Adams-Moulton method

The GRM using the generalized Maxwell-Stefan equations has no closed-form solutions. Numerical solutions were calculated using a computer program based on an implementation of the method of orthogonal collocation on finite elements [29,62,63]. The set of discretized ordinary differential equations was solved with the Adams-Moulton method, implemented in the VODE procedure [64]. The relative and absolute errors of the numerical calculations were 1 x 10 and 1 x 10 , respectively. 

If data at tn+i is included in the interpolation polynomial, implicit methods, known as Adams-Moulton methods, are obtained. The first order method coincides with the implicit Euler method, and the second order method coincides with the trapezoid rule. The third order method is written as ... 

The Adam-Bashforth methods are frequently used as predictors and the Adam-Moulton methods are often used as correctors. The combination of the two formulas results in predictor-corrector schemes. 

The primary advantage of the single-step methods is that they are self starting. We can also vary the step sizes. In contrast, the multistep methods require a single-step formula to start the calculations. Step size variation is difficult. However, the efficiency of both the Milne s and Adams-Moulton methods is about twice that of the single-steps methods. We need two function evaluations per step in the former while four or five are required with the single step. 

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. 

For example, the fourth-order Adams-Moulton method is characterized by the following vector ... 

As we can see, the Adams-Moulton method is implicit even in its multivalue form. 

For example, if the first- or third-order Adams-Moulton method is used and we wish to use the second-order method, we have... 

For example, in the case of the third-order multistep algorithm of the family of explicit Adams-Bashforth methods and of implicit Adams-Moulton methods,... 

Since the predictor-corrector method is particularly suitable (when it works) in terms of computational times and memory allocation (it does not need to store the Jacobian), it is used with nonstiff problems and with algorithms that are not good at solving stiff problems, but with better accuracy featiu-es (usually the Adams-Moulton methods are adopted). 

If we truncate up to the fourth term in the RHS of Eq. 7.105, we obtain the following fourth order Adams-Moulton method. 

The common factor in the implicit Euler, the trapezoidal (Crank-Nicolson), and the Adams-Moulton methods is simply their recursive nature, which are nonlinear algebraic equations with respect to y +j and hence must be solved numerically this is done in practice by using some variant of the Newton-Raphson method or the successive substitution technique (Appendix A). 

In practice, implicit multistep methods are used to improve upon approximations obtained by explicit methods. This combination is the so-called predictor-corrector method. Predictor-corrector methods employ a single-step method, such as the Runge-Kutta of order 4, to generate the starting values to an explicit method, such as an Adams-Bashforth. Then the approximation from the explicit method is improved upon by use of an implicit method, such as an Adams-Moulton method. Also, there are variable step size algorithms associated with the predictor-corrector strategy in the literature [5,25]. 

Using the Runge-Kutta order four method for starting, the Adams-Bashforth method as the predictor, and the Adams-Moulton method as a corrector, approximate the solution to... 

Similar to eq. (4.1.4) this leads to the so-called Adams-Moulton method ... 

Example 4.1.2 For equal (constant) step sizes the Adams-Moulton methods are given by the following formulas... 

Corrollary 4.1.7 Adams-Bashforth methods have order of consistency fc, Adams-Moulton methods have order of consistency fc + 1, and BDF methods have order of consistency k with k defined by (4-1-12). 

Figure 4.1 Absolute value of error constants for different multistep methods. (For fc = 1 the Adams-Moulton method is the implicit Euler method.)...

Prom Eq. (4.1.5) a continuous representation for Adams-Moulton methods can easily be derived ... 

By construction the order of the continuous representation is the order of the Adams-Moulton method, i.e. g = fc -f 1. 

By decreasing the step size this instability is not removed. The two step Adams-Moulton method is like all higher order methods in this class not zero-stable when applied to index-2 or index-3 systems. We will give in Sec. 5.2.3 criteria for zero stability of multistep methods applied to index-2 systems. 

Figure 5.3 The solution of the pendulum problem generated with the two step Adams-Moulton method and h = 0.01...

The index-2 case is the situation where the equations of motion are set up together with constraints on velocity level. We will see that the negative observation concerning the two step Adams-Moulton method holds in general for all higher order Adams-Moulton methods. The central convergence theorem requires oo-stability of the method. 

In contrast, Adams-Moulton methods other than the implicit Euler are not strictly stable at infinity. The two step Adams-Moulton method is not stable at infinity as its a polynomial has a root at -1.76. 

On the other hand, for Adams-Moulton methods with order p> 2 this matrix has eigenvalues larger than one in modulus and the discretization becomes unstable. The discretization scheme is no longer stable at infinity and Theorem 5.2.3 can no longer be applied. 

Example 5.2.6 We consider the equations of motion of the linearized constrained constrained truck in its index-2 formulation. Discretizing these equations by the two step Adams-Moulton method (see Sec. 4-T2) leads to the eigenvalues of the discrete transfer matrix which are given in Table 5.3. One clearly identifies structural and dynamic eigenvalues and the source of instability of AM3 when applied to an index-2 problem, see also Tab. 2.2. 

We note that every integration method which is convergent for index-1 DAEs can be applied when treating overdetermined DAEs together with the implicit state space method, in particular also Adams-Moulton methods. 

A shortcoming of this stabilized formulation is that it consists of an index-2 formulation which can only be solved by numerical methods which are convergent for this class of DAEs. In particular, higher order Adams-Moulton methods are no longer applicable, see Th. 5.2.3. They introduce an instability in p. 

Table 5.3 Eigenvalues of the discrete linear truck example in its index 2 formulation for the two step Adams-Moulton method with step size h = 0.005.

This equation was solved using the LSODE routine from the ODEPACK library. LSODE uses a 12 order Adams-Moulton method to solves a system of ordinary differential equations. 

Neither the Runge-Kutta nor tiie Adams Bashforth methods can handle stiff differential equations well. The Adams-Moulton method is an implicit multistep method that can handle stiff problems better (stiff problems are dicussed later in this chapter). The two-step Adams-Moulton method (third-order accurate) is... 

The implicit multistep methods add stability but require more computation to evaluate the implicit part. In addition, the error coefficient of the Adams-Moulton method of order k is smaller than that of the Adams Bashforth method of the same order. As a consequence, the implicit methods should give improved accuracy. In fact, the error coefficient for the imphcit fourth-order Adams Moulton method is 19/720, and for the explicit fourth-order Adams Bashforth method it is 251/720. The difference is thus about an order of magnitude. Pairs of exphcit and implicit multistep methods of the same order are therefore often used as predictor-corrector pairs. In this case, the explicit method is used to calculate the solution,, at v +i. Furthermore, the imphcit method (corrector) uses y + to calculate /(x +i,y +i), which replaces /(x +i,y +i). This allows the solution, y +i, to be improved using the implicit method. The combination of the Adams Bashforth and the Adams Moulton methods as predictor orrector pairs is implemented in some ODE solvers. The Matlab odel 13 solver is an example of a variable-order Adams Bashforth Moulton multistep solver. 

---