analysis

lobatto deferred correction for stiff two point boundary value problems abstract an iterated deferred correction algorithm based on lobatto runge kutta formulae is developed for the efficient numerical solution of nonlinear stiff two point boundary value problems an analysis of the stability properties of general deferred correction schemes which are based on implicit runge kutta methods is given and results which are analogous to those obtained for initial value problems are derived a revised definition of symmetry is presented and this ensures that each deferred correction produces an optimal increase in order finally some numerical results are given to demonstrate the superior performance of lobatto formulae compared with mono implicit formulae on stiff two point boundary value problems

integral deferred correction methods constructed with high order runge kutta integrators spectral deferred correction sdc methods for solving ordinary differential equations odes were introduced by dutt greengard and rokhlin 2000 it was shown in that paper that sdc methods can achieve arbitrary high order accuracy and possess nice stability properties their sdc methods are constructed with low order integrators such as forward euler or backward euler and are able to handle stiff and non stiff terms in the odes in this paper we use high order runge kutta rk integrators to construct a family of related methods which we refer to as integral deferred correction idc methods the distribution of quadrature nodes is assumed to be uniform and the corresponding local error analysis is given the smoothness of the error vector associated with an idc method measured by the discrete sobolev norm is a crucial tool in our analysis the expected order of accuracy is demonstrated through several numerical examples superior numerical stability and accuracy regions are observed when high order rk integrators are used to construct idc methods

grid adaptation and non iterative defect correction for improved accuracy of numerical solutions of pdes in this work we present a computational approach for improving the order of accuracy of a given finite difference method for solution of linear and nonlinear hyperbolic partial differential equations the methodology consists of analysis of leading order terms in the discretization error of any given finite difference method leading to a modified version of the original partial differential equation singular perturbations of this modified equation are regularized using an adaptive grid distribution and a non iterative defect correction method is used to eliminate the leading order regular perturbation terms in the modified equation implementation of this approach on a low order finite difference scheme not only results in an increase in its order of accuracy but also results in an improvement in its numerical stability due to the regularization of singular perturbations the proposed approach is applied to four different canonical problems including the numerical solution of 1 liouville equation 2 inviscid burgers equation 3 nonlinear reaction advection equation and 4 a system of hyperbolic pdes when compared to exact solutions the numerical results demonstrate the ability of this method in both boosting the accuracy of finite difference schemes up to the desired order and also providing a fully stable numerical solution

uniform fourth order difference scheme for a singular perturbation problem the numerical solution of a nonlinear singularly perturbed two point boundary value problem is studied the developed method is based on hermitian approximation of the second derivative on special discretization mesh numerical examples which demonstrate the effectiveness of the method are presented

staircase matrices and systems difference equations spline approximations and multiperiod linear programs all give rise to linear equation systems that have a characteristic staircase structure a staircase system s variables can be partitioned into a natural sequence of periods in such a way that every equation involves at most the variables of two adjacent periods this paper surveys the numerous properties of staircase systems and of the staircase matrices that comprise their coefficients particular attention is given to reduced and sparse staircase forms and to the varieties of gaussian elimination for solving staircase systems

on the numerical solution of a class of singular two point boundary value problems abstract mono implicit runge kutta mirk formulae used in a deferred correction framework have proved to be a very effective way of solving first order systems of nonlinear two point boundary value problems we examine this general approach in detail and question whether mirk formulae are the most efficient class of runge kutta methods to use we also consider the extension of this deferred correction approach to deal with problems having a certain type of singularity at the end of the range of integration we consider the questions whether it is possible to derive mirk formulae which are applicable to such problems and whether it is in general possible to solve such problems automatically we will answer the first question in the affirmative however we will show that despite claims to the contrary appearing in the literature the answer to the second question is no

thirteen ways to estimate global error various techniques that have been proposed for estimating the accumulated discretization error in the numerical solution of differential equations particularly ordinary differential equations are classified described and compared for most of the schemes either an outline of an error analysis is given which explains why the scheme works or a weakness of the scheme is illustrated

on the method of modified equations iii numerical techniques based on the second equivalent equation for the euler forward difference method new numerical successive correction techniques for ordinary differential equations based on the euler forward explicit method and the first modified or equivalent equation are presented these techniques are similar to iterative updating deferred methods and are based on the application of asymptotic methods to modified equations which do not require initial conditions for the high order derivatives in the truncation terms and which yield stable numerical methods it is shown that depending on the discretization of the high order derivatives in the high order correction equations different methods of as high order of consistency as required can be developed in this paper backward and centered formulas are used but the resulting numerical methods are not self starting it is shown that if the starting procedure is not adequate the numerical order of the method can be smaller than the theoretical one in order to avoid this loss of numerical order a method for consistently starting the asymptotic successive correction technique based on the use of fictitious times is presented and applied to autonomous and non autonomous ordinary differential equations and compared with the results of second and fourth order runge kutta methods it is shown that the fourth order runge kutta method is more accurate than the successive correction techniques for large time steps due to the higher order derivatives in the successive correction but for sufficiently small time steps these techniques have almost the same accuracy as the fourth order runge kutta method