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

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

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

stability of an elastic rod buckling into a soft wall the conjugate point theory of the calculus of variations is extended to apply to the buckling of an elastic rod in an external field we show that the operator approach presented by manning mannin

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

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

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

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

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

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

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

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

theory of boundary effects in invasion percolation we study the boundary effects in invasion percolation ip with and without trapping we find that the presence of boundaries introduces a new set of surface critical exponents as in the case of standard percolation numerical simulations show a fractal dimension for the region of the percolating cluster near the boundary remarkably different from the bulk one in fact on the surface we find a value of for ip with trapping compared with the bulk value of we find a logarithmic crossover from surface to bulk fractal properties as one would expect from the finite size theory of critical systems the distribution of the quenched variables on the growing interface near the boundary self organizes into an asymptotic shape characterized by a discontinuity at a value which coincides with the bulk critical threshold the exponent of the boundary avalanche distribution for ip without trapping is this value is very near to the bulk one then we conclude that only the geometrical properties fractal dimension of the model are affected by the presence of a boundary while other statistical and dynamical properties are unchanged furthermore we are able to present a theoretical computation of the relevant critical exponents near the boundary this analysis combines two recently introduced theoretical tools the fixed scale transformation and the run time statistics which are particularly suited for the study of irreversible self organized growth models with quenched disorder our theoretical results are in rather good agreement with numerical data

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

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