Stable predictor-corrector methods for first order ordinary differential equations

Because of the wide variety of differential equations, there seems to be no numerical method which will affect the solution best for all problems. Predictor-corrector methods have been developed which utilize more ordinates in the predictor and corrector equations in the search for a better method. These methods are compared for stability and convergence with the well known methods of Milne, Adams, and Hamming --Abstract, page ii

A fourth-order one-step block method for solving a stiff ordinary differential equation

In this paper, we propose an A-stable one-step block method of order four for solving a stiff ordinary differential equation. This method will approximate the solutions of a stiff ordinary differential equation at three points simultaneously using a constant step size. The method is similar to the one-step method and it is self-starting but the implementation is based on the predictor-and-corrector formulae. Several problems have been tested in this paper to prove the efficiency and accuracy of this method. Numerical results are presented to illustrate the performance of the proposed method. The results clearly show that the proposed method is able to produce comparable and better results compared to the existing methods when solving stiff differential equations

A One Step Method for the Solution of General Second Order Ordinary Differential Equations

In this paper, an implicit one step method for the numerical solution of second order initial value problems of ordinary differential equations has been developed by collocation and interpolation technique. The introduction of an o step point guaranteed the zero stability and consistency of the method. The implicit method developed was implemented as a block which gave simultaneous solutions, as well as their rst derivatives, at both o step and the step point. A comparison of our method to the predictor-corrector method after solving some sample problems reveals that our method performs better

One step multiderivative methods for first order ordinary differential equations

A family of one-step multiderivative methods based on Padé approximants to the exponential function is developed. The methods are extrapolated and analysed for use in PECE mode. Error constants and stability intervals are calculated and the combinations compared with well known linear multi-step combinations and combinations using high accuracy Newton-Cotes quadrature formulas as correctors. w926020

Exponential integration algorithms applied to viscoplasticity

Four, linear, exponential, integration algorithms (two implicit, one explicit, and one predictor/corrector) are applied to a viscoplastic model to assess their capabilities. Viscoplasticity comprises a system of coupled, nonlinear, stiff, first order, ordinary differential equations which are a challenge to integrate by any means. Two of the algorithms (the predictor/corrector and one of the implicits) give outstanding results, even for very large time steps

An efficient numerical algorithm for the transient analysis of high-frequency non-linear circuits

The paper proposes a new approach for the discrete-time integration of non-linear differential equations that describe the behaviour of high-frequency circuits, in particular those containing complex equivalent-circuit models of microwave transistor devices. The proposed approach reformulates a conventional predictor-corrector method in terms of Pad�© approximates about each function sample. The method is especially suited to the kind of non-linear stiff differential equations that arise frequently in high-frequency analysis

Multiderivative methods for periodic initial value problems

A family of two-step multiderivative methods based on Pade approximants to the exponential function is developed. The methods are analysed and periodicity intervals in PECE mode are calculated. Two of the methods are tested on two problems from the literature and one predictor-corrector combination is tested on two further problems

Optimization of Nordsieck's Method for the Numerical Integration of Ordinary Differential Equations

Stability and accuracy of Nordsieck's integration method can be improved by choosing the zero-positions of the extraneous roots of the characteristic equation in a suitable way. Optimum zero-positions have been found by minimizing the lower bound of the interval of absolute stability and the coefficient of the truncation error. Various properties of the improved methods have been analysed, such as the behaviour of the high-order terms, the equivalence with multistep methods and the damping of perturbations