Time-integration scheme Runge-Kutta

In these equations fi is the coluirm mass of dry air, V is the velocity (u, v, w), and (jf) is a scalar mixing ratio. These equations are discretized in a finite volume formulation, and as a result the model exactly (to machine roundoff) conserves mass and scalar mass. The discrete model transport is also consistent (the discrete scalar conservation equation collapses to the mass conservation equation when = 1) and preserves tracer correlations (c.f. Lin and Rood (1996)). The ARW model uses a spatially 5th order evaluation of the horizontal flux divergence (advection) in the scalar conservation equation and a 3rd order evaluation of the vertical flux divergence coupled with the 3rd order Runge-Kutta time integration scheme. The time integration scheme and the advection scheme is described in Wicker and Skamarock (2002). Skamarock et al. (2005) also modified the advection to allow for positive definite transport. 

Multistep methods can be used for integration of the system of equations in time, such as Runge-Kutta, and the central finite difference scheme for the spatial approximation of first-order derivatives for each grid point (i,j, k) as shown in Figure 6.4 for uniform meshes by simplicity... 

The Runge-Kutta integration scheme advances the solution by step size h — Az from time level n to n + 1. The algorithm is... 

An extended Runge-Kutta integration scheme from time tn to tn + At employs an iterative procedure during time advancement, with the notation Ynj = for the j — th iteration. With similar notation for other functions of time, and with At = h, the following algorithm can be used [21]... 

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. 

Table V shows examples of the gains obtained using the new computational scheme. The typical real-time/computer-time ratio was increased from 36/1 to 180/1. Perhaps, more significant is the fact that the Fade method has allowed us to obtain acceptable solutions in situations where Runge-Kutta either failed to converge or produced spurious solutions. One such instance is the integration of full differential equations for the free-radical species. The Fade method successfully computed solutions without algebraic substitution of stationary-state assumptions whereas Runge-Kutta failed to produce any solution.

A variety of explicit (Dufort-Frankel, Lax-Wendroff, Runge-Kutta) and implicit (approximate factorization, LU-SGS) or hybrid schemes have been employed for integration in time. Because of the complexity of the incompressible Navier-Stokes equations, stability analyses to determine critical time steps are difficult. As a general rule, the allowable time step for an explicit method is proportional to the ratio of the smallest grid size to the largest convective velocity (or the wave propagation speed for an artificial compressibility method). 

The results shown in Figs. 3.1 and 3.2 were obtained with the scheme of Eq. (3.24) with At = 0.01 s and a Runge-Kutta integration of second order with variable step size (cf. [9]). The difference between the two procedures may be neglected in this case. It is obvious that the thermal explosion (runaway) leaves but little time for emergency interventions. 

In our early work on multimode vibronic dynamics, a fourth-order predictor-corrector method has been used to integrate the time-dependent Schrodinger equation. Later, FOD schemes and a fourth-order Runge-Kutta method have also been employed. These techniques proved to be superior to the predictor-corrector method for example, the FOD scheme was found to be 3-5 times faster than the SOD integrator (the latter... 

Figure 3- Comparison of the average error versus computation time, for the modified Euler, 3 d 4 order Runge-Kutta, and Adams-Moulton integration methods, applied to an ECEE scheme [(a) k = 10 s l, scan rate = 200 mV/s (b) k =100 s l, scan rate =1000 mV/s].

Similarly, one could attempt to improve the 3C/3T discretisation (other than by using Runge-Kutta integration). In effect, the Crank-Nicolson scheme does this by specifying a central difference approximation at T+J 8T. The same can be done at T by using the Richardson (1911) formula (denoting time steps by the index k) ... 

Prominent representatives of the first class are predictor-corrector schemes, the Runge-Kutta method, and the Bulir-sch-Stoer method. Among the more specific integrators we mention, apart from the simple Taylor-series expansion of the exponential in equation (57), the Cayley (or Crank-Nicholson) scheme, finite differencing techniques, especially those of second or fourth order (SOD and FOD, respectively) the split-operator, method and, in particular, the Chebychev and the shoit-time iterative Lanczos (SIL) integrators. Some of the latter integration schemes are norm-conserving (namely Cayley, split-operator, and SIL) and thus accumulate only... 

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