Numerical integration examples

Essential Dynamics In most applications details of individual MD trajectories are of only minor interest. An illustrative example due to Grubmuller [10] is documented in Figure 3. It describes the dynamics of a polymer chain of 100 CH2 groups. Possible stepsizes for numerical integration are confined... 

Using splitting schemes of the exponential function allows for a generation of numerical integrators. For example [24, 22] ... 

Example 1 - Numerical integration of the Stress-Strength Interference (SSI) equation... 

This improved procedure is an example of the Runge-Kutta method of numerical integration. Because the derivative was evaluated at two points in the interval, this is called a second-order Runge-Kutta process. We chose to evaluate the mean derivative at points Pq and Pi, but because there is an infinite number of points in the interval, an infinite number of choices for the two points could have been made. In calculating the average for such choices appropriate weights must be assigned. 

These calculations employ a grid of points in space in order to perform the numerical integration. Grids are specified as a number of radial shells around each atom, each of which contains a set number of integration points. For example, in the (75,302) grid, 75 radial shells each contain 302 points, resulting in a total of 22,650 integration points. 

Intermediate values for C m can be obtained from a numerical integration of equation (10.158). When all are put together the complete heat capacity curve with the correct limiting values is obtained. As an example, Figure 10.13 compares the experimental Cy, m for diamond with the Debye prediction. Also shown is the prediction from the Einstein equation (shown in Figure 10.12), demonstrating the improved fit of the Debye equation, especially at low temperatures. 

Free Enzymes in Flow Reactors. Substitute t = zju into the DDEs of Example 12.5. They then apply to a steady-state PFR that is fed with freely suspended, pristine enzyme. There is an initial distance down the reactor before the quasisteady equilibrium is achieved between S in solution and S that is adsorbed on the enzyme. Under normal operating conditions, this distance will be short. Except for the loss of catalyst at the end of the reactor, the PFR will behave identically to the confined-enzyme case of Example 12.4. Unusual behavior will occur if kfis small or if the substrate is very dilute so Sj Ej . Then, the full equations in Example 12.5 should be (numerically) integrated. 

One Important aspect of the supercomputer revolution that must be emphasized Is the hope that not only will It allow bigger calculations by existing methods, but also that It will actually stimulate the development of new approaches. A recent example of work along these lines Involves the solution of the Hartree-Fock equations by numerical Integration In momentum space rather than by expansion In a basis set In coordinate space (2.). Such calculations require too many fioatlng point operations and too much memory to be performed In a reasonable way on minicomputers, but once they are begun on supercomputers they open up several new lines of thinking. 

Since Inn is a slowly varying function of n, the error goes essentially as n. This is the typical behaviour of a discretization error for a numerical integration [23], but is atypical for the examples that we want to study. 

This parameter can be obtained by numerical integration, for example using the trapezium rule, between time 0 and the time T when the last plasma sample has been taken. The remaining tail of the curve (between T and infinity) must be estimated from an exponential model of the slowest descending part of the observed plasma curve ((3-phase) as shown in Fig. 39.15. The area under the curve AUC can thus be decomposed into a tmncated and extrapolated part ... 

Thus Y1 is obtained not as the result of the numerical integration of a differential equation, but as the solution of an algebraic equation, which now requires an iterative procedure to determine the equilibrium value, Xj. The solution of algebraic balance equations in combination with an equilibrium relation has again resulted in an implicit algebraic loop. Simplification of such problems, however, is always possible, when Xj is simply related to Yi, as for example... 

The combination of the two mass balance equations, together with an explicit form of equilibrium relationship gives a system that is very easily solvable by direct numerical integration, as demonstrated in the simulation example BSTILL. 

Furthermore, the implementation of the Gauss-Newton method also incorporated the use of the pseudo-inverse method to avoid instabilities caused by the ill-conditioning of matrix A as discussed in Chapter 8. In reservoir simulation this may occur for example when a parameter zone is outside the drainage radius of a well and is therefore not observable from the well data. Most importantly, in order to realize substantial savings in computation time, the sequential computation of the sensitivity coefficients discussed in detail in Section 10.3.1 was implemented. Finally, the numerical integration procedure that was used was a fully implicit one to ensure stability and convergence over a wide range of parameter estimates. 

Export processes are often more complicated than the expression given in Equation 7, for many chemicals can escape across the air/water interface (volatilize) or, in rapidly depositing environments, be buried for indeterminate periods in deep sediment beds. Still, the majority of environmental models are simply variations on the mass-balance theme expressed by Equation 7. Some codes solve Equation 7 directly for relatively large control volumes, that is, they operate on "compartment" or "box" models of the environment. Models of aquatic systems can also be phrased in terms of continuous space, as opposed to the "compartment" approach of discrete spatial zones. In this case, the partial differential equations (which arise, for example, by taking the limit of Equation 7 as the control volume goes to zero) can be solved by finite difference or finite element numerical integration techniques. 

As an alternative to this traditional procedure, which involves, in effect, linear regression of equation 5.3-18 to obtain kf (or a corresponding linear graph), a nonlinear regression procedure can be combined with simultaneous numerical integration of equation 5.3-17a. Results of both these procedures are illustrated in Example 5-4. If the reaction is carried out at other temperatures, the Arrhenius equation can be applied to each rate constant to determine corresponding values of the Arrhenius parameters. 

To characterize the performance of a PFR subject to an axial gradient in temperature, the material and energy balances must be solved simultaneously. This may require numerical integration using a software package such as E-Z Solve. Example 15-4 illustrates the development of equations and the resulting profile for fA with respect to... 

This example can also be solved by numerical integration of equation (A) using the E-Z Solve software (file exl5-6.msp). For variable density, equation (B) is used to substitute for q. For constant density, q = qg. 

For other reaction networks, a similar set of equations may be developed, with the kinetics terms adapted to account for each reaction occurring. To determine the conversion and selectivity for a given bed depth, Ljh equations 23.4-11 and -14 are numerically integrated from x = 0 to x = Lfl, with simultaneous solution of the algebraic expressions in 23.4-12, -13, -15, and -16. The following example illustrates the approach for a series network. 

When integrating between numerical limits it may be more convenient to proceed directly with a numerical integration rather than through an analytical integration, particularly when the coefficients and integration limits are long numbers, for example, the integral... 

The simultaneous solution strategy offers several advantages over the sequential approach. A wide range of constraints may be easily incorporated and the solution of the optimization problem provides useful sensitivity information at little additional cost. On the other hand, the sequential approach is straightforward to implement and also has the advantage of well-developed error control. Error control for numerical integrators (used in the sequential approach) is relatively mature when compared, for example, to that of orthogonal collocation on finite elements (a possible technique for a simultaneous approach). 

The numerical integration in this example uses a trilinear resistancc-dcflection curve, j thus several additional values are needed ... 

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