Predictor-corrector techniques

Equation [68] is based on methods such as the Verlet algorithm or the predictor-corrector technique. Binder suggests the use of time steps in the range of 0.002-0.008t, where t obeys... 

A compromise between the explicit and implicit methods is the predictor-corrector technique, where the explicit method is used to obtain a first estimate of and this estimated y +i is then used in the RHS of the implicit formula. The result is the corrected y + i, which should be a better estimate to the true y +i than the first estimate. The corrector formula may be applied several times (i.e., successive substitution) until the convergence criterion (7.49) is achieved. Generally, predictor-corrector pairs are chosen such that they have truncation errors of approximately the same degree in h but with a difference in sign, so that the truncation errors compensate one another. 

A nh). This extra effort may well be rewarded by the increased accuracy of the solution obtained. A more effective approach, which is frequently used in modern software packages, is to combine an explicit method with an implicit one to give a predictor-corrector technique. An explicit formula like eq. (7.5) or eq. (7.6) is used, not to give a final result, but to provide an initial guess of the solution, Ap nh), at f = nh. We can then insert this value of A into the differential equation (7.4) to obtain the predicted value of the derivative A p nh)= f Ap nh)). These values of A and A are then used in the implicit method to yield the final, corrected solution A nh). A secondary advantage of predictor-corrector methods is that the difference between Ap and A provides a reliable estimate of the error, typically (Acton, 1970),... 

This paper deals with thermal wave behavior during frmisient heat conduction in a film (solid plate) subjected to a laser heat source with various time characteristics from botii side surfaces. Emphasis is placed on the effect of the time characteristics of the laser heat source (constant, pulsed and periodic) on tiiermal wave propagation. Analytical solutions are obtained by memis of a numerical technique based on MacCormack s predictor-corrector scheme to solve the non-Fourier, hyperbolic heat conduction equation. 

Heat waves have been theoretically studied in a very thin film subjected to a laser heat source and a sudden symmetric temperature change at two side walls. The non-Fourier, hyperbolic heat conduction equation is solved using a numerical technique based on MacCormak s predictor-corrector scheme. Results have been obtained for ftie propagation process, magnitude and shape of thermal waves and the range of film ftiickness Mid duration time wiftiin which heat propagates as wave. 

As mentioned above, the goal of MD is to compute the phase-space trajectories of a set of molecules. We shall just say a few words about numerical technicalities in MD simulations. One of the standard forms to solve these ordinary differential equations i.s by means of a finite difference approach and one typically uses a predictor-corrector algorithm of fourth order. The time step for integration must be below the vibrational frequency of the atoms, and therefore it is typically of the order of femtoseconds (fs). Consequently the simulation times achieved with MD are of the order of nanoseconds (ns). Processes related to collisions in solids are only of the order of a few picoseconds, and therefore ideal to be studied using this technique. 

Finite-difference techniques were used to compute numerical solutions as column-breakthrough curves because of the nonlinear Freundlich isotherm in each transport model. Along the column, 100 nodes were used, and 10 nodes were used in the side-pore direction for the profile model. A predictor-corrector calculation was used at each time step to account for nonlinearity. An iterative solver was used for the profile model whereas, a direct solution was used for the mixed side-pore and the rate-controlled sorption models. 

The Runge-Kutta technique is different from the predictor-corrector method in the sense that, instead of using a number of points from backward it uses a number of functional evaluations between the current point t and the desired point f + i. 

The dynamic structural solution for the thrust loading is obtained using a modified-predictor-corrector-integration technique and normal mode theory. 

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... 

State bifurcations, arise. Two widely used packages that implement numerical continuation methods are AUTO (Doedel et al., 1991) and CONT (Marek and Schreiber, 1991). These programs trace out curves of steady states and follow the properties of limit cycles by using techniques similar to the predictor-corrector approach described above to improve their computational efficiency. An example is shown in Figure 7.8. 

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... 

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