Adams-Moulton

These equations are semi-implicit second order in time typically called Adams-Moulton (AM2) method or Crank-Nicholson (CN), when applied to diffusion problems, and due to the implicit nature of the procedure, the scheme is also unconditionally stable. 

The method increases the order but the stability is compromized due to the extrapolation done by the the linear approximation between the previous times. This stability issue can be improved by adding an extra implicit step using an Adams-Moulton (AM2) as follows... 

When the quadrature of eq 2 cannot be performed analytically the integration should be carried out numerically by robust routines such as the Runge-Kutta, Adams-Moulton predictor-corrector or Bulirsch-Stoer methods with step size and error control [53, 55, 56], These routines can also be found in computer codings at Netlib and in standard books on computer codes [53]. 

The initial value problem, Eqs. 1-3, can be integrated by any marching algorithm which is based on the Runge-Kutta or Adams-Moulton techniques. Based on the calculated space profiles of C,... 

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... 

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. 

Adams-Moulton Fourth-Step Method Predictor... 

The predictor calls for four previous values in Adams-Moulton and Milne s algorithms. We obtain these by the fourth-order Runge-Kutta method. Also, we can reduce the step size to improve the accuracy of these methods. Milne s method is unstable in certain cases because the errors do not approach zero as we reduce the step size, h. Because of this instability, the method of Adams-Moulton is more widely used. 

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. 

Adams-Moulton Family of Methods One-step Adams-Moulton, backward Euler s rule... 

Interval of absolute stability (-6,0) Four-step Adams-Moulton... 

For the solution ol y , a, is known from the previous solutions. The problem thus consists of an iterative solution of the equation system (10.68). This can be achieved by the Newton-Raphson method. The coefficients (q and pj for the Adams-Moulton and backward difference networks of various orders are listed in Tables 10.3 and 10.4 respectively. 

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 trapezium algorithm, also called the Crank-Nicolson or the second-order Adams-Moulton ... 

The third-order implicit Adams-Moulton algorithm ... 

To better understand this new way of looking at the problem, we can take the example of a particular algorithm, such as the fourth-order Adams-Moulton algorithm in its multivalue version. 

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

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