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

optimal a priori error estimates for the hp version of the local discontinuous galerkin method for convection diffusion problems we study the convergence properties of the hp version of the local discontinuous galerkin finite element method for convection diffusion problems we consider a model problem in a one dimensional space domain we allow arbitrary meshes and polynomial degree distributions and obtain upper bounds for the energy norm of the error which are explicit in the mesh width h in the polynomial degree p and in the regularity of the exact solution we identify a special numerical flux for which the estimates are optimal in both h and p the theoretical results are confirmed in a series of numerical examples

unification of boolean monotone anti monotone and tensor independence and levy processes the unification of the boolean and the tensor product of quantum probability spaces given by lenczewski len98 is modified to include the monotone product and its mirror image the anti monotone product and used to reduce the theories of boolean monotone and anti monotone levy processes on dual semi groups to the theory of levy processes on involutive bialgebras this leads to a representation theorem for these processes further applications are a proof of the schoenberg correspondence for boolean monotone and anti monotone convolution semi groups a realization of boolean monotone and anti monotone creation annihilation and conservation operators on boson fock spaces and a discussion of the markov structure of these levy processes

remarks on the r and δ convolutions in this paper we study the properties of the r deformation introduced in b1 we observe that the associated convolution coming from the conditionally free convolution is associative only for r 1 and r 0 we give the realization of some r gaussian random variables and obtain haagerup pisier buchholz type inequalities we also study another convolution defined with the use of the r deformation through a moment cumulant formula ky1 and show that it is associative and in general not positive

piecewise linearized methods for initial value problems piecewise linearized methods for the solution of initial value problems in ordinary differential equations are developed by approximating the right hand sides of the equations by means of a taylor polynomial of degree one the resulting approximation can be integrated analytically to obtain the solution in each interval and yields the exact solution for linear problems three adaptive methods based on the norm of the jacobian matrix maintaining constant the value of the approximation errors incurred by the linearization of the right hand sides of the ordinary differential equations and richardson s extrapolation are developed numerical experiments with some nonstiff first and second order ordinary differential equations indicate that the accuracy of piecewise linearized methods is in general superior to those of the explicit modified second order accurate euler method and the implicit trapezoidal rule but lower than that of the explicit fourth order accurate runge kutta technique it is also shown that piecewise linearized methods do not exhibit computational i e spurious modes for the relaxation oscillations of the van der pol oscillator and for those systems of equations which satisfy certain conservation principles conserve more accurately the invariants than the trapezoidal rule an error bound for piecewise linearized methods is provided for ordinary differential equations whose right hand sides satisfy certain lipschitz conditions

discontinuous galerkin methods for ordinary differential equations a class of galerkin methods derived from discontinuous piecewise polynomial spaces is analyzed for polynomials of degree k these methods lead to a family of one step schemes generating approximations up to order 2k 2 for the solution of an ordinary differential equation

fourth order time stepping for stiff pdes a modification of the exponential time differencing fourth order runge kutta method for solving stiff nonlinear pdes is presented that solves the problem of numerical instability in the scheme as proposed by cox and matthews and generalizes the method to nondiagonal operators a comparison is made of the performance of this modified exponential time differencing etd scheme against the competing methods of implicit explicit differencing integrating factors time splitting and fornberg and driscoll s sliders for the kdv kuramoto sivashinsky burgers and allen cahn equations in one space dimension implementation of the method is illustrated by short matlab programs for two of the equations it is found that for these applications with fixed time steps the modified etd scheme is the best

solving linear ordinary differential equations by exponentials of iterated commutators a sequence of transformations of a linear system of ordinary differential equations is investigated it is shown that these transformations produce new systems which represent progressively smaller perturbations of the original set of equations the transformations are implemented as a basis of a numerical method order stability and error control of this method are analyzed numerical examples demonstrate the potential of this approach

locally linearized runge kutta method of dormand and prince in this paper the effect that produces the local linearization of the embedded runge kutta formulas of dormand and prince for initial value problems is studied for this embedded locally linearized runge kutta formulas are defined and their performance is analyzed by means of exhaustive numerical simulations for a variety of well known test equations with different dynamics the simulation results show that the locally linearized formulas exhibit significant higher accuracy than the original ones which implies a substantial reduction of the number of time steps and consequently a sensitive reduction of the overall computational cost of their adaptive implementation

numerical simulation of nonlinear dynamical systems driven by commutative noise the local linearization ll approach has become an effective technique for the numerical integration of ordinary random and stochastic differential equations one of the reasons for this success is that the ll method achieves a convenient trade off between numerical stability and computational cost besides the ll method reproduces well the dynamics of nonlinear equations for which other classical methods fail however in the stochastic case most of the reported works has been focused in stochastic differential equations sde driven by additive noise this limits the applicability of the ll method since there is a number of interesting dynamics observed in equations with multiplicative noise on the other hand recent results show that commutative noise sdes can be transformed into a random differential equation rde by means of a random diffeomorfism conjugacy this paper takes advantages of such conjugacy property and the ll approach for defining a ll scheme for sdes driven by commutative noise the performance of the proposed method is illustrated by means of numerical simulations

nodal superconvergence of the local discontinuous galerkin method for singularly perturbed problems abstract in this paper a superconvergence of order ln n n 2 k 1 for the numerical traces of the ldg approximation to a one dimensional singularly perturbed convection diffusion reaction problem is proved the ldg method is applied on a shishkin mesh with 2 n elements and we use polynomials of degree at most k on each element this result puts the numerical finding reported in xie and zhang 2007 xie et al 2009 on firm mathematical ground

asymptotically exact a posteriori ldg error estimates for one dimensional transient convection diffusion problems in this paper new a posteriori error estimates for the local discontinuous galerkin ldg formulation applied to transient convection diffusion problems in one space dimension are presented and analyzed these error estimates are computationally simple and are computed by solving a local steady problem with no boundary conditions on each element we first show that the leading error term on each element for the solution is proportional to a p 1 degree right radau polynomial while the leading error term for the solution s derivative is proportional to a p 1 degree left radau polynomial when polynomials of degree at most p are used these results are used to prove that for smooth solutions these error estimates at a fixed time converge to the true spatial errors in the l 2 norm under mesh refinement more precisely we prove that our ldg error estimates converge to the true spatial errors at o h p 5 4 rate finally we prove that the global effectivity indices in the l 2 norm converge to unity at o h 1 2 rate our computational results indicate that the observed numerical convergence rates are higher than the theoretical rates

differential systems with fuchsian linear part correction and linearization normal forms and multiple orthogonal polynomials abstract differential systems with a fuchsian linear part are studied in regions including all the singularities in the complex plane of these equations such systems are not necessarily analytically equivalent to their linear part they are not linearizable and obstructions are found as a unique nonlinear correction after which the system becomes formally linearizable more generally normal forms are found the corrections and the normal forms are found constructively expansions in multiple orthogonal polynomials and their generalization to matrix valued polynomials are instrumental to these constructions

a higher order local linearization method for solving ordinary differential equations the local linearization ll method for the integration of ordinary differential equations is an explicit one step method that has a number of suitable dynamical properties however a major drawback of the ll integrator is that its order of convergence is only two the present paper overcomes this limitation by introducing a new class of numerical integrators called the llt method that is based on the addition of a correction term to the ll approximation in this way an arbitrary order of convergence can be achieved while retaining the dynamic properties of the ll method in particular it is proved that the llt method reproduces correctly the phase portrait of a dynamical system near hyperbolic stationary points to the order of convergence the performance of the introduced method is further illustrated through computer simulations

efficient simulation of unsaturated flow using exponential time integration abstract we assess the performance of an exponential integrator for advancing stiff semidiscrete formulations of the unsaturated richards equation in time the scheme is of second order and explicit in nature but requires the action of the matrix function a a 1 e a i on a suitability defined vector v at each time step when the matrix a is large and sparse a v can be approximated by krylov subspace methods that require only matrix vector products with a we prove that despite the use of this approximation the scheme remains second order furthermore we provide a practical variable stepsize implementation of the integrator by deriving an estimate of the local error that requires only a single additional function evaluation numerical experiments performed on two dimensional test problems demonstrate that this implementation outperforms second order variable stepsize implementations of the backward differentiation formulae

convergence rate of weak local linearization schemes for stochastic differential equations with additive noise there exists a diversity of weak local linearization ll schemes for the integration of stochastic differential equations with additive noise which differ in the algorithms employed for the numerical implementation of the weak local linear discretizations despite convergence results for these discretizations have been already developed the convergence of the weak ll schemes has not been considered up to date in this work a general result concerning the convergence rate of the weak ll schemes is presented as well as specificities for a number of particular schemes as an application the convergence of weak ll schemes for equations driven by poisson processes is presented in addition

a projection based error analysis of hdg methods we introduce a new technique for the error analysis of hybridizable discontinuous galerkin hdg methods the technique relies on the use of a new projection whose design is inspired by the form of the numerical traces of the methods this renders the analysis of the projections of the discretization errors simple and concise by showing that these projections of the errors are bounded in terms of the distance between the solution and its projection our studies of influence of the stabilization parameter are reduced to local analyses of approximation by the projection we illustrate the technique on a specific hdg method applied to a model second order elliptic problem

efficient computation of phi functions in exponential integrators abstract a crucial point for the computational efficiency of the exponential type integrators for differential equations is the effective evaluation of the integrals involving matrix exponential called phi functions focusing on exponential integrators for differential equations of small dimensions for which the krylov based approximations for the phi functions are not viable in this article refined algorithms are introduced for speeding up both the evaluation of several phi functions simultaneously and the computation of a linear combination of phi functions times matrices the algorithms are derived from a meticulous approximation to the exponential of certain partitioned matrix by adapting the conventional pade method to the special structure of this matrix numerical simulations are provided to evaluate the performance of the proposed algorithms and compare them with others in the literature

pade approximation for the exponential of a block triangular matrix abstract in this work we obtain improved error bounds for pade approximations to e a when a is block triangular as a result improved scaling strategies ensue which avoid some common overscaling difficulties

rate of convergence of local linearization schemes for initial value problems there is a large variety of local linearization ll schemes for the numerical integration of initial value problems which differ with respect to the algorithm that is used in the numerical implementation of the local linear discretization however in contrast with the ll discretization the order of convergence of the ll schemes have not been studied so far in this paper a general result about that matter is given in addition a brief survey of the main implementations of the ll method is also presented

exponential integrators for large systems of differential equations we study the numerical integration of large stiff systems of differential equations by methods that use matrix vector products with the exponential or a related function of the jacobian for large problems these can be approximated by krylov subspace methods which typically converge faster than those for the solution of the linear systems arising in standard stiff integrators the exponential methods also offer favorable properties in the integration of differential equations whose jacobian has large imaginary eigenvalues we derive methods up to order 4 which are exact for linear constant coefficient equations the implementation of the methods is discussed numerical experiments with reaction diffusion problems and a time dependent schrodinger equation are included

decompositions of the free additive convolution we introduce and study a new type of convolution of probability measures denoted and called the s free additive convolution which is defined by the subordination functions associated with the free additive convolution we derive an alternating decomposition of for compactly supported and using another convolution called orthogonal additive convolution this decomposition leads to two types of complete alternating decompositions of the free additive convolution more importantly we develop an operatorial approach to the subordination property and introduce the associated notion of s free independence moreover we establish relations between convolutions associated with the main notions of noncommutative independence free monotone and boolean finally our result leads to natural decompositions of the free product of rooted graphs

on krylov subspace approximations to the matrix exponential operator krylov subspace methods for approximating the action of matrix exponentials are analyzed in this paper we derive error bounds via a functional calculus of arnoldi and lanczos methods that reduces the study of krylov subspace approximations of functions of matrices to that of linear systems of equations as a side result we obtain error bounds for galerkin type krylov methods for linear equations namely the biconjugate gradient method and the full orthogonalization method for krylov approximations to matrix exponentials we show superlinear error decay from relatively small iteration numbers onwards depending on the geometry of the numerical range the spectrum or the pseudospectrum the convergence to exp tau a v is faster than that of corresponding krylov methods for the solution of linear equations i tau a x v which usually arise in the numerical solution of stiff ordinary differential equations odes we therefore propose a new class of time integration methods for large systems of nonlinear differential equations which use krylov approximations to the exponential function of the jacobian instead of solving linear or nonlinear systems of equations in every time step

quadrature methods for stiff ordinary differential systems the quadrature methods are based upon a substitution of an explicit a stable first approximation into a generalized convolution formula they are a stable explicit and of arbitrarily high order the generalized convolution formula is derived and its order raising properties examined two families of explicit a stable first approximations are developed generalizing results of lawson and n0rsett various aspects of the numerical implementation are discussed numerical results supplement the paper and exemplify the various merits and weaknesses of the quadrature methods

exponential time differencing for stiff systems we develop a class of numerical methods for stiff systems based on the method of exponential time differencing we describe schemes with second and higher order accuracy introduce new runge kutta versions of these schemes and extend the method to show how it may be applied to systems whose linear part is nondiagonal we test the method against other common schemes including integrating factor and linearly implicit methods and show how it is more accurate in a number of applications we apply the method to both dissipative and dispersive partial differential equations after illustrating its behavior using forced ordinary differential equations with stiff linear parts

the discontinuous petrov galerkin method for elliptic problems abstract we propose a novel discontinuous mixed finite element formulation for the solution of second order elliptic problems fully discontinuous piecewise polynomial finite element spaces are used for the trial and test functions the discontinuous nature of the test functions at the element interfaces allows to introduce new boundary unknowns that on the one hand enforce the weak continuity of the trial functions and on the other avoid the need to define a priori algorithmic fluxes as in standard discontinuous galerkin methods static condensation is performed at the element level leading to a solution procedure based on the sole interface unknowns the resulting family of discontinuous dual primal mixed dpm finite element methods is presented in the one and two dimensional cases in the one dimensional case we show the equivalence of the method with implicit runge kutta schemes of the collocation type exhibiting optimal behavior and we develop the stability analysis numerical experiments in one and two dimensions demonstrate the order accuracy of the new method confirming the results of the analysis

explicit exponential runge kutta methods for semilinear parabolic problems the aim of this paper is to analyze explicit exponential runge kutta methods for the time integration of semilinear parabolic problems the analysis is performed in an abstract banach space framework of sectorial operators and locally lipschitz continuous nonlinearities we commence by giving a new and short derivation of the classical nonstiff order conditions for exponential runge kutta methods but the main interest of our paper lies in the stiff case by expanding the errors of the numerical method in terms of the solution we derive new order conditions that form the basis of our error bounds for parabolic problems we show convergence for methods up to order four and we analyze methods that were recently presented in the literature these methods have classical order four but they do not satisfy some of the new conditions therefore an order reduction is expected we present numerical experiments which show that this order reduction in fact arises in practical examples based on our new conditions we finally construct methods that do not suffer from order reduction

a matricial exponentially fitted scheme for the numerical solution of stiff initial value problems abstract we present an exponentially fitted scheme for the numerical integration of stiff systems of initial value problems the basic algorithm consists of an explicit one step method which requires the computation of the exponential of a matrix this task is achieved using ward s scaling and squaring algorithm 1 the results of some numerical experiments are presented which allow an appraisal of the proposed approach with other methods

a discontinuous hp finite element method for convection diffusion problems this paper presents a new method which exhibits the best features of both finite volume and finite element techniques special attention is given to the issues of conservation flexible accuracy and stability the method is elementwise conservative the order of polynomial approximation can be adjusted element by element and the stability is not based on the introduction of artificial diffusion but on the use of a very particular finite element formulation with discontinuous basis functions the method supports h p and hp approximations and can be applied to any type of meshes including non matching grids a priori error estimates and numerical experiments on representative model problems indicate that the method is robust and capable of delivering high accuracy

collocation and relaxed collocation for the fer and the magnus expansions we consider the fer and the magnus expansions for the numerical solution of the nonlinear matrix lie group ode y gamma t y y y 0 y 0 where y evolves in a matrix lie group g and gamma t y is in the lie algebra hbox fr g ghe departing from a geometrical approach that distinguishes between those operations performed in the group and those performed in the tangent space we construct lie group invariant methods based on collocation we prove that as long as the two expansions are correctly truncated the collocation nodes c 1 c 2 ldots c nu yield numerical methods whose order is the same as in the classical setting we also relax the collocation conditions thereby devising explicit methods of order 3 to conclude we discuss the proposed methods in a numerical experiment that arises in hamiltonian mechanics comparing the results with projection techniques

analysis of a family of discontinuous galerkin methods for elliptic problems the one dimensional case in this paper we analyze a family of discontinuous galerkin methods parameterized by two real parameters for elliptic problems in one dimension our main results are 1 a complete inf sup stability analysis characterizing the parameter values yielding a stable scheme and energy norm error estimates as a direct consequence thereof 2 an analysis of the error in l2 where the standard duality argument only works for special parameter values yielding a symmetric bilinear form and different orders of convergence are obtained for odd and even order polynomials in the nonsymmetric case the analysis is consistent with numerical results and similar behavior is observed in two dimensions

cumulants in noncommutative probability theory i noncommutative exchangeability systems cumulants linearize convolution of measures we use a formula of good to define noncommutative cumulants in a very general setting it turns out that the essential property needed is exchangeability of random variables roughly speaking the formula says that cumulants are moments of a certain discrete fourier transform of a random variable this provides a simple unified method to understand the known examples of cumulants like classical free and various q cumulants

free products of c algebras small spatial free products of c algebras are constructed under certain conditions they have properties similar to those proved by paschke and salinas for the algebras c gi g2 where g1 g2 are discrete groups the freeproduct analogs of noncommutative bernoulli shifts are discussed 0 introduction let k be a field consider the category of unital algebras over k it is well known that this category admits coproducts free products of algebras 2 heuristically the free product of algebras is the algebra generated by them with no relations except for the identification of unit elements if k c the complex numbers and we consider unital algebras we can easily define a operation on the free products let a b be unital c algebras and a b their free product which is a unital algebra the question arises in what ways may one define a pre c norm on a b that extends the norms on a and b guided by analogy with tensor products we expect to have a choice among many pre c norms giving rise to many c free products of a and b one natural norm is 1c i i sup 1it c ih st representation of a b the representations of a b are in 1 1 correspondence with pairs of representations of a and b which act on the same hilbert space let a b be the completion of a b in this norm it is easy to see that this construction defines a coproduct in the category of c algebras and that a b is the biggest free product of a and b analogous to the biggest tensor product a 0 b if g 1 g2 are discrete groups we obtain c g1 c g2 c gi g2 where g1 g2 is the free product group and this is analogous to the relation c g1 0 c g2 c g1 x g2 this paper is motivated by the question is there a smallest c product a b in analogy to the smallest tensor product a b satisfying a relation cr gi g2 cr gi cr g2 received by the editors january 28 1981 and in revised form may 19 1981 1980 mathematics subject classification primary 46l05 secondary 46l55 1 982 american mathematical society 0002 9947 82 00001022 04 25

dynamic properties of the local linearization method for initial value problems some dynamic properties of the local linearization ll scheme for the numerical integration of initial value problems in ordinary differential equations odes are investigated specifically the general conditions under which this scheme preserves the stationary points and periodic orbits of the odes and the local stability at these steady states are studied these dynamic properties are also examined by means of numerical experiments and the results are compared with those achieved by other numerical schemes in addition a brief review of the computational implementations of the ll scheme is also presented

on a class of free levy laws related to a regression problem abstract the free meixner laws arise as the distributions of orthogonal polynomials with constant coefficient recursions we show that these are the laws of the free pairs of random variables which have linear regressions and quadratic conditional variances when conditioned with respect to their sum we apply this result to describe free levy processes with quadratic conditional variances and to prove a converse implication related to asymptotic freeness of random wishart matrices

operator khintchine inequality in non commutative probability abstract for the operator t sum n k 1 a k otimes r k where a k belongs to the schatten class s 2n and where r k are non commutative random variables with mixed moments satisfying a specific condition we prove the following khintchine inequality vert t vert 2n leq d 2n left left vert left sum n k001 a ka k right frac 1 2 right vert s 2n quad quad left vert left sum n k 1 a k a k right frac 1 2 right vert s 2n right we find the optimal constants d 2n in the case when r k are the q gaussian and circular random variables moreover we show that the moments of any probability symmetric measure appear as the optimal constants for some random variables

an a priori error analysis of the local discontinuous galerkin method for elliptic problems in this paper we present the first a priori error analysis for the local discontinuous galerkin ldg method for a model elliptic problem for arbitrary meshes with hanging nodes and elements of various shapes we show that for stabilization parameters of order one the l2 norm of the gradient and the l2 norm of the potential are of order k and k 1 2 respectively when polynomials of total degree at least k are used if stabilization parameters of order h 1 are taken the order of convergence of the potential increases to k 1 the optimality of these theoretical results is tested in a series of numerical experiments on two dimensional domains

singular perturbation analysis of time dependent convection diffusion equations in a circle abstract we study singularly perturbed time dependent convection diffusion equations in a circular domain considering suitable compatibility conditions we present convergence results and provide as well approximation schemes and error estimates to resolve the oscillations of classical numerical solutions due to the stiffness of our problem we construct via a specific boundary layer analysis the so called boundary layer elements which absorb the boundary layer singularities using a p1 classical finite element space enriched with the boundary layer elements we obtain an accurate numerical solution using a quasi uniform mesh that is without refinement of the mesh in the boundary layer

a discontinuoushpfinite element method for diffusion problems we present an extension of the discontinuous galerkin method which is applicable to the numerical solution of diffusion problems the method involves a weak imposition of continuity conditions on the solution values and on fluxes across interelement boundaries within each element arbitrary spectral approximations can be constructed with different orderspin each element we demonstrate that the method is elementwise conservative a property uncharacteristic of high order finite elements for clarity we focus on a model class of linear second order boundary value problems and we developpriorierror estimates convergence proofs and stability estimates the results of numerical experiments onh andp convergence rates for representative two dimensional problems suggest that the method is robust and capable of delivering exponential rates of convergence

uniform superconvergence analysis of the discontinuous galerkin method for a singularly perturbed problem in 1 d it has been observed from the authors numerical experiments 2007 that the local discontinuous galerkin ldg method converges uniformly under the shishkin mesh for singularly perturbed two point boundary problems of the convection diffusion type especially when using a piecewise polynomial space of degree k the ldg solution achieves the optimal convergence rate k 1 1 under the l 2 norm and a superconvergence rate 2k 1 1 for the one sided flux uniformly with respect to the singular perturbation parameter in this paper we investigate the theoretical aspect of this phenomenon under a simplified ode model in particular we establish uniform convergence rates 1nn n k 1 for the l 2 norm and ln n n 2k 1 for the one sided flux inside the boundary layer region here n even is the number of elements

local linearization method for numerical integration of delay differential equations in this paper a new approach for the numerical computation of delay differential equations ddes is introduced the essential idea consists of obtaining numerical integrators that use a code expressly developed for linear ddes in contrast with the conventional approach of using a code for ordinary differential equations specifically two numerical schemes of this new class of integrators are proposed and their numerical viability analyzed it includes the estimation of the convergence rate the evaluation of the computational cost of the schemes and a simulation study it is proved that these one step explicit integrators converge uniformly with order two to the solution of nonlinear ddes and are able to integrate stiff equations in a satisfactory way with low computational cost

a superconvergence result for discontinuous galerkin methods applied to elliptic problems this paper presents a theoretical and numerical study of a class of discontinuous galerkin methods that shows the approximation of the gradient superconverges at the zeros of the legendre polynomials on a model 1d elliptic problem numerical experiments validate the theoretical results

letter to the editor computing multiple integrals involving matrix exponentials in this paper a generalization of a formula proposed by van loan computing integrals involving the matrix exponential ieee trans automat control 23 1978 395 404 for the computation of multiple integrals of exponential matrices is introduced in this way the numerical evaluation of such integrals is reduced to the use of a conventional algorithm to compute matrix exponentials the formula is applied for evaluating some kinds of integrals that frequently emerge in a number classical mathematical subjects in the framework of differential equations numerical methods and control engineering applications

a uniformly convergent continuous discontinuous galerkin method for singularly perturbed problems of convection diffusion type abstract in this paper we introduce a coupled approach of local discontinuous galerkin and standard finite element method for solving singularly perturbed convection diffusion problems on shishkin mesh with linear elements a rate o n 1 ln n in an associated norm is established where n is the number of elements numerical experiments complement the theoretical results moreover a rate o n 2 ln 2 n in a discrete l norm and o n 2 in l2 norm are observed numerically on the shishkin mesh

on the numerical approximation of phase portraits near stationary points we show that the phase portrait of a dynamical system near a stationary hyperbolic point is reproduced correctly by numerical methods such as one step methods or multi step methods satisfying a strong root condition this means that any continuous trajectory can be approximated by an appropriate discrete trajectory and vice versa to the correct order of convergence and uniformly on arbitrarily large time intervals in particular the stable and unstable manifolds of the discretization converge to their continuous counterparts

atomft solving odes and daes using taylor series abstract taylor series methods compute a solution to an initial value problem in ordinary differential equations by expanding each component of the solution in a long taylor series the series terms are generated recursively using the techniques of automatic differentiation the atomft system includes a translator to transform statements of the system of odes into a fortran 77 object program that is compiled linked with the atomft runtime library and run to solve the problem we review the use of the atomft system for nonstiff and stiff odes the propagation of global errors and applications to differential algebraic equations arising from certain control problems to boundary value problems to numerical quadrature and to delay problems

a discontinuous galerkin method for higher order ordinary differential equations in this paper we propose a new discontinuous finite element method to solve initial value problems for ordinary differential equations and prove that the finite element solution exhibits an optimal o tp 1 convergence rate in the l2 norm we further show that the p degree discontinuous solution of differential equation of order m and its first m 1 derivatives are o t2p 2 m superconvergent at the end of each step we also establish that the p degree discontinuous solution is o tp 2 superconvergent at the roots of p 1 m degree jacobi polynomial on each step finally we present several computational examples to validate our theory and construct asymptotically correct a posteriori error estimates

superconvergence of the numerical traces of discontinuous galerkin and hybridized methods for convection diffusion problems in one space dimension in this paper we uncover and study a new superconvergence property of a large class of finite element methods for one dimensional convection diffusion problems this class includes discontinuous galerkin methods defined in terms of numerical traces discontinuous petrov galerkin methods and hybridized mixed methods we prove that the so called numerical traces of both variables superconverge at all the nodes of the mesh provided that the traces are conservative that is provided they are single valued in particular for is local discontinuous galerkin method we show that the superconvergence is order 2p 1 when polynomials of degree at most p are used extensive numerical results verifying our theoretical results are displayed