Higher-order integrators

The method of Ishida et al [84] includes a minimization in the direction in which the path curves, i.e. along (g/ g -g / gj), where g and g are the gradient at the begiiming and the end of an Euler step. This teclmique, called the stabilized Euler method, perfomis much better than the simple Euler method but may become numerically unstable for very small steps. Several other methods, based on higher-order integrators for differential equations, have been proposed [85, 86]. 

In vivo effects of a chemical are due to an alteration in the higher-order integration of an intact animal system, which cannot be reflected in a less complex system. 

Higher-order integration schemes in time and space can be used to improve on the accuracy of the calculations, for example, using the higher order convection schemes presented above. For reactor simulations, any appropriate second- and third-order approximations are recommended. 

The higher-order integrals can be obtained from those of lower order by differentiation differentiating Eq. (4.37) with respect to yields... 

For the same reasons, higher order integration schemes would become unwieldy. 

For more discussion of higher order integrators, refer to [139, 164, 227, 259, 260, 262, 277,290]. Of particular relevance for molecular dynamics is the excellent numerical study of [153]. 

Explicit formulae for higher order integrals could be expressed as linear combinations of lower order integrals. Section 4 develops formulae for the integrals (jj jj), making it possible to use explicitly the formula presented above. 

Therefore only two-electron integrals, as in the case of the Cl method, and triangle integrals have to be computed. This fact will be extremely helpful when extending the application of Hy-CI method to larger systems. In our code, the same computer memory is needed for Cl and Hy-CI calculations. Note, that in the Hylleraas-CI method for any electron number no higher order integrals than four-electron ones appear. 

We now use this formalism to demonstrate the derivation of a higher order integration method than the explicit Euler one. In the explicit Euler method we neglect the time-variation of a and b over the time step. This is particularly bad for the second integral as dWi is of order and thus the explicit Euler method is only 1/2-order accurate for predicting the actual trajectory. Thus, let us increase die order of accuracy of this term by using a t accurate expansion ofbmtk[Pg.345]

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