Derivative Multistep Methods Research Articles

Solving Volterra's Population Model Using New Second Derivative Multistep Methods

In this study new second derivative multistep methods (denoted SDMM) are used to solve Volterra's model for population growth of a species within a closed system. This model is a nonlinear integro-differential where the integral term represents the effect of toxin. This model is first converted to a nonlinear ordinary differential equation and then the new SDMM, which has good stability and accuracy properties, are applied to solve this equation. We compare this method with the others and show that new SDMM gives excellent results.

Stiffly stable continuous extension of second derivative linear multi-step methods with an off-step point for initial value problems in ordinary differential equations.

In this paper, we introduce a continuous extension of second derivative linear multi-step methods with a hybrid point for the numerical solution of initial valued stiff ordinary differential equations. The continuous extension is based on the Gear's fixed step size backward differential methods [7]. The intervals of absolute stability of methods of step number k&#88047 are determined using the root locus plot. Numerical results of the methods solving a non-linearly stiff initial value problem in ordinary differential equations are compared with that from the state-of-the-art ordinary differential equations code of MATLAB discussed in Higham et al [9]. JONAMP Vol. 11 2007: pp. 175-190

New second derivative multistep methods for stiff systems

In this paper we present details of a new class of second derivative multistep methods. Stability analysis is discussed and an improvement in stability region is obtained. With a simple modification we take advantage of calling for the solution of algebraic equations with the same coefficient matrix in each step. Moreover, using IOM to solve the resulting linear systems, the coefficient matrix is not needed explicitly. Some numerical experiments and comparison with several popular codes are given, showing strong superiority of this new class of methods.

New efficient second derivative multistep methods for stiff systems

The aim of this paper is to select from the large family of possible general linear methods, just a single class which has considerable potential for efficient implementation. This class has possible applications depending on stiff nature of a problem to be performed. A special class of second derivative multistep method (SDMM) is derived. The stability analysis of this class which is depending on free parameters is discussed. The stability regions are plotted for certain choices of parameters. A good comparison between the results of this class and the results due to Gear and Enright is recommended during some numerical tests.

On fitting stability regions of second derivative multi-step methods

A class of second-derivative two-step methods, S(2), with order 5 and depending on two parameters – which are used to adapt the stability region while minimizing the truncation error coefficient – is constructed. An analysis of two subclasses of S(2)–A(0) stable methods and methods with finite stability interval - with their stability and accuracy properties is presented. Numerical examples are also included.

A class of contractive second derivatives multistep methods

In this paper we study some properties of contractivity regions of second derivative multistep methods. These properties lead to the design of classes of A o and A contractive methods.

Stiffly stable second derivative multistep methods with higher order and improved stability regions

New stiffly stable second derivative multistep formulas with better stability regions and higher order are presented. These formulas of order as high as thirteen and with optimized stability regions are developed by using arbitrary parameters suitably implanted into the formulas. The stability properties of the formulas suggest their use especially for problems with Jacobians possessing eigenvalues close to the imaginary axis. The high order formulas will be specially useful when the accuracy requirement is stringent.

Polynomial Formulation of Second Derivative Multistep Methods

Following the work of Enright [3] there has been interest in studying second derivative methods for solving stiff ordinary differential equations. Successful implementations of second derivative methods have been reported by Enright [3], Sacks-Davis [9], [10] and Addison[l]. Wallace and Gupta [13] have suggested a polynomial formulation of the usual first-derivative multistep methods. Recently Skeel [11] has shown the equivalence of several formulations of multistep methods. The work of Wallace and Gupta [13] was extended to second derivative methods by Gupta [8]. The present work includes results obtained regarding the stability and truncation error of second derivative methods using the polynomial formulation.

Polynomial formulation of second derivative multistep methods

Following the work of Enright [3] there has been interest in studying second derivative methods for solving stiff ordinary differential equations. Successful implementations of second derivative methods have been reported by Enright [3], Sacks-Davis [9], [10] and Addison[l]. Wallace and Gupta [13] have suggested a polynomial formulation of the usual first-derivative multistep methods. Recently Skeel [11] has shown the equivalence of several formulations of multistep methods. The work of Wallace and Gupta [13] was extended to second derivative methods by Gupta [8]. The present work includes results obtained regarding the stability and truncation error of second derivative methods using the polynomial formulation.

Second Derivative Multistep Methods for Stiff Ordinary Differential Equations

The difficulty associated with the numerical solution of stiff ordinary differential equations is considered and the stability requirements of methods suitable for stiff equations are described. A class of second derivative formulas is developed and the stability of these formulas is investigated. These k-step $(k + 2)$nd order formulas are shown to be suitable for stiff equations for $k \leqq 7$. These formulas have been implemented in a variable order, variable-step method and some numerical results are presented.