Initial Value Problems For ODEs Research Articles

Energy-Conservative Algorithm for the Numerical Solution of Initial-Value Hamiltonian System Problems

The numerical treatment of ODE initial-value problems has been intensively researched. Energy-conservative algorithms are very important to dynamic systems. For the Hamiltonian system the symplectic algorithms are very effective. Powerful computers and algebraic software enable the creation of efficient numerical algorithms for solving ODE initial-value problems. In this paper, we propose an adaptive energy-conservative numerical-analytical algorithm for Hamiltonian systems. This algorithm is adaptable to initial-value problems where some quantities are preserved. The algorithm and its efficiency are presented for solving two-body and linear oscillator problems.

The BiM code for the numerical solution of ODEs

In this paper we present the code BiM, based on blended implicit methods (J. Comput. Appl. Math. 116 (2000) 41; Appl. Numer. Math. 42 (2002) 29; Recent Trends in Numerical Analysis, Nova Science Publ. Inc., New York, 2001, pp. 81.), for the numerical solution of stiff initial value problems for ODEs. We describe in detail most of the implementation strategies used in the construction of the code, and report numerical tests comparing the code BiM with some of the best codes currently available. The numerical tests show that the new code compares well with existing ones. Moreover, the methods implemented in the code are characterized by a diagonal nonlinear splitting, which makes its extension for parallel computers very straightforward.

Precisely A(α)-Stable One-Leg Multistep Methods

One-Leg Multistep (OLM) methods for initial value problems in ODEs use a nonlinear multistep formula to compute the solution at the next integration point. This paper shows that there exists an evaluation point t* which gives an OLM formula more precise than BDF's and (almost) precisely A(alpha)-stable for a k-step method (k less than or equal to 6), and whose stability angle is essentially similar to BDF's. The stability region can be further improved by applying the corrector idea of Klopfenstein.

Energy Conservative Methods for Numerical Solution of ODE‐s Based on Taylor series Algorithm

AbstractThe numerical solution of the ODE initial value problems is an intensively researched field. We have a lot of methods, eg. R‐K, Adams, symplectic but it isn't known well, how the approximate solutions behave from point of wiev of conservative quantities, for example energy [3], impulse etc. In this work we present an energy conservative algorithm based on the Taylor series method which was realized by MapleV system [2].

A remark on nondecidabilities of initial value problems of ODEs

A remark on nondecidabilities of initial value problems of ODEs

Automatic control and adaptive time-stepping

Adaptive time-stepping is central to the efficient solution of initial value problems in ODEs and DAEs. The error committed in the discretization method primarily depends on the time-step size h, which is varied along the solution in order to minimize the computational effort subject to a prescribed accuracy requirement. This paper reviews the recent advances in developing local adaptivity algorithms based on well established techniques from linear feedback control theory, which is introduced in a numerical context. Replacing earlier heuristics, this systematic approach results in a more consistent and robust performance. The dynamic behaviour of the discretization method together with the controller is analyzed. We also review some basic techniques for the coordination of nonlinear equation solvers with the primary stepsize controller in implicit time-stepping methods.

Unique solvability of nonlinear systems arising in the multiderivative block methods

We deal with the systems of nonlinear equations arising in the application of multiderivative block methods for initial value problems of ODEs. We give sufficient conditions for the unique solvability of the systems, and show that previous results become to special case of ours. A non-trivial example is given for the established result.

Initial value problems for ODEs in problem solving environments

A program is presented for solving initial value problems for ODEs numerically in Maple. We draw upon our experience with a number of closely related solvers to illustrate the differences between solving such problems in general scientific computation and in the problem solving environments Maple and M ATLAB.

Modified extended backward differentiation formulae for the numerical solution of stiff initial value problems in ODEs and DAEs

For many years the methods of choice for the numerical solution of stiff initial value problems and certain classes of differential algebraic equations have been the well-known backward differentiation formulae (BDF). More recently, however, new classes of formulae which can offer some important advantages over BDF have emerged. In particular, some recent large-scale independent comparisons have indicated that modified extended backward differentiation formulae (MEBDF) are particularly efficient for general stiff initial value problems and for linearly implicit DAEs with index ⩽3. In the present paper we survey some of the more important theory associated with these formulae, discuss some of the practical applications where they are particularly effective, e.g., in the solution of damped highly oscillatory problems, and describe some significant recent extensions to the applicability of MEBDF codes.

Interval analysis: theory and applications

We give an overview on applications of interval arithmetic. Among others we discuss verification methods for linear systems of equations, nonlinear systems, the algebraic eigenvalue problem, initial value problems for ODEs and boundary value problems for elliptic PDEs of second order. We also consider the item software in this field and give some historical remarks.

Dynamical Systems and Adaptive Timestepping in ODE Solvers

Initial value problems for ODEs are often solved numerically using adaptive timestepping algorithms. These algorithms are controlled by a user-defined tolerance which bounds from above the estimated error committed at each step. We formulate a large class of such algorithms as discrete dynamical systems which are discontinuous and of higher dimension than the underlying ODE. By assuming sufficiently strong finite-time convergence results on some neighbourhood of an attractor of the ODE we prove existence and upper semicontinuity results for a nearby numerical attractor as the tolerance tends to zero.

Highly Continuous Interpolants for One-Step ODE Solvers and their Application to Runge--Kutta Methods

We suggest a general method for the construction of highly continuous interpolants for one-step methods applied to the numerical solution of initial value problems of ODEs of arbitrary order. For the construction of these interpolants one uses, along with the numerical data of the discrete solution of a problem provided by a typical one-step method at endstep points, high-order derivative approximations of this solution. This approach has two main advantages. It allows an easy way of construction of high-order Runge--Kutta and Nystrom interpolants with reduced cost in additional function evaluations that also preserve the one-step nature of the underlying discrete ODE solver. Moreover, for problems which are known to possess a solution of high smoothness, the approximating interpolant resembles this characteristic, a property that on occasion might be desirable. An analysis of the stability behavior of such interpolatory processes is carried out in the general case. A new numerical technique concerning the accurate determinationof the stability behavior of numerical schemes involving higher order derivatives and/or approximations of the solution from previous grid-points over nonequidistant meshes is presented. This technique actually turns out to be of a wider interest, as it allows us to infer, in certain cases, more accurate results concerning the stability of, for example, the BDF formulas over variable stepsize grids. Moreover it may be used as a framework for analyzing more complex (and supposedly more promising) types of methods, as they are the general linear methods for first- and second-order differential equations. Many particular variants of the new method for first-order differential equations that have good prospects of finding a practical implementation are fully analyzed with respect to their stability characteristics. A detailed application concerning the construction of $C^2$ and $C^3$ continuous extensions for some fifth- and sixth-order Runge--Kutta pairs, supplemented by a detailed study of the local truncation error characteristics of a class of interpolants of this type, is also provided. Various numerical examples show, in these cases, several advantages of the newly proposed technique with respect to function evaluation cost and global error behavior, in comparison with others currently in use.

Parallel implementation of block boundary value methods for ODEs

The parallel solution of initial value problems for ODEs has been the subject of much research in the last thirty years, and different approaches to the problem have been devised. In this paper we examine the parallel methods derived by block boundary value methods (BVMs), recently introduced for approximating Hamiltonian problems. Here we restrict the analysis of the methods when applied to linear problems, since their nonlinear parallel implementation deserves further study. However, for linear problems, the methods can reach a high parallel efficiency. Some of these solvers can also be adapted for approximating continuous two-point boundary value problems. Numerical tests carried out on a distributed memory parallel computer are reported.

Parallel ODE solvers based on block BVMs

In this paper we deal with Boundary Value Methods (BVMs), which are methods recently introduced for the numerical approximation of initial value problems for ODEs. Such methods, based on linear multistep formulae (LMF), overcome the stability limitations due to the well-known Dahlquist barriers, and have been the subject of much research in the last years. This has led to the definition of a new stability framework, which generalizes the one stated by Dahlquist for LMF. Moreover, several aspects have been investigated, including the efficient stepsize control [17,25,26] and the application of the methods for approximating different kinds of problems such as BVPs, PDEs and DAEs [7,23,41]. Furthermore, a block version of such methods, recently proposed for approximating Hamiltonian problems [24], is able to provide an efficient parallel solver for ODE systems [3].

Boundary value methods for the solution of differential-algebraic equations

Boundary value techniques for the solution of initial value problems of ODEs, despite their apparent higher cost, present some important advantages over initial value methods. Among them, there is the possibility to have greater accuracy, to control the global error, and to have an efficient parallel implementation. In this paper, the same techniques are applied to the solution of linear initial value problems of DAEs. We have considered three term numerical methods (Midpoint, Simpson, and an Adams type method) in order to obtain a block tridiagonal linear system as a discrete problem. Convergence results are stated in the case of constant coefficients, and numerical examples are given on linear time-varying problems.

Stability properties of some boundary value methods

The boundary value methods (BVMs) are a class of numerical methods for solving initial value problems for ODEs. So far they did not have a broad diffusion, essentially for the following two reasons: their stability properties were not clearly understood and, moreover, they were considered too expensive. In this paper we shall analyze the stability properties of three particular BVMs when used for solving linear systems of ODEs. An efficient implementation of these methods will be described in a companion paper [6].

A parallel preconditioning technique for boundary value methods

The boundary value methods (BVMs) are a class of numerical methods for solving initial value problems for ODEs [3,7,14]. One reason that prevented their diffusion in the past years was their higher cost, with respect to the standard initial value methods. We show that BVMs may become competitive when they are efficiently implemented on parallel or vector computers. Since they require the solution of large, sparse block linear systems, usually obtained by using an iterative method, a preconditioning technique is needed for their efficient implementation. In this paper we introduce and study a new preconditioning technique. Some numerical tests on a distributed memory parallel computer are reported.

Optimal solution of ordinary differential equations

We survey some recent optimality results for the numerical solution of initial value problems for ODE. We assume that information used by an algorithm about a right-hand-side function is partial. Two settings of information-based complexity are considered: the worst case and asymptotic. Upper and lower bounds on the error are presented for three types of information: standard, linear, and nonlinear continuous. In both settings, minimum error algorithms are exhibited.

Matrix-Free Methods for Stiff Systems of ODE’s

We study here a matrix-free method for solving stiff systems of ordinary differential equations (ODE’s). In the numerical time integration of stiff ODE initial value problems by BDF methods, the resulting nonlinear algebraic system is usually solved by a modified Newton method and an appropriate linear system algorithm. In place of that, we substitute Newton’s method (unmodified) coupled with an iterative linear system method. The latter is a projection method called the Incomplete Orthogonalization Method (IOM), developed mainly by Y. Saad. A form of IOM, with scaling included to enhance robustness, is studied in the setting of Inexact Newton Methods. The implementation requires no Jacobian matrix storage whatever. Tests on several stiff problems, of sizes up to 16,000, show the method to be quite effective and much more economical, in both computational cost and storage, than standard solution methods, at least when the problem has a certain amount of clustering in its spectrum.

33 years of numerical instability, Part I

BIT has played and plays a great role in the development of concepts concerning numerical (in)stability in initial value problems forODE's and related questions. This development is here seen through the looking-glass of the author, who experienced much of its pains and pleasures. The article is based on a talk given in 1981 at the Zurich symposium to commemorate the tenth anniversary of the death of the eminent Swiss numerical analyst, Heinz Rutishauser. The presentation is mainly chronological with a few digressions. Part I ends at the beginning of the stiff epoch.