Finite-difference algorithm

Nicholls, A., Honig, B. A rapid finite difference algorithm, utilizing successive over-relaxation to solve the Poisson-Boltzmann equation. J. Comp. Chem. 12 (1991) 435-445. 

Computer simulation of the reactor kinetic hydrodynamic and transport characteristics reduces dependence on phenomenological representations and idealized models and provides visual representations of reactor performance. Modem quantitative representations of laminar and turbulent flows are combined with finite difference algorithms and other advanced mathematical methods to solve coupled nonlinear differential equations. The speed and reduced cost of computation, and the increased cost of laboratory experimentation, make the former increasingly usehil. 

By breaking time down into small intervals dt, the equations of motion can then be solved directly using finite difference algorithms [48]. In the simplest form of MD the total energy of the molecular system is a conserved quantity. However, it is equally possible to carry out MD at constant temperature by employing one of a number of available thermostat algorithms [51]. When... 

Beam, R. M., and Wanning, R. R, An implicit finite difference algorithm for hyperbolic systems in conservation-law form. J. Comp. Phys. 22(1), 87 (1977). 

A. Nicholls and B. Honig, /. Comput. Chem., 12, 435 (1991). A Rapid Finite Difference Algorithm, Utilizing Successive Over-Relaxation to Solve the Poisson-Boltzmann Equation. 

A simple way to solve Eqs. (7.22)-(7.29) is with the use of the finite-difference algorithm (Eq. (7.30)), assuming that the right-hand side of Eq. (7.22) is zero, and the numerical dispersion of the finite-difference algorithm gives the plate count for that component. 

This subroutine calculates the nonzero elements of fpj= df /dOj, for use by DDAPLUS in constructing a column of the matrix B t). Bsub is executed only if MAIN sets lnfo(12)>0 and lnfo(14)=l before calling DDAPLUS. If dEjdOj is nonzero, leave lres=0 to activate DDAPLUS s finite-difference algorithm for this column of... 

This system of equations is an extension of the classical equilibrium-dispersive model to problems with two spadal dimensions, e.g., to the cases of a column having a cylindrical symmetry. It has no analytical solution but it is possible to write simple computational schemes for the calculation of its numerical solutions, using finite difference algorithms (see Chapter 10) [60]. 

Figure 11.6 Comparison of profiles calculated with OCFE and a finite difference method. N = 1000 plates. Line 1, forward-backward finite difference algorithm, Courant number = 2 line 2, OCFE, fp = 5 s. (a) 1 9 mixture, a = 1.5. (b) 1 1 mixture, a. = 1.5. (c) 5 1 mixture, a = 1.2.

Jaulmes and Vidal-Madjar [51] studied the influence of the mass transfer kinetics on band profiles, using a Langmuir second-order kinetics, and a constant axial dispersion coefficient, D. They derived numerical solutions using a finite difference algorithm. The influence of the rate constant on the band profile at various sample sizes is illustrated in Figure 14.18. As the mass transfer kinetics slows down, the band broadens and the shock layer becomes thicker. When the sample size increases, however, the influence of thermodynamics on the profile becomes more dominant, as shown by the change in shock layer thickness which decreases with increasing sample size. 

The finite difference algorithm is obtained by replacing the time derivatives by a forward difference, using an implicit rule to evaluate F(Y) at time t , and setting h = At. The result is... 

To solve the multi-fluid model version without the population balance, a relatively simple generalization of the two-fluid IPSA algorithms to multiple dispersed phases [21] are often used [96, 142]. In the referred three-fluid models only two dispersed phases were included in the simulations. Tomiyama and Shimada [190] made a similar four-fluid model which was solved by a pressure-based finite difference algorithm similar to the single phase SMAC [2] or SOLA [73] methods. In recent years, coupled solver technology is employed in some codes solving the three-fluid models [21, 6]. 

An expression for solute concentration versus angular displacement at the column outlet requires inversion of this solution back to the 0 domain, a procedure which cannot be performed analytically. A fast Fourier transform algorithm was used to perform the inversion numerically (21). Equations 1 and 2 were also solved using a finite difference algorithm. 

It is apparent that as the momentum p increases, the finite difference spectrum deviates more and more from the correct value. It is usually assumed that acceptable accuracy with the FD method is obtained when at least 10 points are used per wave period. This means also using 10 points per unit volume in phase space. The finite difference algorithms are based on a local polynomial approximation of the wave function and therefore the convergence of the method follows a power law of the form (Aq)n, where n is the order of the finite difference approximation. This semilocal description leads to a poor spectral representation of the kinetic energy operator, which will be true as well, for other banded representations of the kinetic energy operator such as the... 

Ermakov, S. V., Finite-difference algorithm for convection diffusion equation applied to electrophoresis problem, Informatica, 3, 173,1992. 

Wang, J., Cai, Q., Li, Z.-L, Zhao, H.-K., and Luo, R. (2009). Achieving energy conservation in Poisson-Boltzmann molecular dynamics Accuracy and precision with finite-difference algorithms, Chem. Phys. Lett. 468, pp. 112-118. 

J. A. Alden and R. G. Compton. A comparison of finite difference algorithms for the simulation of microband electrode problems with and without convective flow, J. Electroanal. Chem,. 402, 1-10 (1996). 

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