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

split step orthogonal spline collocation methods for nonlinear schrodinger equations in one two and three dimensions abstract split step orthogonal spline collocation osc methods are proposed for one two and three dimensional nonlinear schrodinger nls equations with time dependent potentials firstly the nls equation is split into two nonlinear equations and one or more one dimensional linear equations commonly the nonlinear subproblems could be integrated directly and accurately but it fails when the time dependent potential cannot be integrated exactly in this case we propose three approximations by using quadrature formulae but the split order is not reduced discrete time osc schemes are applied for the linear subproblems in numerical experiments many tests are carried out to prove the reliability and efficiency of the split step osc ssosc methods solitons in one two and three dimensions are well simulated and conservative properties and convergence rates are demonstrated we also apply the ways of solving the nonlinear subproblems to the split step finite difference ssfd methods and the time splitting spectral tssp methods and the approximate ways still work well finally we apply the ssosc methods to solve some problems of bose einstein condensates

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

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

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 smooth approximation for the solution of a fourth order boundary value problem based on nonpolynomial splines we develop a method of second order for the continuous approximation of the solution of a two point boundary value problem involving a fourth order differential equation via mixed spline functions consisting of a polynomial part of degree three and a trigonometric part it will be shown that the angular frequency k of the trigonometric part can be used to raise the order of accuracy of the obtained scheme two numerical examples are given to sustain our autonomous parametric spline method and to compare it with a quintic polynomial spline method of second order

evolution of a multimodal map induced by an equivariant vector field it has been shown that the topological characterization of an equivariant system should preferably be achieved by working in a fundamental domain generated by the symmetry properties appearing in the phase space in this paper we discuss the case when the equivariance of the studied system is taken into account to study the evolution of the population of periodic orbits when a control parameter is varied the burke shaw system is considered here as an example it is shown that the equivariance of this system may be used to reduce the multimodal first return map in a poincare section to a unimodal map a relationship between four symbol sequences and two symbol sequences is given the non trivial evolution of the orbit spectrum of a multimodal map is then predicted from the much simpler unimodal map to which the multimodal map reduces

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

conditioning of collocation matrices and discrete green s functions the conditioning of spline collocation matrices is considered in the case of a boundary value problem for an mth order equation for bases with continuity built in which corresponds to most current implementations of splines for solving odes the collocation matrix a is related to the corresponding multiple shooting matrix for a first order system this gives the explicit form for a 1 as a discrete green function we briefly discuss the implications for mesh selection and how our results relate to previous ones for finite differences

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

solution of sixth order boundary value problems using non polynomial spline technique abstract non polynomial spline is used for the numerical solutions of the 12th order linear special case boundary value problems the end conditions are derived for the definition of spline siddiqi and twizell s s siddiqi e h twizell spline solutions of linear twelfth order boundary value problems int j comput math 78 1998 345 362 presented the solutions of 12th order boundary value problems using 12th degree spline where some unexpected results for the solution and its derivatives were obtained near the boundaries of the interval no such situation is observed in this method near the boundaries of the interval and the results are better in the whole interval out of three two examples compared with that considered by siddiqi and twizell 1998 show that the method developed in the paper is more efficient

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

the solution of nonlinear schrodinger equations using orthogonal spline collocation abstract an orthogonal spline collocation semidiscretization previously applied in the numerical solution of the cubic schrodinger equation is extended to the solution of schrodinger equations which possess general power nonlinearities and nonlinear terms that contain derivatives the accuracy of the method is examined as well as its ability to conserve approximations to certain invariants associated with the theoretical solution the results of numerical experiments are presented in which the integration in time is performed using a routine from a software library particular attention is paid to the value and limitations of conservation as an indicator of accuracy

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

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

high order accuracy nonpolynomial spline solutions for 2μth order two point boundary value problems sixth order accurate method based on quintic nonpolynomial spline function for the numerical solution of 2 th order two point bvps where is a positive integer is presented by transforming the problem into a system of second order two point bvps the nonpolynomial spline function is used to drive some consistency relations for computing approximation to the solution of this problem the proposed approach gives better approximations than existing polynomial spline and finite difference methods and has a lower computational cost convergence analysis of the proposed method is discussed numerical examples are included to illustrate the practical usefulness of our method

a note on conditioning stability and collocation matrices the concept of a well conditioned bvp as considered in recent literature essentially corresponds to the concept of a uniformly stable ivp but this terminology suggests a shift from absolute to relative quantities which makes it more difficult to understand the conditioning of numerical discretizations especially for higher order differential operators in this note we wish to make it clear that the notion of well conditioning of bvps is not always a direct analogue of the u the condition number of a discretization of a well conditioned bvp generally grows unboundedly when higher accuracy is desired because it contains a discretization of a differential operator this condition number gets large particularly rapidly if a direct discretization of a higher order differential operator is attempted for collocation an implementation which uses a local runge kutta solution representation produces a much better conditioning because the repeated numerical differentiation is avoided instead the highest derivative is approximated locally by a discontinuous piece wise polynomial and local integrations yield lower solution derivatives upon imposing the collocation equations the highest derivative approximation can be locally eliminated resulting in a sophisticated discretization of the corresponding first order system

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

covering dynamical systems twofold covers we study the relation between a dynamical system which is unchanged equivariant under a discrete symmetry group g and another locally identical dynamical system with no residual symmetry we also study the converse mapping lifting a dynamical system without symmetry to a multiple cover which is equivariant under g this is done in r 3 for the two element rotation and inversion groups comparisons are done for the equations of motion the strange attractors that they generate and the branched manifolds that classify these strange attractors a dynamical system can have many inequivalent multiple covers all equivariant under the same symmetry group g these are distinguished by the value of a certain topological index many examples are presented a new global bifurcation the peeling bifurcation is described

numerical solution of a system of fourth order boundary value problems using cubic non polynomial spline method abstract a cubic non polynomial spline technique is developed for the numerical solutions of a system of fourth order boundary value problems associated with obstacle unilateral and contact problems the end conditions consistent with the bvp are derived corresponding to the boundary conditions in terms of not only second derivatives but first derivatives as well the present method has less computational cost and gives better approximations than those produced by other collocation finite difference and spline methods the method developed is compared with those developed by khan et al a khan m a noor t aziz parametric quintic spline approach to the solution of a system of fourth order boundary value problems journal of optimization theory and applications 122 2 2004 309 322 al said and noor e a al said m a noor computational methods for fourth order obstacle boundary value problems communications in applied nonlinear analysis 2 1995 73 83 and siddiqi and ghazala s s siddiqi g akram solution of the system of fourth order boundary value problems using non polynomial spline technique applied mathematics and computation 185 2007 128 135 through different examples

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

the conditioning of linear boundary value problems investigation is made into the sensitivity of solutions of linear boundary value problems to perturbations of the boundary condition we derive a useful quantity to decide for well or ill conditioning from this it is deduced which kind of requirements the boundary conditions should meet in order to have a well conditioned problem it is shown that this quantity can fruitfully be used to explain why e g the multiple shooting technique is stable in contrast to the single shooting one for certain well posed problems having solutions with different growth behavior the results are sustained by a number of examples

global vector field reconstruction by using a multivariate polynomial l 2 approximation on nets a multivariate polynomial mathit l 2 approximation on nets is designed for global vector field reconstructions of time continuous dynamical systems the technique is tested by investigating standard forms of the r ossler band the lorenz mask and a chaotic attractor produced by a simple model of thermal lens oscillations

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

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

analytical results and feedback circuit analysis for simple chaotic flows simple jerk systems are very useful for combining analytical computations and dynamical analysis in phase space this is particularly relevant since there is still no direct link between the algebraic structure of ordinary differential equations and the topology of the chaotic attractors which they generate in this paper particular analytical solutions are identified for three simple chaotic flows it is shown that these solutions have varying effects on the bifurcation diagrams moreover a feedback circuit analysis is used to exhibit the similarities between the three simple systems such analysis also exhibits the relevant role of double nullcline in the topology of the attractor

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

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

fast direct solvers for piecewise hermite bicubic orthogonal spline collocation equations fast direct methods are proposed for the solution of linear systems arising when orthogonal spline collocation with piecewise hermite bicubics is employed for the approximate solution of poisson s

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

orthogonal cubic spline collocation method for the extended fisher kolmogorov equation a second order splitting combined with orthogonal cubic spline collocation method is formulated and analysed for the extended fisher kolmogorov equation with the help of lyapunov functional a bound in maximum norm is derived for the semidiscrete solution optimal error estimates are established for the semidiscrete case finally using the monomial basis functions we present the numerical results in which the integration in time is performed using radau 5 software library

the close relationships between methods for solving two point boundary value problems the well conditioning of a boundary value problem is shown to be related to bounding two quantities one involving the boundary conditions and the other involving the green s function in the case of a solution dichotomy they are related to known stability results these results easily explain why difficulties arise using superposition and reduced superposition it is shown how multiple shooting overcomes these difficulties by relating its matrix conditioning to the underlying boundary value problem some factorization methods for the multiple shooting method are considered these show the close relationships between multiple shooting the stabilized march and invariant imbedding and one factorization leads to an efficient new way to implement each of these three methods many of the results apply for collocation and finite difference methods too finally it is shown both theroetically and computationally when matrix compactification can lead to difficulty

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

a computational study of finite element methods for second order linear two point boundary value problems a computational study of five finite element methods for the solution of a single second order linear ordinary differential equation subject to general linear separated boundary conditions is described in each method the approximate solution is a piecewise polynomial expressed in terms of a b spline basis and is determined by solving a system of linear algebraic equations with an almost block diagonal structure the aim of the investigation is twofold to determine if the theoretical orders of convergence of the methods are realized in practice and to compare the methods on the basis of cost for a given accuracy in this study three parametrized families of test problems containing problems of varying degrees of difficulty are used the conclusions drawn are rather straightforward collocation is the cheapest method for a given accuracy and the easiest to implement also for solving the linear algebraic equations the use of a special purpose solver which takes advantage of the structure of the equations is advisable 1 introduction finite element methods for two point boundary value problems for a single ordinary differential equation may take many forms the simplest type of method is probably collocation 3 where an approximate solution is sought in some finite space subject to the constraint that it satisfy the differential equation at certain specified points this type of method has been applied very successfully in the case of mixed order systems of boundary value problems see for example 1 other methods can be described based on the standard l2 galerkin approximation 9 or on a combination of this and the collocation approach 7 13 25 in addition different weak formulations of the boundary value problem lead to two other techniques the hx and galerkin methods 12 14 15 17 how one would apply these methods to a system of differential equations with the exception of the collocation case is not immediately clear in addition it is not apparent what the relative advantages are of these various methods even when applied to a single equation in this paper we restrict our attention to the case of a single linear differential equation our aim is to investigate numerically five finite element techniques for a second order linear two point boundary value problem with separated boundary conditions of a general nature in particular we consider i the organization of the methods ii the treatment of the boundary conditions

an alternating direction implicit method for orthogonal spline collocation linear systems an alternating direction implicit method is analyzed for the solution of linear systems arising in high order tensor product orthogonal spline collocation applied to some separable second order linear elliptic partial differential equations in rectangles on annxn partition with jordan s selection of the acceleration parameters the method requireso n 2 ln 2 n arithmetic operations to produce an approximation whose accuracy in theh 1 norm is that of the collocation solution

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

linear stability analysis of a boundary crisis in a minimal chaotic flow abstract we present a mechanism that explains the boundary crisis presented by the first malasoma class p minimal chaotic flow which is equivalent to the simplest sprott flow the treatment followed resorts to the existence of two invariant sets of points of the flow in addition to the chaotic attractor to which linear stability theory is applied

second order splitting combined with orthogonal cubic spline collocation method for the kuramoto sivashinsky equation in this paper a second order splitting method is applied to the kuramoto sivashinsky equation and then an orthogonal cubic spline collocation procedure is employed to the approximate resulting system this semidiscrete method yields a system of differential algebraic equations daes of index 1 error estimates in l 2 and l c norms are obtained for the semidiscrete approximation for the time discretization the time integrator radau5 is used the results of numerical experiments are presented to validate the theoretical findings

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

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

some simple chaotic flows a systematic examination of general three dimensional autonomous ordinary differential equations with quadratic nonlinearities has uncovered 19 distinct simple examples of chaotic flows with either five terms and two nonlinearities or six terms and one nonlinearity the properties of these systems are described including their critical points lyapunov exponents and fractal dimensions

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

numerical methods for singular perturbation problems consider the two point boundary value problem for a stiff system of ordinary differential equations an adaptive method to solve these problems even when turning points are present is discussed

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

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

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

exponentially fitted methods applied to fourth order boundary value problems fourth order boundary value problems are solved by means of exponentially fitted methods of different orders these methods which depend on a parameter can be constructed following a six step flow chart of ixaru and vanden berghe special attention is paid to the expression for the error term and to the choice of the parameter in order to make the error as small as possible some numerical examples are given to illustrate the practical implementation issues of these methods

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

a preconditioned conjugate gradient method for nonselfadjoint or indefinite orthogonal spline collocation problems we study the computation of the orthogonal spline collocation solution of a linear dirichlet boundary value problem with a nonselfadjoint or an indefinite operator of the form lu sum a ij x u x ix j sum b i x u x i c x u we apply a preconditioned conjugate gradient method to the normal system of collocation equations with a preconditioner associated with a separable operator and prove that the resulting algorithm has a convergence rate independent of the partition step size we solve a problem with the preconditioner using an efficient direct matrix decomposition algorithm on a uniform n n partition the cost of the algorithm for computing the collocation solution within a tolerance epsilon is o n 2 ln n ln epsilon

unimodal order in the image of the simplest equivariant chaotic system the simplest equivariant chaotic dynamics is investigated in terms of its image i e under the 2 1 mapping allowing one to obtain a projection of the dynamics without any residual symmetry the inversion symmetry is therefore deleted the bifurcation diagram can thus be predicted from the unimodal order although the first return map computed in the original phase space exhibits three critical points this feature is the same as the one observed on the burke and shaw system although this latter system has a rotation symmetry

a smooth approximation based on exponential spline solutions for nonlinear fourth order two point boundary value problems methods of order two four and six based on exponential spline functions consisting of a polynomial part of degree three and an exponential part is developed to find approximation of linear and nonlinear fourth order two point boundary value problems it is shown that the free parameter k of the exponential part can be used to raise the order of accuracy of the new scheme convergence analysis of these methods is shown along with numerical examples each for linear and nonlinear are included to illustrate the practical usefulness of our methods

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 class of methods based on non polynomial spline functions for the solution of a special fourth order boundary value problems with engineering applications we use a quintic non polynomial spline functions to develop a numerical method for computing approximations to the solution of a system of fourth order boundary value problems associated with plate deflection theory we show that the present family of methods gives better approximations and generalize all the existing finite difference and spline functions based methods up to order six convergence of the methods is shown through standard convergence analysis numerical exampls are given to illustrate the applicability and efficiency of the new method

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

multilevel preconditioners for non self adjoint or indefinite orthogonal spline collocation problems efficient numerical algorithms are developed and analyzed that implement symmetric multilevel preconditioners for the solution of an orthogonal spline collocation osc discretization of a dirichlet boundary value problem with a non self adjoint or an indefinite operator the osc solution is sought in the hermite space of piecewise bicubic polynomials it is proved that the proposed additive and multiplicative preconditioners are uniformly spectrally equivalent to the operator of the normal osc equation the preconditioners are used with the preconditioned conjugate gradient method and numerical results are presented that demonstrate their efficiency

on stability constants and condition numbers of discretisation methods discretizations lead to systems of algebraic equations of steadily growing size if the accuracy of the approximation is increasing inverse stability constants and condition numbers describe the sensitivity of the solution of these equations with respect to perturbations of the data starting from an abstract result in this paper the inverse stability constants of finite difference and collocation methods for bvodes as well as of ritz galerkin methods for elliptic variational problems are shown to converge to those of their continuous counterparts as the discretization becomes finer and finer for two choices of basis systems the exact asymptotic behavior of the collocation method condition number is given

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

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

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

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

orthogonal spline collocation methods for schr o dinger type equations in one space variable we examine the use of orthogonal spline collocation for the semi discreti za tion of the cubic schr o dinger equation and the two dimensional parabolic equation of tappert in each case an optimal order l 2 estimate of the error in the semidiscrete approximation is derived for the cubic schr o dinger equation we present the results of numerical experiments in which the integration in time is performed using a routine from a software library