Finite difference procedure

Numerical Solution Equations 6.40 and 6.41 represent a nonlinear, coupled, boundary-value system. The system is coupled since u and V appear in both equations. The system is nonlinear since there are products of u and V. Numerical solutions can be accomplished with a straightforward finite-difference procedure. Note that Eq. 6.41 is a second-order boundary-value problem with values of V known at each boundary. Equation 6.40 is a first-order initial-value problem, with the initial value u known at z = 0. 

Ehrlich (El) uses the Crank-Nicholson (C16) finite-difference procedure for the integration of the diffusion equation, with a three-point approximation of the space derivatives on either side of the moving... 

The finite difference method (FDM) is probably the easiest and oldest method to solve partial differential equations. For many simple applications it requires minimum theory, it is simple and it is fast. When a higher accuracy is desired, however, it requires more sophisticated methods, some of which will be presented in this chapter. The first step to be taken for a finite difference procedure is to replace the continuous domain by a finite difference mesh or grid. For example, if we want to solve partial differential equations (PDE) for two functions 4> x) and w(x, y) in a ID and 2D domain, respectively, we must generate a grid on the domain and replace the functions by functions evaluated at the discrete locations, iAx and jAy, (iAx) and u(iAx,jAy), or 4>i and u%3. Figure 8.1 illustrates typical ID and 2D FDM grids. 

An implicit finite-difference procedure for solving the laminar boundary layer equations was discussed in this chapter. Discuss how these boundary layer equations could be solved using an explicit procedure. In such a procedure, the terms duldy and d1uldyl are evaluated on the (i - 1) line. The continuity equation is trea ed in the same way in both procedures. Write a computer program based on this procedure and show by numerical experimentation with this program that instability develops if ... 

An explicit finite-difference procedure will be used here in dealing with the momentum and energy equations. Consider the nodal points shown in Fig. 4.27. It will be seen that a uniform grid spacing is used in the -direction. 

Because an explicit finite-difference procedure is being used to solve the momentum and energy equations, the solution can become unstable, i.e., as the solution proceeds it can diverge increasingly from the actual solution as indicated in Fig. 4.29. 

As previously mentioned, the solution to the above set of equations will here be obtained using a forward-marching, explicit finite-difference procedure. The solution starts with the known conditions on the inlet plane and marches forward in the -direction from grid line to grid line as indicated in Fig. 8.19. Consider the nodal points shown in Fig. 8.20. 

A number of computer programs are discussed in this book. These are all based on relatively simple finite-difference procedures that are developed in the book. While the numerical methods used are relatively simple, it is believed that if the students gain a good understanding of these methods and are exposed to the power of even simple numerical solution procedures, they will have little difficulty in understanding and using more advanced numerical methods. Examples of the use of the computer programs are included in the text. 

Fukui function is usually computed by a finite differences procedure,... 

Anomalies (such as V) come from irregularities in the shell filling process around the element. For the spin-orbit splitting (lower right), maxima do not appear at P, As, Sb, or Mn and Tc (half filled shells), but do appear at Sc, Ar, Zn, Kr, Cd (full shells) while minima appear at Sr and Xe. The variations of this property present more anomalies because its values result from small differences between large values, which cumulates experimental errors and anomalies due to the finite-difference procedure. 

Given partial third energy derivative information, further contributions to the coefficients in Eq (6) can be determined [7]. For example, the third order coefficient in Eq (6) requires the derivative of the force constant matrix with respect to s. This third derivative information can be estimated by a simple finite difference procedure if successive force constant matrices have been determined along the path. 

The use of a systems analog to improve the performance of a DTA apparatus and also to study the thermal effects in the DTA curve was investigated by Wilburn et al. (29,30). A finite-difference procedure was used to relate the thermal gradients within the samples and to generate or absorb heat according to a known equation. The influence of such physical properties on the shape and peak temperature of a typical DTA curve was calculated on an ICT Model 1909 computer. 

Thus closing the finite difference procedure defined in Section 7.6.1 for each 0 (and, hence, t) the values y, C, and also n from Eq. (7.47) can be calculated. 

Thus, Eq. 12.131, Eq. 12.130 for i = 2,3,..., IV — 2, and Eq. 12.132 will form a set of N - 1 equations with AT - 1 unknowns (y, y2, , yv- )- This set of coupled ordinary differential equations can be solved by any of the integration solvers described in Chapter 7. Alternately, we can apply the same finite difference procedure to the time domain, for a completely iterative solution. This is essentially the Euler, backward Euler and the Trapezoidal rule discussed in Chapter 7, as we shall see. 

A finite difference procedure using the marching technique is used to determine the pressure field, Pij- Boundary pressures of zero at the bearing ends and along the grooves are assumed. Both the pressure and the pressure gradient are set to zero at the end of the full film. Flows and loads are calculated from the pressure field. 

Nguyen TV, White R (1987) A finite difference procedure for solving coupled, nonlinear elliptic partial differentitil equations. Comput Chem Eng 11 543-546... 

Morf WE, Pretsch E, De Rooij NF (2007) Computer simulation of ion-selective membrane electrodes and related systems by finite-difference procedures. J Electroanal Chem 602 43-54... 

A finite difference procedure is first applied to the equation. The first derivative of C with respect to z is equivalent to the finite difference in the z-direction, i.e.,... 

Yurchenko et al. [80] reported flrst full-dimensional description of the electric-dipole moments of NH in their theoretical study of electric-dipole and Raman spectra of this ion. The ab initio DMSs were computed at the RCCSD(T)/aug-cc-pVTZ level of theory in the frozen-core approximation, referred to as ATZfc. Dipole moment values were computed in a numerical finite-difference procedure with an added external dipole field of 0.001 a.u. Relative to previously published theoretical results [84, 85] for NH, a noticeable improvement in the reproduction of the extant experimental data was achieved, with the electronic-property surfaces being more accurate than those previously available and therefore more useful for the prediction of spectroscopic properties, especially for transitions involving highly excited states. 

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