Adams-Bashforth method

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. 

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. 

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

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,... 

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... 

Euler s and RK methods are also known as one-step techniques which use function values only in a single step, that is, in the preceding step. However, in the multistep techniques, evaluation of each step requires function values from more than one of the preceding steps. The benefit of the multistep techniques is the use of additional information to obtain more accurate solutions. The Adams-Bashforth methods for explicit solution of Equation 11.1 are multi-step in nature and are given in second and fourth orders in Equations 11.19 and 11.20, respectively, as follows ... 

Inside each fluid, p and p are constants. Equations (95) and (96) were solved by using a finite difference method on a fixed two- or three-dimensional grid. The spatial terms were discretized by second-order finite differences on a staggered Eulerian grid. The discretization of time was achieved by an expKdt Euler method or a second order Adams Bashforth method. The boundary conditions used in their study were either periodic or full sKp in the horizontal directions and rigid, stress-free on the top and bottom. 

The well-known predictor-corrector Adams-Bashforth method of algebraic order four. 

A second-order Adams-Bashforth method [50] was also used for the explicit treatment of the nonlinear term (the sum of the bulk free energy density and template surface free energy differentials). The equation can be discretized into the form. 

Example 4.1.1 For equal (constant) step sizes the Adams-Bashforth 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). 

Example 4.1.11 Finally, we demonstrate zero stability of the two step Adams-Bashforth method (AB-2) by discretizing the unconstrained truck. In Fig. 4 4 seven largest eigenvalues of h) are traced as a function of the step size. 

Among the explicit multistep methods, the Adams-Bashforth methods are the most widely used. The second-order (global error) Adams-Bashforth two-step method is... 

The fourth-order, four-step Adams-Bashforth method,... 

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. 

---