Predictor-corrector integration schemes

When a Heun (predictor-corrector) integration scheme is used, a higher accuracy is obtained but at the cost of many supplementary calculations. This is due to... 

Figure 6 Two algorithms to conduct Gibbs-Duhem integrations. The pressure route (upper-left) corresponds to integration of Clapey-ron .s equation and entails iVPr-ensemble simulations of both phases (depicted here by the two boxes). The chemical potential route (upper-right) entails p.VT simulations. In both cases, a trapezoidal predictor-corrector integration. scheme is illustrated here the integration advances from point 0 to point 1...

Higher derivatives can be obtained by successive differentiation of equation (10), using the derivatives y, y, etc., which are routinely calculated in predictor-corrector integration schemes. Thus equations (8)-(10) provide the means to calculate the derivatives etc., in equation (7). Once... 

Our discussion so far has considered the use of SHAKE with the Verlet algorithm Versions have also been derived for other integration schemes, such as the leap-froj algorithm, the predictor-corrector methods and the velocity Verlet algorithm. In the cast of the velocity Verlet algorithm, the method has been named RATTLE [Anderson 1983]... 

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 ... 

The solution of Equations 47, 48, and 49 requires numerical techniques. For such nonlinear equations, it is usually wise to employ a simple numerical integration scheme which is easily understood and pay the price of increased computational time for execution rather than using a complex, efficient, numerical integration scheme where unstable behavior is a distinct possibility. A variety of simple methods are available for integrating a set of ordinary first order differential equations. In particular, the method of Huen, described in Ref. 65, is effective and stable. It is self-starting and consists of a predictor and a corrector step. Let y = f(t,y) be the vector differential equation and let h be the step size. 

Many predictor-corrector formulas are available. The original implementation of the GDI method suggested a set of constant step-size formulas of order that increased as the integration proceeded. We agree with the assessment of Escobedo and de Pablo [49] that lower-order schemes should be preferred because they are more stable, are easier to implement with a variable step size, and do not compromise accuracy—in most cases it is the uncertainty in the simulation averages that limits the overall accuracy of the integration. Escobedo and de Pablo present two second-order corrector formulas that allow for a variable step size. Both are of the form [49]... 

Systematic errors associated with the predictor-corrector scheme of numerical integration... 

One of the most common numerical methods used in molecular dynamics to solve Newton s equations of motions is the Velocity Verlet integrator. This is typically implemented as a second order method, and we find that it can become numerically tmstable dtuing the course of hyperthermal collision events, where the atom velocities are often far from equilibrium. As an alternative, we have implemented a fifth/ sixth order predictor-corrector scheme for our calculations. Specifically, the driver we chose utilizes the Adams-Bashforth predictor method together with the Adams-Moulton corrector method for approximating the solution to the equations of motion. 

In our early work on multimode vibronic dynamics, a fourth-order predictor-corrector method has been used to integrate the time-dependent Schrodinger equation. Later, FOD schemes and a fourth-order Runge-Kutta method have also been employed. These techniques proved to be superior to the predictor-corrector method for example, the FOD scheme was found to be 3-5 times faster than the SOD integrator (the latter... 

To integrate the ordinary differential equations resulting fi om space discretization we tried the modified Euler method (which is equivalent to a second-order Runge-Kutta scheme), the third and fourth order Runge-Kutta as well as the Adams-Moulton and Milne predictor-corrector schemes [7, 8]. The Milne method was eliminated from the start, since it was impossible to obtain stability (i.e., convergence to the desired solution) for the step values that were tried. 

The last part of the process is now to solve (integrate) the equation system 5.129. We could use the simple Runge-Kutta method, probably going to a fourth-order scheme since we (hopefully) are not limited here by the second-order discretisation error inherent in system 5.13. it is more common to employ a more sophisticated technique. Whiting and Carr (1977) suggest Hamming s modified predictor-corrector method, Villadsen and Michelsen (1978) that of Caillaud and Padmanabhan (1971) (and provide the actual subroutines). There are other methods. The criterion will always... 

There are, of course, a lot of other integration schemes, predominantly the predictor-corrector algorithms. Of these methods the Stoermer-Verlet algorithm has proved to be the best and simplest one. 

Prominent representatives of the first class are predictor-corrector schemes, the Runge-Kutta method, and the Bulir-sch-Stoer method. Among the more specific integrators we mention, apart from the simple Taylor-series expansion of the exponential in equation (57), the Cayley (or Crank-Nicholson) scheme, finite differencing techniques, especially those of second or fourth order (SOD and FOD, respectively) the split-operator, method and, in particular, the Chebychev and the shoit-time iterative Lanczos (SIL) integrators. Some of the latter integration schemes are norm-conserving (namely Cayley, split-operator, and SIL) and thus accumulate only... 

The constants depend on the order of the integration scheme. For a fourth-order predictor corrector scheme, Allen and Tildesley calculate the constants to have the following values ... 

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