Coupled solution approach

The result of the discretization process is a finite set of coupled algebraic equations that need to be solved simultaneously in every cell in the solution domain. Because of the nonlinearity of the equations that govern the fluid flow and related processes, an iterative solution procedure is required. Two methods are commonly used. A segregated solution approach is one where one variable at a time is solved throughout the entire domain. Thus, the x-component of the velocity is solved on the entire domain, then the y-component is solved, and so on. One iteration of the solution is complete only after each variable has been solved in this manner. A coupled solution approach, on the other hand, is one where all variables, or at a minimum, momentum and continuity, are solved simultaneously in a single... 

The relativistic coupled cluster method starts from the four-component solutions of the Drrac-Fock or Dirac-Fock-Breit equations, and correlates them by the coupled-cluster approach. The Fock-space coupled-cluster method yields atomic transition energies in good agreement (usually better than 0.1 eV) with known experimental values. This is demonstrated here by the electron affinities of group-13 atoms. Properties of superheavy atoms which are not known experimentally can be predicted. Here we show that the rare gas eka-radon (element 118) will have a positive electron affinity. One-, two-, and four-components methods are described and applied to several states of CdH and its ions. Methods for calculating properties other than energy are discussed, and the electric field gradients of Cl, Br, and I, required to extract nuclear quadrupoles from experimental data, are calculated. 

A microscopic calculation of the size-dependent diffusion is presented here. The calculation is based on the well-known mode coupling theoretical approach. The theoretical calculation is shown to give an excellent agreement with the simulation results and provides a physical interpretation of the enhanced diffusion. It is found that the enhanced diffusion of smaller solutes arises from the decoupling of the solute motion from the density mode of the solvent. 

Another attempt to go beyond the cell model proceeds with the Debye-Hiickel-Bjerrum theory [38]. The linearized PB equation is used as a starting point, however ion association is inserted by hand to correct for the non-linear couplings. This approach incorporates rod-rod interactions and should thus account for full solution properties. For the case of added salt the theory predicts an osmotic coefficient below the Manning limiting value, which is much too low. The same is true for a simplified version of the salt free case. 

C. L. anssen and H. F. Schaefer, Theor. Chim. Acta, 79,1 (1991). The Automated Solution of Second Quantization Equations with Applications to the Coupled-Cluster Approach. 

Generally, the design of modern solution algorithms for fluid flow problems is associated with the choice of primitive variables, the grid arrangement, and the solution approach [133]. In the class of pressure-based solution algorithms, both fully coupled and segregated approaches have been proposed. In the coupled approach, the discretized forms of the momentum and continuity... 

Fig. E.2. The zero-momentum exciton binding energies in units of Ej in the weak-coupling limit for a regularized Coulomb potential versus a/aj. Even parity (odd n) states (solid curves) and odd (even n) parity states (dashed curves). The energies of the odd parity solutions approach the Rydberg series as a/aj 0, while the energy of the n = 1 solution diverges.

CFX software focuses on one approach to solve the governing equations of motion (coupled algebraic multigrid), while the FLUENT product offers several solution approaches (density-segregated- and coupled-pressure-based methods). 

---