Integration Of Initial Value Problems Research Articles

Comparing the Performance of Two Butcher-Based Block Hybrid Algorithms for the Solution of Initial Value Problems

In this paper, we compare the performances of two Butcher-based block hybrid methods for the numerical integration of initial value problems. We compare the condition numbers of the linear system of equations arising from both methods and the absolute errors of the solution obtained. The results of the numerical experiments illustrate that the better conditioned method outperformed its less conditioned counterpart based on the absolute errors. In addition, after applying our method on some examples, it was discovered that the absolute errors in this work were better than those of a recent study in the literature. Hence, we recommend this method for the numerical solution of stiff and non-stiff initial value problems.

Generalized TASE-RK methods for stiff problems

A family of Time-Accurate and Stable Explicit (TASE) methods for the numerical integration of Initial Value Problems in stiff Ordinary Differential Equations (ODEs) y′(t)=f(t,y) was recently introduced in [1]. The main idea was to make local extrapolation of a stabilized Euler method. More recently, in [3] a similar approach by considering the stabilization of arbitrary explicit Runge-Kutta methods (TASE-RK) was analyzed. In this case the explicit Runge-Kutta method integrates a transformed ODE obtained by multiplying the vector field f(t,y) by a certain operator which approximates the identity mapping up to a given order p. The main inconvenience of both approaches is that to reach order p the solution of p2 linear systems plus the evaluation of p derivatives are required per integration step.In order to substantially reduce the computational costs of the former approaches in the linear system solution, but maintaining the good accuracy and stability properties, a new family of TASE-RK methods which allow to introduce a few more free parameters are considered. The formulation of the methods was conceived to be implemented not only in sequential mode but it admits parallelism in a straightforward way. Furthermore, since these methods are linearly implicit, connections to the class of W-methods [19] are properly established. The order conditions for the new class of methods are widely studied by using the rooted tree theory. For p=3,4, new methods with p sequential stages and order p are derived and compared on semidiscrete 1D and 2D Partial Differential Equations (PDEs) to those in [1,3] and other standard Rosenbrock and W-methods in the literature.

Optimized Hybrid Block Adams Method for Solving First Order Ordinary Differential Equations

Multistep integration methods are being extensively used in the simulations of high dimensional systems due to their lower computational cost. The block methods were developed with the intent of obtaining numerical results on numerous points at a time and improving computational efficiency. Hybrid block methods for instance are specifically used in numerical integration of initial value problems. In this paper, an optimized hybrid block Adams block method is designed for the solutions of linear and nonlinear first-order initial value problems in ordinary differential equations (ODEs). In deriving the method, the Lagrange interpolation polynomial was employed based on some data points to replace the differential equation function and it was integrated over a specified interval. Furthermore, the convergence properties along with the region of stability of the method were examined. It was concluded that the newly derived method is convergent, consistent, and zero-stable. The method was also found to be A-stable implying that it covers the whole of the left/negative half plane. From the numerical computations of absolute errors carried out using the newly derived method, it was found that the method performed better than the ones with which we compared our results with. The method also showed its superiority over the existing methods in terms of stability and convergence.

Subspace adaptivity in Rosenbrock–Krylov methods for the time integration of initial value problems

The Rosenbrock–Krylov family of time integration schemes is an extension of Rosenbrock-W methods that employs a specific Krylov based approximation of the linear system solutions arising within each stage of the integrator. This work proposes an extension of Rosenbrock–Krylov methods to address stability questions which arise for methods making use of inexact linear system solution strategies. Two approaches for improving the stability and efficiency of Rosenbrock–Krylov methods are proposed, one through direct control of linear system residuals and the second through a novel extension of the underlying Krylov space to include stage right hand side vectors. Rosenbrock–Krylov methods employing the new approaches show a substantial improvement in computational efficiency relative to prior implementations.

Composition Methods for Dynamical Systems Separable into Three Parts

New families of fourth-order composition methods for the numerical integration of initial value problems defined by ordinary differential equations are proposed. They are designed when the problem can be separated into three parts in such a way that each part is explicitly solvable. The methods are obtained by applying different optimization criteria and preserve geometric properties of the continuous problem by construction. Different numerical examples exhibit their improved performance with respect to previous splitting methods in the literature.

Variable step size commutator free Lie group integrators

We introduce variable step size commutator free Lie group integrators, where the error control is achieved using embedded Runge–Kutta pairs. These are schemes for the integration of initial value problems posed on homogeneous spaces by means of Lie group actions. The focus is on commutator free methods, in which the approximation evolves by composing flows generated by Lie group exponentials. Such methods are encoded by a generalization of Butcher’s Runge–Kutta tableaux, but it is known that more order conditions must be satisfied to obtain a scheme of a given order than are required for classical RK schemes. These extra considerations complicate the task of designing embedded pairs. Moreover, whilst the computational cost of RK schemes is typically dominated by function evaluations, in most situations, the dominant cost of commutator free Lie group integrators comes from computing Lie group exponentials. We therefore give Butcher tableaux for several families of methods of order 3(2) and 4(3), designed with a view to minimizing the number of Lie group exponentials required at each time step, and briefly discuss practical error control mechanisms. The methods are then applied to a selection of examples illustrating the expected behaviour.

Numerical Strategies for the System of First Order IVPs Using Block Hybrid Extended Trapezoidal Multistep Method of Second Kind for Stiff ODEs

A Modified Three Step Block Hybrid Extended Trapezoidal Multistep Method of Second Kind (BHETR₂s) with two off-grid points, one at interpolation and another at collocation point yielding uniform order six (6, 6, 6, 6, 6)^T for the Numerical Integration of initial value problems of stiff Ordinary Differential Equations was developed. The main method and additional equations were obtained from the same continuous formulation through interpolation and collocation procedures. The stability properties of the method was discussed and from the stability region obtained, the method is suitable for the solution Stiff Ordinary Differential Equations. Three numerical examples were considered to illustrate the efficiency and accuracy.

Predictor – Corrector Methods of High Order for Numerical Integration of Initial Value Problems

Predictor – Corrector Methods of High Order for Numerical Integration of Initial Value Problems

Projector based integration of DAEs with the Taylor series method using automatic differentiation

Automatic (or Algorithmic) Differentiation (AD) opens new possibilities to analyze and solve DAEs by projector based methods. In this paper, we present a new approach to compute consistent initial values and integrate DAEs up to index two, considering the nonlinear DAE in each time-step as a nonlinear system of equations for Taylor expansions. These systems will be solved by the Newton–Kantorowitsch method, whereas the resulting linear systems are decoupled using the splitting techniques related to the tractability index concept. This approach provides a description of the inherent ODE that allows an application of the classical Taylor series method to the integration of initial value problems. Linear and nonlinear DAEs with index up to two are examined and solved numerically.

Optimal stability polynomials for numerical integration of initial value problems

We consider the problem of finding optimally stable polynomial approximations to the exponential for application to one-step integration of initial value ordinary and partial differential equations. The objective is to find the largest stable step size and corresponding method for a given problem when the spectrum of the initial value problem is known. The problem is expressed in terms of a general least deviation feasibility problem. Its solution is obtained by a new fast, accurate, and robust algorithm based on convex optimization techniques. Global convergence of the algorithm is proven in the case that the order of approximation is one and in the case that the spectrum encloses a starlike region. Examples demonstrate the effectiveness of the proposed algorithm even when these conditions are not satisfied.

Two-step modified collocation methods with structured coefficient matrices

In the context of the numerical integration of initial value problems based on ordinary differential equations, it is the purpose of this paper to introduce a modification of two-step collocation methods, in order to obtain coefficient matrices with a structured shape, to get an efficient implementation. Our aim is the development of new collocation-based methods having high order of convergence and strong stability properties (e.g. A-stability and L-stability). We present the constructive technique, discuss the order of convergence and the stability properties of the resulting methods and provide some numerical results confirming the theoretical expectations.

New 4(3) Pairs Diagonally Implicit Runge-Kutta-Nyström Method for Periodic IVPs

New 4(3) pairs Diagonally Implicit Runge-Kutta-Nyström (DIRKN) methods with reduced phase-lag are developed for the integration of initial value problems for second-order ordinary differential equations possessing oscillating solutions. Two DIRKN pairs which are three- and four-stage with high order of dispersion embedded with the third-order formula for the estimation of the local truncation error. These new methods are more efficient when compared with current methods of similar type and with the L-stable Runge-Kutta pair derived by Butcher and Chen (2000) for the numerical integration of second-order differential equations with periodic solutions.

A class of collocation methods for numerical integration of initial value problems

For the numerical solution of initial value problems a general procedure to determine global integration methods is derived and studied. They are collocation methods which can be easily implemented and provide a high order accuracy. They further provide globally continuous differentiable solutions. Computation of the integrals which appear in the coefficients are generated by a recurrence formula and no integrals are involved in the calculation. Numerical experiments provide favorable comparisons with other existing methods.

Analysis of a Chebyshev-based backward differentiation formulae and relation with Runge–Kutta collocation methods

The construction of a class of backward differentiation formulae based on intra-step Chebyshev–Gauss–Lobatto nodal points, suitable for the approximate numerical integration of initial-value problems of first-order ordinary differential equations, is presented. Formulae of this new family are A 0-stable and L(α)-stable for any orders, and, particularly, for orders 1 and 2 they are L-stable. Regions of absolute stability and stability measures make this class very promising. We prove that this family of methods may be considered as Runge–Kutta collocation methods where the abscissae are obtained from the Chebyshev–Gauss–Lobatto points.

A Collocation Method for Direct Numerical Integration of Initial Value Problems in Higher Order Ordinary Differential Equations

A Collocation Method for Direct Numerical Integration of Initial Value Problems in Higher Order Ordinary Differential Equations This paper concerns the solution of initial value problems (IVPs) in ordinary differential equations (ODEs) of orders higher than unity. The Chebyshev polynomials is hereby adopted as basis function in a multi-step collocation technique for the derivation of continuous integration schemes for direct solution of these ODEs without recourse to the conventional approach of first reducing such to their equivalent first order differential systems.

New embedded explicit pairs of exponentially fitted Runge–Kutta methods

Two new embedded pairs of exponentially fitted explicit Runge–Kutta methods with four and five stages for the numerical integration of initial value problems with oscillatory or periodic solutions are developed. In these methods, for a given fixed ω the coefficients of the formulae of the pair are selected so that they integrate exactly systems with solutions in the linear space generated by { sinh ( ω t ) , cosh ( ω t ) } , the estimate of the local error behaves as O ( h 4 ) and the high-order formula has fourth-order accuracy when the stepsize h → 0 . These new pairs are compared with another one proposed by Franco [J.M. Franco, An embedded pair of exponentially fitted explicit Runge–Kutta methods, J. Comput. Appl. Math. 149 (2002) 407–414] on several problems to test the efficiency of the new methods.

ON NAIVE TAYLOR MODEL INTEGRATION METHOD

Interval methods for verified integration of initial value problems (IVPs) for ODEs have been used for more than 40 years. For many classes of IVPs, these methods have the ability to compute guaranteed error bounds for the flow of an ODE, where traditional methods provide only approximations to a solution. Overestimation, however, is a potential drawback of verified methods. For some problems, the computed error bounds become overly pessimistic, or integration even breaks down. The dependency problem and the wrapping effect are particular sources of overestimations in interval computations. Berz (see [1]) and his co-workers have developed Taylor model methods, which extend interval arithmetic with symbolic computations. The latter is an effective tool for reducing both the dependency problem and the wrapping effect. By construction, Taylor model methods appear particularly suitable for integrating nonlinear ODEs. In this paper, we analyze Taylor model based integration of ODEs and compare Taylor model with traditional enclosure methods for IVPs for ODEs. More advanced Taylor model integration methods are discussed in the algorithm (1). For clarity, we summarize the major steps of the naive Taylor model method as algorithm 1.

ON NAIVE TAYLOR MODEL INTEGRATION METHOD

Interval methods for verified integration of initial value problems (IVPs) for ODEs have been used for more than 40 years. For many classes of IVPs, these methods have the ability to compute guaranteed error bounds for the flow of an ODE, where traditional methods provide only approximations to a solution. Overestimation, however, is a potential drawback of verified methods. For some problems, the computed error bounds become overly pessimistic, or integration even breaks down. The dependency problem and the wrapping effect are particular sources of overestimations in interval computations. Berz (see [1]) and his co-workers have developed Taylor model methods, which extend interval arithmetic with symbolic computations. The latter is an effective tool for reducing both the dependency problem and the wrapping effect. By construction, Taylor model methods appear particularly suitable for integrating nonlinear ODEs. In this paper, we analyze Taylor model based integration of ODEs and compare Taylor model with traditional enclosure methods for IVPs for ODEs. More advanced Taylor model integration methods are discussed in the algorithm (1). For clarity, we summarize the major steps of the naive Taylor model method as algorithm 1.

Robust and reliable defect control for Runge-Kutta methods

The quest for reliable integration of initial value problems (IVPs) for ordinary differential equations (ODEs) is a long-standing problem in numerical analysis. At one end of the reliability spectrum are fixed stepsize methods implemented using standard floating point, where the onus lies entirely with the user to ensure the stepsize chosen is adequate for the desired accuracy. At the other end of the reliability spectrum are rigorous interval-based methods, that can provide provably correct bounds on the error of a numerical solution. This rigour comes at a price, however: interval methods are generally two to three orders of magnitude more expensive than fixed stepsize floating-point methods. Along the spectrum between these two extremes lie various methods of different expense that estimate and control some measure of the local errors and adjust the stepsize accordingly. In this article, we continue previous investigations into a class of interpolants for use in Runge-Kutta methods that have a defect function whose qualitative behavior is asymptotically independent of the problem being integrated. In particular the point, in a step, where the maximum defect occurs as h → 0 is known a priori. This property allows the defect to be monitored and controlled in an efficient and robust manner even for modestly large stepsizes. Our interpolants also have a defect with the highest possible order given the constraints imposed by the order of the underlying discrete formula. We demonstrate the approach on three Runge-Kutta methods of orders 5, 6, and 8, and provide Fortran and preliminary Matlab interfaces to these three new integrators. We also consider how sensitive such methods are to roundoff errors. Numerical results for four problems on a range of accuracy requests are presented.

On Taylor Model Based Integration of ODEs

Interval methods for verified integration of initial value problems (IVPs) for ODEs have been used for more than 40 years. For many classes of IVPs, these methods are able to compute guaranteed error bounds for the flow of an ODE, where traditional methods provide only approximations to a solution. Overestimation, however, is a potential drawback of verified methods. For some problems, the computed error bounds become overly pessimistic, or the integration even breaks down. The dependency problem and the wrapping effect are particular sources of overestimations in interval computations. Berz and his coworkers have developed Taylor model methods, which extend interval arithmetic with symbolic computations. The latter is an effective tool for reducing both the dependency problem and the wrapping effect. By construction, Taylor model methods appear particularly suitable for integrating nonlinear ODEs. We analyze Taylor model based integration of ODEs and compare Taylor model methods with traditional enclosure methods for IVPs for ODEs.