Runge classical, fourth-order

Simos and Williams106 have considered the following Runge-Kutta methods, based on the well known classical fourth-order Runge-Kutta method. The free parameters of the above methods are defined in order to minimize the phase-lag, and are given by ... 

BzzOdeRK. Classical fourth-order Runge-Kutta method with check of the error obtained by means of a reintegration with a halved step. 

A more accurate technique used for numerical integration is the classical fourth-order Runge-Kutta (RK) method. In this... 

Coefficients au and b, are determined in order that the algorithm possesses some qualities such as stability, accuracy, etc. A classical explicit fourth-order Runge—Kutta algorithm is defined by the values... 

Numerical professionals, when using the term Runge-Kutta , usually mean fourth-order RK, and the classical scheme, here RK4, is /.q as above, then k-A as in (4.15), then... 

Because of the apparent chaos in Fig. 6.5, simple analytical solutions of the driven SSE system probably do not exist, neither for the classical nor for the quantum mechanical problem. Therefore, if we want to investigate the quantum dynamics of the SSE system, powerful numerical schemes have to be devised to solve the time dependent Schrddinger equation of the microwave-driven SSE system. While the integration of classical trajectories is nearly trivial (a simple fourth order Runge-Kutta scheme, e.g., is sufficient), the quantum mechanical treatment of microwave-driven surface state electrons is far from trivial. In the chaotic regime many SSE bound states are strongly coupled, and the existence of the continuum and associated ionization channels poses additional problems. Numerical and approximate analytical solutions of the quantum SSE problem are proposed in the following section. 

Equations 10.106 to 10.111 constitute a set of algebraic equations and first order ordinary differential equations. The two algebraic Eqs. 10.104 and 10.105 are solved using the classical procedure described by Villadsen [65]. The set of ordinary differential equations is easy to solve with the fourth-order Runge-Kutta method. 

The Runge-Kutta method, which was so popular for so long that it is known as the classical Runge-Kutta, was the fourth-order method ... 

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