Runge-Kutta time integration

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. 

Initial numerical simulations of population density dynamics incorporated experimental data of Figures 2 and 4 and Equations 16, 20, 23, and 27 into a Runge-Kutta-Gill integration algorithm (21). The constant k.. was manipulated to obtain an optimum fit, both with respect to sample time and to degree of polymerization. Further modifications were necessary to improve the numerical fit of the population density distribution surface. 

Figure 4 shows the output of this program, which consists of concentrations of o-methylol, p-methylol and methylene ether groups at various reaction times. Although many integration routines can be used in CSMP calculations, the variable interval Runge-Kutta method was used in this case since that is the option selected when no other method is specified. 

Softwares for numerical integration of equations include the calculator HP-32SII, POLYMATH, CONSTANTINIDES AND CHAPRA CANALE. The last of these also can handle tabular data with variable spacing. POLYMATH fits a polynomial to the tabular data and then integrates. A comparison is made in problem PI.03.03 of the integration of an equation by the trapezoidal and Runge-Kutta rules. One hundred intervals with the trapezoidal rule takes little time and the result is usually accurate enough, so it is often convenient to standardize on this number. 

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

With hmax = ma,x(xi+i — x/), the global error order of the classical Runge-Kutta method is of order 4, or 0(h/nax), provided that the solution function y of (1.13) is 5 times continuously differentiable. The global error order of a numerical integrator measures the maximal error committed in all approximations of the true solution y(xi) in the computed y values y. Thus if we use a constant step of size h = 10 3 for example and the classical Runge-Kutta method for an IVP that has a sufficiently often differentiable solution y, then our global error satisfies... 

To analyze this phenomenon further, 2D numerical simulations of (49) and (50) were performed using a central finite difference approximation of the spatial derivatives and a fourth order Runge-Kutta integration of the resulting ordinary differential equations in time. Details of the simulation technique can be found in [114, 119]. The material parameters of the polymer blend PDMS/PEMS were used and the spatial scale = (K/ b )ll2 and time scale r = 2/D were established from the experimental measurements of the structure factor evolution under a homogeneous temperature quench. 

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]... 

In Fig. 10, the transients exhibit quite different behavior from opal A to opal CT. In particular, a bi-exponential decay (Eq. 2) failed to reproduce the kinetics of opal CT. In this material, the emission is red-shifted towards 2.6 eV and the PL is strongly quenched at shorter time delays, with an unusual, non-linear kinetics in semi-log scale, indicating a complex decay channel either involving multi-exponential relaxation or exciton-exciton annihilations. Runge-Kutta integration of Eq. 5 seems to confirm the latter assumption with satisfactory reproduction of the observed decays. The lifetimes and annihilation rates are Tct = 9.3 ns, ta = 13.5 ns, 7ct o = 650 ps-1 and 7 0 = 241 ps-1, for opal CT and opal A, respectively. 

This method often requires very small integration step sizes to obtain a desired level of accuracy. Runge-Kutta integration has a higher level of accuracy than Euler. It is also an explicit integration technique, since the state values at the next time step are only a function of the previous time step. Implicit methods have state variable values that are a function of both the beginning and end of the current... 

First, without explaining the details [15], we will develop an Excel spreadsheet for the numerical integration of the reaction mechanism 2as seen in Figure 7.13. The fourth-order Runge-Kutta method requires four evaluations of concentrations and derivatives per step. This appears to be a serious disadvantage, but as it turns out, significantly larger step sizes can be taken for the same accuracy, and the overall computation times are much shorter. We will comment on the choice of appropriate step sizes after this description. 

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