Adams-Bashforth

There are several methods that we can use to increase the order of approximation of the integral in eqn. (8.72). Two of the most common higher order explicit methods are the Adams-Bashforth (AB2) and the Runge-Kutta of second and fourth order. The Adams-Bashforth is a second order method that uses a combination of the past value of the function, as in the explicit method depicted in Fig. 8.19, and an average of the past two values, similar to the Crank-Nicholson method depicted in Fig. 8.21, and written as... 

In 26 the authors have developed a new trigonometrically-fitted predictor-corrector (P-C) scheme based on the Adams-Bashforth-Moulton P-C methods. In particular, the predictor is based on the fifth algebraic order Adams-Bashforth scheme and the corrector on the sixth algebraic order Adams-Moulton scheme. More specifically the new developed scheme integrates exactly any linear combination of the functions ... 

In 30 the authors have developed trigonometrically fitted Adams-Bashforth-Moulton predictor-corrector (P-C) methods. It is the first time in the literature that these methods are applied for the efficient solution of the resonance problem of the Schrodinger equation. The new trigonometrically fitted P-C schemes are based on the well known Adams-Bashforth-Moulton methods. In particular, they are based on the fourth order Adams-Bashforth scheme (as predictor) and on the fifth order Adams-Moulton scheme (as corrector). More... 

In 37 the authors have developed trigonometrically fitted Adams-Bashforth-Moulton predictor-corrector (P-C) methods. The new trigonometrically fitted P-C schemes are based on the well known Adams-Bashforth-Moulton... 

In order to construct higher-order approximations one must use information at more points. The group of multistep methods, called the Adams methods, are derived by fitting a polynomial to the derivatives at a number of points in time. If a Lagrange polynomial is fit to /(t TO, V )i. .., f tn, explicit method of order m- -1. Methods of this tirpe are called Adams-Bashforth methods. It is noted that only the lower order methods are used for the purpose of solving partial differential equations. The first order method coincides with the explicit Euler method, the second order method is defined by ... 

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. 

For an in-depth discnssion of the Adams-Bashforth family of methods, see Johnson and Riess (1982). 

Interval of absolnte stability (-2,0) Two-step Adams-Bashforth... 

The two equations, [53] and [55], form a system of coupled ODEs with the variable z playing the role of the independent variable. Given initial conditions at a point Zq these equations can be solved by standard numerical routines such as those discussed in the previous section. Because much computational effort is required to evaluate each p, at each increment of the independent variable z, a method that does not require too many evaluations of the right hand side of the iterative equation is desirable. Usually, a simple forward Euler routine is quite adequate for these purposes. If a multistep algorithm is used, the Adams-Bashforth method has been recommended by Kubicek and Marek the first-order Adams-Bashforth algorithm is, in fact, equivalent to the simple forward Euler algorithm. 

The third-order exphcit multistep algorithm by Adams-Bashforth ... 

Figure 2.5 Stability region of the third-order Adams-Bashforth algorithm.

For example, the fourth-order Adams-Bashforth method applied to (2.38) generates the following fourth-order finite-difference equation ... 

For example, the following third-order Adams-Bashforth algorithm is explicit ... 

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,... 

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