Four step implicit block method of Runge-Kutta type for solving first order ordinary differential equations

In this paper, a four step implicit block method for solving first order ordinary differential equations (ODEs) is proposed. The method approximates the solutions of initial value problems at four-point mesh simultaneously using variable step size. This four step implicit method is of the multistep type but it is implemented as the Runge-Kutta type. The stability regions of the method are also studied. Numerical results are presented to show the efficiency of the proposed block method

Parallel Block Methods for Solving Ordinary Differential Equations

In this thesis, new and efficient codes are developed for solving Initial Value Problems (IVPs) of first and higher order Ordinary Differential Equations (ODEs) using variable step size. The new codes are based on the implicit multistep block methods formulae. Subsequently, a more structured and efficient algorithm comprising the block methods was constructed for solving systems of first order ODEs using variable step size and order. The new codes were then used for the parallel implementation in solving large systems of first and higher order ODEs. The sequential programs of these methods were executed on DYNIXlptx operating system. The parallel programs were run on a Sequent Symmetry SE30 parallel computer.The Cq stability in the multistep method was introduced and the focused was on the error propagation from a more practical angle. The numerical results showed that the sequential implementation of the new codes could reduce the total number of steps and execution times even when solving small systems of first and higher order ODEs compared with the 1-point method and the existing 2PBVSO code in Omar (1 999). The parallel implementation of the codes was found to be most appropriate in solving large systems of first and higher order ODEs. It was also discovered that the maximum speed up of the parallel methods improved as the dimension of the ODEs systems increased. In conclusion, the new codes developed in this thesis are suitable for solving systems of first and higher order ODE

Variable Step Variable Order Two Point Block Fully Implicit Method for Solving Ordinary Differential Equations

The aim of this paper is to investigate the performance of the developed two point block methods of order 5, 7 and 9 for solving first order Ordinary Differential Equations (ODEs) using variable step size and order. The code will combine three proposed block methods i.e the 2-point one block fully implicit block method of order 5, the 2-point two block fully implicit block method of order 7 and the 2-point three block fully implicit block method of order 9. These methods will estimate the numerical solution at two equally spaced points simultaneously within a block. The existence multistep method involves the computations of the divided differences and integration coefficients when using the variable step size or variable step size and order. The block method developed will be presented in the simple form of the Adams Moulton type. The performances of the code will be compared in terms of maximum error, total number of steps and execution times with the existence non block method and 2-point block method of variable step size and order code

Blended General Linear Methods based on Boundary Value Methods in the GBDF family

Among the methods for solving ODE-IVPs, the class of General Linear Methods (GLMs) is able to encompass most of them, ranging from Linear Multistep Formulae (LMF) to RK formulae. Moreover, it is possible to obtain methods able to overcome typical drawbacks of the previous classes of methods. For example, order barriers for stable LMF and the problem of order reduction for RK methods. Nevertheless, these goals are usually achieved at the price of a higher computational cost. Consequently, many efforts have been made in order to derive GLMs with particular features, to be exploited for their efficient implementation. In recent years, the derivation of GLMs from particular Boundary Value Methods (BVMs), namely the family of Generalized BDF (GBDF), has been proposed for the numerical solution of stiff ODE-IVPs. In particular, this approach has been recently developed, resulting in a new family of L-stable GLMs of arbitrarily high order, whose theory is here completed and fully worked-out. Moreover, for each one of such methods, it is possible to define a corresponding Blended GLM which is equivalent to it from the point of view of the stability and order properties. These blended methods, in turn, allow the definition of efficient nonlinear splittings for solving the generated discrete problems. A few numerical tests, confirming the excellent potential of such blended methods, are also reported.Comment: 22 pages, 8 figure

A 5-Step Block Predictor and 4-Step Corrector Methods for Solving General Second Order Ordinary Differential Equations

A 5-step block predictor and 4-step corrector methods aimed at solving general second order ordinary differential equations directly will be constructed and implemented on non-stiff problems. This method, which extends the work of block predictor-corrector methods using variable step size technique possess some computational advantages of choosing a suitable step size, deciding the stopping criteria and error control. In addition, some selected theoretical properties of the method will be investigated as well as determination of the region of absolute stability. Numerical results will be given to show the efficiency of the new metho

One step hybrid block methods with generalised off-step points for solving directly higher order ordinary differential equations

Real life problems particularly in sciences and engineering can be expressed in differential equations in order to analyse and understand the physical phenomena. These differential equations involve rates of change of one or more independent variables. Initial value problems of higher order ordinary differential equations are conventionally solved by first converting them into their equivalent systems of first order ordinary differential equations. Appropriate existing numerical methods will then be employed to solve the resulting equations. However, this approach will enlarge the number of equations. Consequently, the computational complexity will increase and thus may jeopardise the accuracy of the solution. In order to overcome these setbacks, direct methods were employed. Nevertheless, most of these methods approximate numerical solutions at one point at a time. Therefore, block methods were then introduced with the aim of approximating numerical solutions at many points simultaneously. Subsequently, hybrid block methods were introduced to overcome the zero-stability barrier occurred in the block methods. However, the existing one step hybrid block methods only focus on the specific off-step point(s). Hence, this study proposed new one step hybrid block methods with generalised off-step point(s) for solving higher order ordinary differential equations. In developing these methods, a power series was used as an approximate solution to the problems of ordinary differential equations of order g. The power series was interpolated at g points while its highest derivative was collocated at all points in the selected interval. The properties of the new methods such as order, error constant, zero-stability, consistency, convergence and region of absolute stability were also investigated. Several initial value problems of higher order ordinary differential equations were then solved using the new developed methods. The numerical results revealed that the new methods produced more accurate solutions than the existing methods when solving the same problems. Hence, the new methods are viable alternatives for solving initial value problems of higher order ordinary differential equations directly

Block Milne’s Implementation For Solving Fourth Order Ordinary Differential Equations

Block predictor-corrector method for solving non-stiff ordinary differential equations (ODEs) started with Milne’s device. Milne’s device is an extension of the block predictor corrector method providing further benefits and better results. This study considers Milne’s devise for solving fourth order ODEs. A combination of Newton’s backward difference interpolation polynomial and numerical integration method are applied and integrated at some selected grid points to formulate the block predictor-corrector method. Moreover, Milne’s devise advances the computational efficiency by applying the chief local truncation error] (CLTE) of the block predictor-corrector method after establishing the order. The numerical results were exhibited to attest the functioning of Milne’s devise in solving fourth order ODEs. The complete results were obtained with the aid of Mathematica 9 kernel for Microsoft Windows. Numerical results showcase that Milne’s device is more effective than existent methods in terms of design new step size, determining the convergence criteria and maximizing errors at all examined convergence levels

Numerical algorithm of block method for general second order ODEs using variable step size

This paper outlines an alternative algorithm for solving general second order ordinary differential equations (ODEs). Normally, the numerical method was designed for solving higher order ODEs by converting it into an n-dimensional first order equations with implementation of constant step length. Nevertheless, this involved a lot of computational complexity which led to consumption a lot of time. Consequently, a direct block multistep method with utilization of variable step size strategy is proposed. This method was developed for computing the solution at four points simultaneously and the derivation based on numerical integration as well as using interpolation approach. The convergence of the proposed method is justified under suitable conditions of stability and consistency. Five numerical examples are considered and some comparisons are made with the existing methods for demonstrating the validity and reliability of the proposed algorithm

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

Numerical Solution of ODEs and the Columbus' Egg: Three Simple Ideas for Three Difficult Problems

On computers, discrete problems are solved instead of continuous ones. One must be sure that the solutions of the former problems, obtained in real time (i.e., when the stepsize h is not infinitesimal) are good approximations of the solutions of the latter ones. However, since the discrete world is much richer than the continuous one (the latter being a limit case of the former), the classical definitions and techniques, devised to analyze the behaviors of continuous problems, are often insufficient to handle the discrete case, and new specific tools are needed. Often, the insistence in following a path already traced in the continuous setting, has caused waste of time and efforts, whereas new specific tools have solved the problems both more easily and elegantly. In this paper we survey three of the main difficulties encountered in the numerical solutions of ODEs, along with the novel solutions proposed.Comment: 25 pages, 4 figures (typos fixed