Extended one-step methods for the numerical solution of ordinary differential equations

A class of one-step finite difference formulae for the numerical solution of first-order differential equations is considered. The accuracy and stability properties of these methods are investigated. By judicious choice of the coefficients in these formulae a method is derived which is both A-stable and third-order convergent. Moreover the new method is shown to be L-stable and so is appropriate for the solution of certain stiff equations. Numerical results are presented for several test problems.     

A parallel multistep predictor-corrector algorithm for solving ordinary differential equations

In this paper we give a generalized predictor-corrector algorithm for solving ordinary differential equations with specified initial values. The method uses multiple correction steps which can be carried out in parallel with a prediction step. The proposed method gives a larger stability interval compared to the existing parallel predictor-corrector methods. A method has been suggested to implement the algorithm in multiple processor systems with efficient utilization of all the processors.     

A unified computer program for the solution of ordinary differential equations

This paper describes a compact program for the solution of ordinary differential equations. The program generalizes all one-step methods including Euler's Heun's, Ralston's, Runge-Kutta, Gill's and other approaches. The idea is to consider any individual method as a particular case of a more general ‘weighted-Average-Slope’ method. A list of the program in BASIC along with an example problem to illustrate its versatility are included.     

A new operator splitting method for the numerical solution of partial differential equations II

We extend the applications of a new method for splitting operators in partial differential equations introduced by us (A. Rouhi and J. Wright, A new operator splitting method for the numerical solution of partial differential equations, Comput. Phys. Commun. 85 (1995) 18–28, and Spectral implementation of a new operator splitting method for solving partial differential equations, Comput. Phys. (1995), to be published.) to equations in two spatial dimensions, and show how the method allows the use of explicit time stepping methods in some instances when other methods require implicit time stepping. This odd-even splitting method also enables one to increase the order of accuracy of time stepping in a straightforward manner. Our main examples will be the two-dimensional Navier-Stokes equations and the shallow water equations. In the first example we show how the pressure term can be dealt with in simple geometries. We will then discuss the treatment of the diffusion term. Next we will discuss how fast waves can be treated by explicit methods using the odd-even splitting, while retaining all stability and accuracy advantages of usual implicit methods. Our example here will be the shallow water equations in two dimensions.     

A method of iteration for boundary value solution of ordinary differential equations

A globally stable iterative search algorithm for finding the unknown initial values in the two-point boundary-value problem (TPBVP) arising in ordinary differential equations has been developed. The problem can be embedded in a discrete-time feedback control system, the steady state of which is the required solution. Algorithms are found through the construction of globally stable digital controllers.     

An efficient algorithm for solving stiff ordinary differential equations

The Brayton-Gustavson-Hatchel (BGH) method for solving stiff ordinary differential equations belongs to the group of backward difference formulas methods. Basic details of the BGH method are presented. A new implementation with original modifications is described. Special attention is paid to the reduction of operation count and improvement of error control. Two examples including mildly stiff and stiff equation systems prove spectacular superiority of the BGH method with respect to the classic Gear method. The software presented in this paper is scalable and has been ported without any problems from the PC/DOS platform to two UNIX environments.     

The evaluation of numerical techniques for solution of stiff ordinary differential equations arising from chemical kinetic problems

Ordinary differential equations with widely scattered eigenvalues (stiff O.D.E.'s) occur often in the studies of reaction network problems. Five numerical methods, including two methods based on Backward differentiation formulas, a modified Runge-Kutta-Fehlberg method, a method based on PECE Adams formulas, and an improved semi-implicit Euler method are evaluated by comparing their performance when applied to test systems. The test systems represent different combinations of linearity and nonlinearity, small and large dimension, real and complex eigenvalues, and slightly stiff and very stiff problems. The relative merits and dificiencies of the methods are discussed.     

New stability theorems concerning one-step numerical methods for ordinary differential equations

A measure of the stability properties of numerical integration methods for ordinary differential equations is provided by their stability region, which is that region in the complex (Δtλ) plane for which a given method is stable when applied to the differential equation

$\text{d}\text{y}\text{d}\text{t=λy}$

with a time-step Δt.Free parameters which exist in numerical integration algorithms may be used to maximize, in some sense, the size of the stability region, rather than increasing the order of accuracy, as is usually done. We derive new results which set theoretical limits to this maximization process for one step, explicit methods.Specifically, if K is the number of function evaluation invoked, then:(i) we prove (Theorem 1) that if ? is the radius of the largest circle, tangent to the imaginary axis at the origin of the complex plane that is contained in the stability region S, then ? cannot exceed K.(ii) we also prove (Theorem 2) that the imaginary stability boundary S_I (or maximum stable value of ∣Δtλ∣ with λ imaginary) cannot exceed (K ? 1).While Theorem 1 is to our knowledge new, a limited form of theorem 2 (K odd only) had been established in v.d. Houwen (1977). That the maximum imaginary boundary S_I = (K ? 1) is attainable had been shown (constructively) for K odd. We show that this maximum is also reached for K = 2 and K = 4, and correct in the process an erroneous result in the above reference.     

An implicit method for numerical solution of differential equations

The paper presents an implicit method for numerical solution of differential equations, based on the use of the derivatives of the right-hand side jointly with a directed motion in the discrepancy space on passing to the next integration layer. Two examples of solving equations by the proposed method are considered.     

Evaluation of a class of iterative methods for the numerical solution of Boundary value problems for ordinary differential equations

The aim of this paper is an experimental evaluation and comparison among several iterative methods proposed for approximate solution of two-point boundary value problems for ordinary differential equations.     

Uniqueness for ordinary differential equations

Various criteria are known for assuring uniqueness of the solution of a system ofn ordinary differential equations,x = f(t, x), with initial conditionx(t ₀) = x₀. Most of these involve some sort of relaxed Lipschitz condition onf(t, x), with respect tox, valid on an open setD R ¹⁺ⁿ which contains the point (t ₀, x₀). The present paper generalizes (and unifies) a number of known uniqueness criteria to cover cases when (t ₀, x₀) lies on the boundary ofD. Research partially supported by NSF Grant GP-37838.     

The shooting method for the solution of ordinary differential equations: A control-theoretical perspective

A discrete space state representation is used to provide a feedback control-theoretical formulation for the iterative shooting method used to solve two-point boundary value problems in ordinary differential equations. Preconditioning matrices, error analysis and convergence conditions for these iterative methods are put into a feedback control perspective as well.     

Unsupervised kernel least mean square algorithm for solving ordinary differential equations

In this paper a novel method is introduced based on the use of an unsupervised version of kernel least mean square (KLMS) algorithm for solving ordinary differential equations (ODEs). The algorithm is unsupervised because here no desired signal needs to be determined by user and the output of the model is generated by iterating the algorithm progressively. However, there are several new approaches in literature to solve ODEs but the new approach has more advantages such as simple implementation, fast convergence and also little error. Furthermore, it is also a KLMS with obvious characteristics. In this paper the ability of KLMS is used to estimate the answer of ODE. First a trial solution of ODE is written as a sum of two parts, the first part satisfies the initial condition and the second part is trained using the KLMS algorithm so as the trial solution solves the ODE. The accuracy of the method is illustrated by solving several problems. Also the sensitivity of the convergence is analyzed by changing the step size parameters and kernel functions. Finally, the proposed method is compared with neuro-fuzzy [21] approach.