Linear multistep methods

1 Two-step Methods. - The exponentially-fitted methods of this category are of the form 

Using (12) a group of exponentially-fitted methods has been produced. These are found in  

It is instructive to note that Numerov s method integrates exactly 

The following general algorithm applies to the solution of ODEs [5-7] 

Depending on the values of Ki and K2, either the Adams-Moulton (AM) or BD method is obtained [7]  

Applying the conditions (Equations A2.12 and A2.13) to Equation A2.11 yields the AM method, 

Both methods are implicit, since f is usually a nonlinear function of y . Methods A2.14 and A2.15 require the solution of a nonlinear, algebraic equation system of the following kind  

After solving y ,a is known from the earlier solutions the problem thus comprises an iterative solution of equation system A2.16. This is best facilitated by the Newton-Raphson method 

---

For simultaneous solution of (16), however, the equivalent set of DAEs (and the problem index) changes over the time domain as different constraints are active. Therefore, reformulation strategies cannot be applied since the active sets are unknown a priori. Instead, we need to determine a maximum index for (16) and apply a suitable discretization, if it exists. Moreover, BDF and other linear multistep methods are also not appropriate for (16), since they are not self-starting. Therefore, implicit Runge-Kutta (IRK) methods, including orthogonal collocation, need to be considered. 

The initial-value problem is solved using new, highly accurate formulas of the linear multistep method. 

F. Mazzia, A. Sestini and D. Trigiante, BS linear multistep methods on non-uniform meshes, JNAIAM J. Numer. Anal. Indust. Appl. Math., 2006, 1(1), 131-144. 

D. S. Vlachos and T. E. Simos, Partitioned Linear Multistep Method for Long Term Integration of the N-Body Problem, Appl. Num. Anal. Comp. Math., 2004,1(2), 540-546. 

In 55 the authors present a new procedure for construction of P-stable linear multistep methods for periodic initial-value problems. This procedure is based on the requirement the characteristic equations produced by the methods to have roots of specific form. 

For stiff differential equations, the backward difference algorithm should be preferred to the Adams-Moulton method. The well-known code LSODE with different options was published in 1980 s by Flindmarsh for the solution of stiff differential equations with linear multistep methods. The code is very efficient, and different variations of it have been developed, for instance, a version for sparse systems (LSODEs). In the international mathematical and statistical library, the code of Hindmarsh is called IVPAG and DIVPAG. 

In [215] the author developed a three-step seventh algebraic order hybrid linear multistep method (HLMM) with three non-step points for the approximate solution special of special second order initial value problems. The author produced the main method and additional methods from the same scheme derived via interpolation and collocation techniques. The local truncation error is presented, the zero stability and the convergence and consistency properties are studied. Numerical experiments show the efficiency of the proposed algorithm. 

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 ... 

For the families of symmetric linear multistep methods developed in this review we have that the polynomials p and o are given by (66) where A = 8 and... 

In Figures 1-5 and based on the above theory and on the coefficients given above we present the stabihty polynomial for the Linear Multistep Methods LMMI - LMMM... 

Table 1 Properties of Symmetric Linear Multistep Methods...

Numerical Illustrations for Linear Multistep Methods and Dissipative Methods... 

As a conclusion we can say that symmetric (non-dissipative) methods are more efficient because they have non-empty interval of periodicity and because they are symplectie (in the case of linear symmetric multistep methods). We note also that symmetric linear multistep methods are very simple in programming and have very low computational cost (only one function evaluation per step). 

These moment equations are typically integrated by linear multistep methods, such as the Adams method and the backward differentiation formula (Petzold, 1983). The right-hand sides of Eqs. (10.6), (10.7) are directly related to the selected reaction scheme. As an example. Scheme 10.1 shows the main reaction steps for a simplified chain-growth radical polymerization. 

As for the microscale models, typically linear multistep methods are used for the numerical integration of the obtained set of equations. Alternatively, a kMC approach can be selected. 

The authors presented the error analysis of A(alpha)-stable parallel multistep hybrid methods (PHMs) for the initial value problems of ordinary differential equations in singular perturbation form (see above). From these results one can see that the convergence results of the present methods are similar to those of linear multistep methods and so no order reduction occurs. 

F. Mazzia, A. Sestini, D. Trigiante, The continuous extension of the B-spline linear multistep methods for BVPs on non-uniform meshes, Applied Numerical Mathematics 59(3-4)-Special Issue, 723-738(2009). 

Dahlquist GG (1963) A special stability problem for linear multistep methods. BIT Numer Math 3 27 3... 

In this section we will present the definitions for the stability of symmetric linear multistep methods according to Lambert and Watson theory as well as some definitions from the paper of Coleman and Ixaru for the stability of methods with variable coefficients. " ... 

The general form of a linear multistep method reads... 

Theorem 4.1.6 A linear multistep method has the order of consistency p if the following p+1 conditions on its coefficients are met ... 

---