Runge

Cheung A S C, Yoshino K, Freeman D E, Friedman R S, Dalgarno A and Parkinson W H 1989 The Schumann-Runge absorption-bands of 0 0 in the wavelength region 175-205 nm and spectroscopic constants of isotopic oxygen molecules J. Mol. Speotroso. 134 362-89... 

The chemical-activation step is between one and two orders of magnitude faster than the subsequent collisional deactivation of vibrationally excited O2. Finally, the population of individual vibrational levels v" of O2 is probed tluough LIF in the Schiunann-Runge band Oi X E") after exciting the oxygen... 

Ackermann M and Biaume F 1970 Structure of the Schumann-Runge bands from the 0-0 to the 13-0 band J. Mol. Spectrosc. 35 73-82... 

Fig. 2. The time evolution of the total energy of four water molecules (potential-energy details are given in [48]) as propagated by the symplectic Verlet method (solid) and the nonsymplectic fourth-order Runge-Kutta method (dashed pattern) for Newtonian dynamics at two timestep values.

The IE and IM methods described above turn out to be quite special in that IE s damping is extreme and IM s resonance patterns are quite severe relative to related symplectic methods. However, success was not much greater with a symplectic implicit Runge-Kutta integrator examined by Janezic and coworkers [40],... 

D. Janezic and B. Orel. Implicit Runge-Kutta method for molecular dynamics integration. J. Chem. Info. Comp. Sd., 33 252-257, 1993. 

Janezic, D., Orel, B. Implicit Runge-Kutta Method for Molecular Dynamics Integration. J. Chem. Inf. Comput. Sci. 33 (1993) 252-257 Janezic, D., Orel, B. Improvement of Methods for Molecular Dynamics Integration. Int. J. Quant. Chem. 51 (1994) 407-415... 

Thus we find that the choice of quaternion variables introduces barriers to efficient symplectic-reversible discretization, typically forcing us to use some off-the-shelf explicit numerical integrator for general systems such as a Runge-Kutta or predictor-corrector method. 

The reaction rate equations give differential equations that can be solved with methods such as the Runge-Kutta [14] integration or the Gear algorithm [15]. 

Analytical and Test Methods. In addition to the modem spectroscopic methods of detection and identification of pyrroles, there are several chemical tests. The classical Runge test with HCl yields pyrrole red, an amorphous polymer mixture. In addition, all pyrroles with a free a- or P-position or with groups, eg, ester, that can be converted to such pyrroles under acid conditions undergo the Ehrlich reaction with p-(dimethylamino)henzaldehyde to give purple products. 

Runge-Kutta methods are explicit methods that use several function evaluations for each time step. Runge-Kutta methods are traditionally written for/(f, y). The first-order Runge-Kutta method is Euler s method. A second-order Runge-Kutta method is... 

A popular fourth-order Runge-Kutta method is the Runge-Kutta-Feldberg formulas (Ref. Ill), which have the property that the method is fourth-order but achieves fifth-order accuracy. The popular integration package RKF45 is based on this method. 

Runge-Kutta, 2nd order, A At < 2 Runge-Kutta-Feldberg, A At < 3.0... 

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