Successive approximation. Solving differential equations

The correction to the relaxing density matrix can be obtained without coupling it to the differential equations for the Hamiltonian equations, and therefore does not require solving coupled equations for slow and fast functions. This procedure has been successfully applied to several collisional phenomena involving both one and several active electrons, where a single TDHF state was suitable, and was observed to show excellent numerical behavior. A simple and yet useful procedure employs the first order correction F (f) = A (f) and an adaptive step size for the quadrature and propagation. The density matrix is then approximated in each interval by... 

Combining Eq. (24) with Eqs. (12) (14), Chen obtained f(a) for given N and c by the following numerical analysis. Firstly, a zero-th approximation to U(a) was calculated by numerical integration of Eq. (24) using a properly chosen zero-th approximation to f(a), secondly the calculated U(a) was used to obtain a first-order approximation to f(a) by solving the differential equation, Eq. (12), with Eqs. (13) and (14), and finally the process was iterated until the mean-square relative difference between the two successive approximations toT(a) became less than a prescribed small value. 

The value of the parameter n in Eqs. (9.13)—(9.14) is different from unity, and these differential equations are nonlinear and cannot be solved analytically. Therefore, Eqs. (9.13)-(9.16) subject to the conditions of Eqs. (9.17)-(9.21) were solved using numerical analysis techniques. Selim et al. (1976a) used the explicit-implicit finite-difference approximations as the method of solution. This was successfully used by Selim et al. (1975) for steady water flow conditions and by Selim et al. (1976a) for transient... 

The exact form of the matrices Qi and Q2 depends on the type of partial differential equations that make up the system of equations describing the process units, i.e., parabolic, elliptic, or hyperbolic, as well as the type of applicable boundary conditions, i.e., Dirichlet, Neuman, or Robin boundary conditions. The matrix G contains the source terms as well as any nonlinear terms present in F. It may or may not be averaged over two successive times corresponding to the indices n and n + 1. The numerical scheme solves for the unknown dependent variables at time t = (n + l)At and all spatial positions on the grid in terms of the values of the dependent variables at time t = nAt and all spatial positions. Boundary conditions of the Neuman or Robin type, which involve evaluation of the flux at the boundary, require additional consideration. The approximation of the derivative at the boundary by a finite difference introduces an error into the calculation at the boundary that propagates inward from the boundary as the computation steps forward in time. This requires a modification of the algorithm to compensate for this effect. 

Since we have not assumed that ip is normalized, we may take xp0 = 1 with no loss of generality. Substitution of Eq. (6.4) into Eq. (5.1) shows that successive approximations xpk can be found from solving the system of differential equations =. (6,5)... 

Example.—A set of equations resembling those of Ex. (5), of the preceding set of examples is solved in Technics, 1, 514, 1904, by the method of successive approximation, and under the assumption that kx and a are small in comparison with and c. Hint. Differentiate the second of equations (11) multiply out the first and on making the proper substitution... 

In the previous three chapters, we described various analytical techniques to produce practical solutions for linear partial differential equations. Analytical solutions are most attractive because they show explicit parameter dependences. In design and simulation, the system behavior as parameters change is quite critical. When the partial differential equations become nonlinear, numerical solution is the necessary last resort. Approximate methods are often applied, even when an analytical solution is at hand, owing to the complexity of the exact solution. For example, when an eigenvalue expression requires trial-error solutions in terms of a parameter (which also may vary), then the numerical work required to successfully use the analytical solution may become more intractable than a full numerical solution would have been. If this is the case, solving the problem directly by numerical techniques is attractive since it may be less prone to human error than the analytical counterpart. 

The connection among all of these disparate fields of study is the universal system of transport equations solved numerically for appropriate temporal and spatial scales. The solution of transport equations typically includes mass and momentum equations to solve the airflow field, while the addition of partial differential equations to represent scalars, such as temperature and contaminant concentrations, or solid phase for particles are problem specific. The required computational power to directly solve these partial differential equations is enormous. For example, the fastest petaflops supercomputers allow up to approximately 10 grid resolutiou that is only sufficient to solve simple indoor airflows in a single room, where a typical Reynolds number is 10. Directly solving an outdoor airflow problem is impossible, as Reynolds numbers are on the order of 10 . Therefore, the required grid resolution for a simple outdoor airflow problem would be close to 10 . As a compromise, building simulations have to be based on accurate physical models that can be successfully implemented and solved with the available computational power. 

---