Runge-Kutta scheme

When the axial dispersion terms are present, D > Q and E > Q, Equations (9.14) and (9.24) are second order. We will use reverse shooting and Runge-Kutta integration. The Runge-Kutta scheme (Appendix 2) applies only to first-order ODEs. To use it here. Equations (9.14) and (9.24) must be converted to an equivalent set of first-order ODEs. This can be done by defining two auxiliary variables ... 

This equation was solved numerically with a fourth-order Runge-Kutta scheme. It was usually more convenient to recast (12.25) as a differential equation in the rectangular Cartesian coordinates sometimes, however, the advantage was tipped in favor of the polar coordinates. The results shown in Fig. 12.4 were obtained with a mixture of the two approaches. 

These equations are representation independent and are valid for an arbitrary time-dependence of the system Hamiltonian as, for example, in the case of strong laser driving. In a suitable representation, the coupled set of differential equations Eqs. (33) to (35) can be solved numerically, for example, using a Runge-Kutta scheme. As already mentioned in the Introduction, Yan and coworkers [33-35] independently developed a similar approach starting at a correlation function which is assumed to be of the form Eqs. (5) and (6). 

X m, m = 0,..., 32, f = 1,2 with the help of a fourth order Runge-Kutta scheme (see, e.g., Milne (1970)). Every single one of the 62 resulting trajectories was followed over 200 cycles of the microwave field, and the values of I and 6 after every completion of a full cycle of the microwave field were plotted as dots in a 6,1) phase-space diagram. The result is shown in Fig. 6.5. Regular and chaotic regions are clearly visible. 

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. 

Solve the coupled ODEs numerically lor a = 0. 1/2. 1.3/2, and 2 and plot the results. Note that the case for a = 0 provides a check on your numerics. For a 0. r/j —>-oo as f 0. Why (You can use any numerical method that you like, but a fourth-order Runge-Kutta scheme is available on many computer systems or from Numerical Recipes.)... 

Additional research could also be done in applying various kinds of processes that lead to an exponential affine framework in the sense that we either are able to derive the characteristic functions or the moments of the underlying random variable numerically e.g. by applying a Runge-Kutta scheme in order to solve the set of coupled ODE s. Then the price of a bond option could be computed by using the numerically derived characteristic function applying the FRFT approach or by plugging the numerically derived moments in the lEE-scheme. 

Symplectic integrators may be constructed in several ways. First, we may look within standard classes of methods such as the family of Runge-Kutta schemes to see if there are choices of coefficients which make the methods automatically conserve the symplectic 2-form. A second, more direct approach is based on splitting. The idea of splitting methods, often referred to in the literature as Lie-Trotter methods, is that we divide the Hamiltonian into parts, and determine the flow maps (or, in some cases, approximate flow maps) for the parts, then compose the maps to define numerical methods for the whole system. 

Sanz-Sema, J. Runge-Kutta schemes for Hamiltonian systems. BIT 28, 877—883 (1988). doi 10.1007/BF01954907... 

The droplet equations are advanced using a third-order Runge-Kutta scheme. Owing to the disparities in the flowfield timescale (rf), the droplet relaxation... 

We have presented the essence of the Runge-Kutta scheme, and now it is possible to generalize the scheme to a pth order. The Runge-Kutta methods are explicit methods, and they involve the evaluation of derivatives at various... 

Gottlieb S, Shu CW. Total variation diminishing Runge-Kutta schemes. Math Comput 1998 67(221) 73-85. 

To integrate the ordinary differential equations resulting fi om space discretization we tried the modified Euler method (which is equivalent to a second-order Runge-Kutta scheme), the third and fourth order Runge-Kutta as well as the Adams-Moulton and Milne predictor-corrector schemes [7, 8]. The Milne method was eliminated from the start, since it was impossible to obtain stability (i.e., convergence to the desired solution) for the step values that were tried. 

In ref. 164 the authors consider a new BDF fourth-order method for solving stiff initial-value problems, based on Chebyshev approximation. The authors prove that the developed method may be presented as a Runge-Kutta method having stage order four. They examine the stability properties of the method and they presented a strategy for changing the step size based on embedded pair of the Runge-Kutta schemes. 

The hybrid LBM-FDM method is used for the simulation, the convection term is discretized by upwind weighted scheme, and the diffusion term is discretized by central difference scheme. Runge-Kutta scheme is employed for time stepping. 

In the Runge-Kutta scheme, integration of q-, in equations (1) is done in the same way as the integration of pi and thus, for the sake of brevity, only the equations used to integrate p, are presented. A literal implementation of equations (1) would be to select a step size Af, and integrate p, over At by an amount Ap,-, i.e,. 

Similar expressions to those shown in equations (5)-(7) can be derived for higher-order Runge-Kutta schemes. The fourth-order algorithm is widely used to integrate physical systems. ... 

Figure 6.3 Runge-Kutta scheme for free-radical polymerization.

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