Finite difference techniques

Molecular dynamics consists of the brute-force solution of Newton s equations of motion. It is necessary to encode in the program the potential energy and force law of interaction between molecules the equations of motion are solved numerically, by finite difference techniques. The system evolution corresponds closely to what happens in real life and allows us to calculate dynamical properties, as well as thennodynamic and structural fiinctions. For a range of molecular models, packaged routines are available, either connnercially or tlirough the academic conmuinity. 

Finite difference techniques are used to generate molecular dynamics trajectories with continuous potential models, which we will assume to be pairwise additive. The essential idea is that the integration is broken down into many small stages, each separated in time by a fixed time 6t. The total force on each particle in the configuration at a time t is calculated as the vector sum of its interactions with other particles. From the force we can determine the accelerations of the particles, which are then combined with the positions and velocities at a time t to calculate the positions and velocities at a time t + 6t. The force is assumed to be constant during the time step. The forces on the particles in their new positions are then determined, leading to new positions and velocities at time t - - 2St, and so on. 

An alternative metlrod of solution to these analytical procedures, which is particularly useful in computer-assisted calculations, is the finite-difference technique. The Fourier equation describes the accumulation of heat in a thin slice of the heated solid, between the values x and x + dx, resulting from the flow of heat tlirough the solid. The accumulation of heat in the layer is the difference between the flux of energy into the layer at x = x, J and the flux out of the layer at x = x + dx, Jx +Ox- Therefore the accumulation of heat in the layer may be written as... 

M.J. Frits, Two-Dimensional Lagrangian Fluid Dynamics Using Triangular Grids, in Finite-Difference Techniques for Vectorized Fluid Dynamics Calculations (edited by D.L. Book), Springer-Verlag, New York, 1981. 

Many numerical methods have been proposed for this problem, most of them finite-difference methods. Using a finite-difference technique, Brode (1955) analyzed the expansion of hot and cold air spheres with pressures of 2000 bar and 1210 bar. The detailed results allowed Brode to describe precisely the shock formation process and to explain the occurrence of a second shock. 

The derivatives are the financial type, e.g., option spreads. The methods used are implicit finite difference techniques. See Appendix 8.3. 

The matrix elements of the electric dipole and of the operators were determined for the perturbed m wavefunctions. The finite differences technique was applied to evaluate with A/ = 0.005 bohr (see [16] and refs, therein). All... 

This then provides a physical derivation of the finite-difference technique and shows how the solution to the differential equations can be propagated forward in time from a knowledge of the concentration profile at a series of mesh points. Algebraic derivations of the finite-difference equations can be found in most textbooks on numerical analysis. There are a variety of finite-difference approximations ranging from the fully explicit method (illustrated above) via Crank-Nicolson and other weighted implicit forward. schemes to the fully implicit backward method, which can be u.sed to solve the equations. The methods tend to increase in stability and accuracy in the order given. The difference scheme for the cylindrical geometry appropriate for a root is... 

Fig. 2. Schematic representation of the computational domain employed in the finite-difference technique.

DISCUSSION AND CONCLUSIONS In this study a general applicable model has been developed which can predict mass and heat transfer fluxes through a vapour/gas-liquid interface in case a chemical reaction occurs in the liquid phase. In this model the Maxwell-Stefan theory has been used to describe the transport of mass and heat. A film model has been adopted which postulates the existence of a well-mixed bulk and stagnant zones where the principal mass and heat transfer resistances are situated. Due to the mathematical complexity the equations have been solved numerically by a finite-difference technique. In this paper (Part I) the Maxwell-Stefan theory has been compared with the classical theory due to Pick for isothermal absorption of a pure gas A in a solvent containing component B. Component A is allowed to react by a unimolecular chemical reaction or by a bimolecular chemical reaction with... 

Figure 15.2. Region of interest for computing potential based on Laplace or Poisson equations, where (a) a complete rectangular grid is established to cover the region, which may be adapted to finite-difference techniques using (b) a five-point method, or (c) a finite-element approach based on sampling functions.

Temperature variations during the formation of LDPE foam sheet were investigated. A thermal model was coupled with a viscoelastic growth model, and an iterative finite difference technique was used to solve unsteady heat transfer equations and viscoelastic growth equations. The heat transfer characteristic time became comparable to the expansion time when the sheet thickness decreased to the millimetre range, during which foam thickness and density became sensitive to temperature effects. 12 refs. USA... 

There would seem to be little reason for using the tanks-in-series model for flow in empty tubes, since little correspondence exists between the physical picture and the model. However, Coste et al. (C22) found recently that a system with mass and heat dispersion combined with chemical reaction was easier to handle with this model than with the dispersion model. On the other hand, Carberry and Wendel (C8) have solved a similar problem with the dispersion-model by a finite-difference technique different from the one used by Coste et al, and found no difficulties. Thus the question of which model is best computationally is still not answered. 

Cleland and Wilhelm (C18) used a finite-difference technique which could be used for nonlinear reactions, but they limited their study to a first-order reaction. Experiments were also performed to test the results of the theory. In a small reaction tube, the two checked quite well. In a large tube there were differences which were explained by consideration of natural convection effects which were due to the fact that completely isothermal conditions were not maintained. This seems to be the only experimental data in the literature to date, and shows another area in which more work is needed. The preceding discussion considered only isothermal conditions except for Chambre (C12) who presented a general method for nonisothermal reactors. 

A significant step in the numerical solution of packed bed reactor models was taken with the introduction of the method of orthogonal collocation to this class of problems (Finlayson, 1971). Although Finlayson showed the method to be much faster and more accurate than that based on finite differences and to be easily applicable to two-dimensional models with both radial temperature and concentration gradients, the finite difference technique remained the generally accepted procedure for packed bed reactor model solution until about 1977, when the analysis by Jutan et al. (1977) of a complex butane hydrogenolysis reactor demonstrated the real potential of the collocation procedure. 

Although the finite difference technique is generally easily implemented and is quite robust, the procedure often becomes numerically prohibitive for packed bed reactor models since a large number of grid points may be required to accurately define the solution. Thus, since the early 1970s most packed bed studies have used one of the methods of weighted residuals rather than finite differences. 

A more detailed study of fuel cloud dispersion, though one lacking direct exptl verification, was made by Rosenblatt et al (Ref 23). The purpose of their study was to develop and use physically based numerical simulation models to examine the cloud dispersion and cloud detonation with fuel mass densities and particle size distributions as well as the induced air pressures and velocities as the principal parameters of interest. A finite difference 2-D Eulerian code was used. We quote The basic numerical code used for the FAE analysis was DICE, a 2-D implicit Eulerian finite difference technique which treats fluid-particle mixtures. DICE treats par-... 

The parabolic partial differential equation can be solved by separation of variables, although the solution shown in Fig. 4.9 is found by a finite-difference technique. Starting from rest (i.e., zero velocity everywhere), the expected steady-state parabolic velocity profile is reached in a dimensionless time of t 1. 

This is a linear equation whose solution can be determined by the method of separation of variables. Indeed, this is what Graetz did, and the details can be found in several fluid-mechanics texts. Here we will use a relatively simple implicit finite-difference technique to determine the solution in a spreadsheet. 

Dewey et al. (D3) present a numerical scheme for the ablation of an annulus with specified heat fluxes at the outer (ablating) surface and at the inner surface. An implicit finite difference technique is used which permits arbitrary variation of the surface conditions with time, and which allows iterative matching of either heat flux or temperature with external chemical kinetics. The initial temperature may also be an arbitrary function of radial distance. The moving boundary is eliminated by a transformation similar to Eq. (80). In addition a new dependent variable is introduced to... 

---