Numerical integration algorithms

In Chapter 14 we discussed the generation of an initial configuration, (r (0), v(0)). The equations of motion can then be integrated numerically, with this phase space point as the initial condition. 

The integration can be implemented with a variety of finite difference algorithms. These propagate the positions, velocities, accelerations, third derivative of positions with respect to time, and so on, at a later time f -I- Sf, given the values of these variables at time t. The time interval St is the numerical integration time step. 

Typically, we assume that positions and their time derivatives can be written as Taylor series expansions. The positions are written as 

Analogous expansions can be written for the accelerations, q(t), the third derivative of the positions with respect to time, b(t), the fourth derivative of the positions with respect to time, c(t), and so on. 

Various finite difference algorithms exist, identified by their truncation schemes. For example, Euler s method includes only the positions and velocities, ignoring the third and higher order derivatives  

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The user has the same options for handling distances and the same choices of meteorology as described above for point sources, but no complex terrain, elevated simple terrain, building downwash, or fumigation calculations are made for area sources. Distances are measured from the center of the rectangular area. Since the numerical integration algorithm can estimate concentrations within the area source, the user can enter any value for the minimum distance. 

Note that for stable conditions and/or mixing heights greater than or equal to 10,000 m, unlimited mixing is assumed and the summation term is assumed to be zero, as noted by expressions presented earlier in this chapter. Equation (9) is used to model the plume impacts from point sources, flare releases, and volume releases in SCREEN. The SCREEN volume source option uses a virtual point source approach. The model uses a numerical integration algorithm for modeling impacts from area sources. 

As discussed in the introduction to this chapter, the solution of ordinary differential equations (ODEs) on a digital computer involves numerical integration. We will present several of the simplest and most popular numerical-integration algorithms. In Sec, 4.4.1 we will discuss explicit methods and in Sec. 4.4.2 we will briefly describe implicit algorithms. The differences between the two types and their advantages and disadvantages will be discussed. 

Yeh KC, Kwan KC. A comparison of numerical integrating algorithms by trapezoidal, Lagrange, and spline approximation. J Pharmacokinet Biopharm 1978 6 79-98. 

Reacting to the need for extended chemical representations, we have applied Fade approximant techniques to the numerical integration algorithms. Speed increases of at least four or fivefold are obtained in the computation comparing the Fade method for the more complex chemical system with previous methods for the simpler system. For equal-sized chemical matrices, the improvements would most likely be even better. To make these advances available to the community at large, we show the mathematical steps in detail. 

The state of the art procedure for design of cyclic PSA or TSA processes using activated carbon adsorbents is to simultaneously solve the partial differential equations describing the mass, the heat, and the momentum balance equations for each step of the process using the appropriate initial and boundary conditions. These numerical calculations are carried out over many cycles for the process until a cyclic steady-state performance solution is achieved. Many different numerical integration algorithms are available for this purpose. The core input variables for the solution are multicomponent gas adsorption equilibria, heats, and kinetics for the system of interest [37]. 

Indirect response models require differential equations and numerical integration algorithms to describe the nonlinear inhibition or stimulation. Partially integrated solutions for these models have been developed (50, 51), which allow qualitative examination of the relationships between response... 

Yu Z, Tse F L (1995). An evaluation of numerical integration algorithms for the estimation of the area under the curve (AUC) in pharmacokinetic studies. Biopharm. 

The Runge-Kutta-Gill fourth-order correct numerical integration algorithm for coupled ODEs is useful to simulate this double-pipe reactor after temperature-and conversion-dependent kinetic rate laws are introduced for both fluids. The generalized procedure is as follows ... 

The integration of the equations of motion In this step, the forces of constraint and their time derivatives up to order s nax obtained from step 1, are used as input to the selected integration scheme, to generate the constrained coordinates. Note that the particular choice of numerical integration algorithm in step 2 determines the parameter of step 1. 

The default numerical integration algorithm uses a variable step size. Switching to a fixed step size eliminated the oscillation problem. To achieve this, go to Run on the upper toolbar in Aspen Dynamics, select Solver Options and click the Integrator page tab. The window shown in Figure 8.43a opens on which the step size can be changed to Fixed. 

Figure 14.4 Gear and implicit Euler numerical integration algorithms.

There were no dynamic simulation issues experienced in this system. The default Implicit Euler numerical integration algorithm worked well, giving quite short simulation times (1 min of real time to simulate lOh of process time). 

Equations (1), (2), (4), and (5) are the equations of motion for a system of particles with cylindrical symmetry. In the following pages, we shall discuss a number of aspects of the numerical solution of these equations such as the choice of numerical integration algorithm, some shortcuts in the numerical integration, the difficult problem of choosing a potential function for polyatomic molecules, and the calculation of quantities of statistical-mechanical interest from the phase space trajectory. 

The cumulative interval area thus provides estimates of the integrals in Eqs. (6.3 and 6.4). Yu and Tse evaluated the performance of several numerical integration algorithms for calculating A L/C. In practice, it is most common to apply the trapezoidal rule for ascending yi > yi- ) and plateau (y/ yi-i) regions of the curve and the log-trapezoidal rule when concentrations are decreasing Vi < Vi-i)-... 

Despite the fact that the numerous integration algorithms have been developed, the correct... 

MD tracks the temporal evolution of a microscopic model system by integrating the equations of motion for all microscopic degrees of freedom. Numerical integration algorithms for initial value problems are used for this purpose, and their strengths and weaknesses have been discussed in simulation texts [104-106]. 

The explicit expression for defined in equation (9.21) may be derived by requiring that for all k the vanishes at all times. Using a certain numerical integration algorithm, the set of Lagrangian multipliers is no longer evaluated by equation (9.21) but by equations that contain free parameters corresponding to the. These... 

Molecular dynamics (MD) and Monte Carlo (MC) simulation techniques have been used now for decades to characterize aqueous solutions. The most basic elements that underlie these techniques, such as numerical integration algorithms and the Metropolis method, are discussed thoroughly else-where, " so they are not included here. Our intention here is to survey methods used for determining the thermodynamic and structural quantities most closely tied to hydrophobicity. 

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