Exponential Multistep Methods

1 Exponential Multistep Methods. - For the numerical solution of the initial value problem 

Remark 2. (see refs. 1 and 2) Every exponential multistep method of order p corresponds, in a unique way, to an algebraic multistep method of order p (by setting v, = 0 for all i). 

Proposition 1. (For proof see refs. 2 and 4) Consider an operator L with 

1 The Derivation of Exponentially-Fitted Methods for General Problems. -Consider the construction of an exponentially-fitted multistep method (2) which exactly integrates the set of functions exp( v oc). =0. We will use this for the numerical solution of the general problem (1). 

We investigate here the case where is a positive number. Then, from Proposition 1 we have a set of equations  

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A. D. Raptis, Exponential multistep methods for ordinary differential equations. Bull. 

A. D. Raptis, Exponential multistep methods for ordinary differential equations, Bull. Greek Math. Soc., 1984, 25, 113-126. 

We also see that the new methods (Method IX, Method X and Method XI) are more efficient than other exponential multistep methods, or other methods of the same or higher algebraic order or other P-stable methods. However we conclude that higher exponential order crucial when integrating the Schrodinger equation. 

B. Stabilization of a multistep exponentially-fitted methods and their application to the Schrodinger equation... 

In this part of the paper we present the study of the stabilization of the multistep exponentially-fitted methods. More specifically we present a family of singularly P-stable exponentially-fitted four-step methods and a family of six-step exponentially-fitted methods for the numerical solution of the radial Schrodinger equation. 

J. Vigo-Aguiar and T. E. Simos, Family of twelve steps exponential fittingsymmetric multistep methods for the numerical solution of the Schrodinger equation, J. Math. Chem., 2002, 32(3), 257-270. 

In Section 2 we analyse a new category of methods named partitioned multistep methods. More specifically we derived exponentially-fitted partitioned multistep methods. For these methods numerical results on the problem of Stiefel and Bettis is presented. 

First Method of the Partitioned Multistep Method. 2.1.1 Exponential Fitting of First Order. We want the method (2) to integrate exactly any linear combination of the functions ... 

In 50 the author presents a further investigation of the frequency evaluation techniques which are recently proposed by Ixaru et al. for exponentially fitted multistep algorithms for the solution of first-order ordinary differential equations (ODEs). These studies have a scope which is to maximize the benefits of the exponentially-fitted methods via the evaluation of the frequency of the problem. The proposed by Ixaru and co-workers method for frequency... 

In chapter 3, the construction of exponential fitting formulae is presented. In chapter 4, applications of exponential fitting to differentiation, to integration and to interpolation are presented. In chapter 5, application of exponential fitting to multistep methods for the solution of differential equations is presented. Finally, in chapter 6, application of exponential fitting to Runge-Kutta methods for the solution of differential equations is presented. 

Exponentially Fitted and Trigonometrically Fitted Symplectic Linear Symmetric Multistep Methods. - The linear mulstistep methods are very important since they are also symplectic (see Sanz-Sema and Sanz-Sema et al ). We study here the exponentially fitted and trigonometrically fitted linear multistep methods. Consider the following nine-step linear symmetric multistep method ... 

From exponentially fitted and trigonometrically fitted methods the most efficient are the linear symmetric multistep methods. This is because these methods have two additional properties [symmetry (i.e. symplecticness) and non-empty interval of periodicity] to the dissipative methods and one additional property (non-empty interval of periodicity) to the symplectic methods. 

In [11-14,17,18,28,29,33,37,48-56] the first and the second methodology of this paper, the well known exponentially and trigonometrically-fitting is presented. Multistep methods of several orders are developed. 

For all the above cases the authors produced a local truncation error analysis. From this analysis it has been concluded that the exponentially-fitted one-step Obrechkoff method of the third exponential order is the most accurate one. In the same paper a stability analysis was given. From this analysis it has been summarized that the new exponentially-fitted one-step Obrechkoff methods are P-stable. A phase-lag analysis was also presented. The conclusion was that the new developed methods are of sixth phase-lag order (i.e. the same as the algebraic order). Numerical results have shown that the new one-step Obrechkoff methods developed in this paper was much more accurate than multistep methods. This is very interesting since it was known, generally, that one-step methods was less efficient than the corresponding multistep formulae. 

Definition 4 The multistep method (48) is called algebraic or exponential) of order p if the associated linear operator L vanishes for any linear combination of the linearly independent functions l,x,x, . . . ... 

Other methods, which have not yet been used in chemical kinetics, include global or passive extrapolation (see Sect. 4.5.7), averaging methods, multistep, multiderivatives methods, exponential fitting and non-linear methods (see, for example, ref. 176 for references). 

Another version of this static headspace chromatography is what has been called by Kolb multiple headspace extraction (MHE) chromatography. This is a multistep injection technique which was alluded to in the Suzuki publication and more openly developed by McAuliffe. The principle of this method is the following. After the first extraction has been made and the aliquot injected, the gas phase is removed by ventilating the vial and re-establishing the thermodynamic equilibrium. The equilibrium between the analyte in the solid or liquid phase and the gas phase will be displaced each time. After n extractions the analyte content in the liquid or solid phase becomes negligible. It is flien sufficient to sum the peak areas obtained for each extraction (which decrease exponentially) and, from an external calibration curve, determine the amount of RS in the substance. 

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