Solving the Corrector Equations

In this section we discuss the corrector iteration. The corrector iteration described so far has the general form 

By Theorem 3.3.2 a necessary condition for the convergence of this iteration is that the corresponding mapping 

On the other hand, if the Jacobian x) has eigenvalues being large in modulus, 

When applying Newton s method, the nonlinear equation 

Like in the case of fixed point iteration the predictor solution is taken as starting value = The method demands an high computational effort, which is 

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DDAPLUS solves the corrector equations for each t-step by Newton s method, using an iteration matrix G(t 7 to), of order Nstvar, defined by Eq. (B.l-5a). On request, Eq. (B.1-2) is solved similarly with the matrix G(to) defined by Eq. (B.l-5b). If the user does not provide a detailed subroutine Jac (see Section B.6 below), DDAPLUS will approximate G(t) by finite differences. 

Frequently, the equations of motion occur in such a form that the effort for solving the corrector equation (4.1.29) can be reduced significantly. This concerns the solution of the linear systems (4.1.30) and the computation the Jacobian by finite differences. This can be done for all ODEs in first order form originating from higher order ODEs. 

We solve the same equations as in Section 4.1.1. Since a is very close to the threshold value a = 2, the difference between the solution to the effective equation obtained by taking the simple mean, at one side, and the solutions to the original problem and to our upscaled equation, are spectacular. Our model approximates fairly well with the physical solution even without adding the correctors (Table 4). Parameters are given at Table 3. 

Cooley, J.W., An improved eigenvalue corrector formula for solving the Schrodinger equation for central fields. Math. Comput., 15, 363-374, 1961. 

J. W. Cooley, Math. Comput., IS, 363 (1961). An Improved Eigenvalue Corrector Formula for Solving the Schrddinger Equation for Central Fields. 

If j30 = 0, the method is explicit and the computation of is straightforward. If 30 + 0, the method is implicit because an implicit algebraic equation is to be solved. Usually, two algorithms, a first one explicit and called the predictor, and a second one implicit and called the corrector, are used simultaneously. The global method is called a predictor-corrector method as, for example, the classical fourth-order Adams method, viz. 

These equations can be analytically solved. The only concern is that the harmonic approximation is valid within some trast radius. Performance can be improved by including a corrector step, allowing for larger step sizes. Also of note is that this... 

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. 

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 the methods explained so far, to solve a differential equation over an interval (xj, x,+i) only the value of y at the beginning of the interval was required. In the predictor corrector methods, however, four prior values are required for finding the value of y at x,+. A predictor formula is used to predict the value of y at x,+i and then a corrector formula is applied to improve this value. We now explain one such method. 

The differential equations which arise in almost all chemical kinetic studies of complex reaction schemes are "stiff differential equations". In chemical kinetics the stiffness is caused by the huge differences in the reaction rate constants of the various elementary reactions. It is impossible to solve such a system of differential equations by the usual Rung-Kutta methods. Therefore we used a program, described by Gear (J ) as a special multi step predictor-corrector method with self adjusting optimum step size control. 

The use of Eq. 10.98 requires solving linear algebraic equations as opposed to the nonlinear algebraic equations of Eq. 10.90 and thus considerably simplifies the numerical problem. Furthermore, accurate results can be obtained using a predictor-corrector method, as detailed in Section 4.9. 

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