Finite difference techniques numerical solutions

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. 

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... 

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. 

An immense number of analytical solutions for conduction heat-transfer problems have been accumulated in the literature over the past 100 years. Even so, in many practical situations the geometry or boundary conditions are such that an analytical solution has not been obtained at all, or if the solution has been developed, it involves such a complex series solution that numerical evaluation becomes exceedingly difficult. For such situations the most fruitful approach to the problem is one based on finite-difference techniques, the basic principles of which we shall outline in this section. 

These mathematical representations are complex and it is necessary to use numerical techniques for the solution of the initial-boundary value problems associated with the descriptions of fluidized bed gasification. The numerical model is based on finite difference techniques. A detailed description of this model is presented in (11-14). With this model there is a degree of flexibility in the representation of geometric surfaces and hence the code can be used to model rather arbitrary reactor geometries appropriate to the systems of interest. [The model includes both two-dimensional planar and... 

Ryskin, G., and Leal, L. G., Numerical solution of free-boundary problems in fluid mechanics. Part I. The finite difference technique. J. Fluid Mech. 148,1 (1984a). 

The second method for solving the PSD evolution equations is brute-force numerical solution using first-order finite difference. Whereas a solution can always be obtained by this technique, it suffers from numerical instability, from the lack of any automatic check on accuracy, and from requiring large amounts of computer time. 

Solution of these equations with appropriate boundary and initial conditions gives the contaminant concentration in both space and time which is the objective of the exrcise. Highly simplified problems (1) and (2) can have analytical solutions (see for example Ogata and Banks [ 3]) but for most real problems, numerical techniques such as finite differences or finite elements have to be used. For the saturated zone examples discussed below, the solutions were obtained using finite difference techniques on the computer of Delft Soil Mechanics Laboratory. 

Solve this problem using Maple s dsolve numeric command, shooting technique and finite difference technique. Initially choose L = 5 and increase L to make sure that the solution has converged (i.e., change L = 6 dy... 

The coupled equations had been solved by numerical computation using an implicit finite difference technique [34]. It was found from Figs. 4.5 and 4.6 that the fraction of solutes [copper(ll) and nickel(ll)] extracted were higher with a lower initial external phase solutes concentration. This was due to a higher distribution coefficient for a lower initial external phase solutes concenttation. 

Finite-difference techniques were used to compute numerical solutions as column-breakthrough curves because of the nonlinear Freundlich isotherm in each transport model. Along the column, 100 nodes were used, and 10 nodes were used in the side-pore direction for the profile model. A predictor-corrector calculation was used at each time step to account for nonlinearity. An iterative solver was used for the profile model whereas, a direct solution was used for the mixed side-pore and the rate-controlled sorption models. 

An approximative analytical treatment of S-T+-type CIDNP of radical pairs in micelles has recently been given [66]. Comparison with numerical solutions of the stochastic Liouville equation obtained by a finite difference technique showed the accuracy of the approximate solution to be quite good. 

A numerical solution, based on a linear adsorption isotherm, was performed by Ziller et al. (1985) using a finite difference technique. An example of the numeric results is given in Fig. 4.16. The dramatic drop of the surface concentration at x=L/K is explained by the boundary condition, which assumed a 10-fold expansion of the interface. Due to the complexity of the problem, it seems impossible to derive an analytical solution without further serious simplifications. 

We first tested the finite difference technique for a single inclusion, which has a simple analytical representation (in the uniform case) or a cylindrically symmetric, simple numerical solution (the nonuniform case. Section III). We used the approximation of Eq. 

Equations 4.1-4.3 describe an ideal constant flow rate problem. The explicit finite difference technique was applied for the numerical solution of Equations 4.1-4.3. An experimental study was carried out of the flow of carbon nanotube-PEDOT PSS solution (p 8.8 mPa-s and o = 68 mN/m) through a needle of 60 pm internal diameter at a constant flow rate to validate the computer simulations. Flow rates of 60 ml/h resulted in a continuous jet stream, and this was also predicted by the computer simulation, see Figure 4.1. On the other hand, as the flow rate was lowered below Wecriticai = 1 5 drop formation occurred. This was also predicted by the computer simulation as is illustrated in Figure 4.2. 

For those cases of explosive initiation in which the important mechanism of energy transfer is heat conduction, the usual method of numerical solution is by a finite difference technique as is described in Appendix A for the SIN code. The technique for two-dimensional geometries is described in Appendixes B and C. If one is interested only in heat conduction, one can use special purpose codes such as TEPLO described in reference 1. 

The traditional approach to understanding both the steady-state and transient behavior of battery systems is based on the porous electrode models of Newman and Tobias (22), and Newman and Tiedermann (23). This is a macroscopic approach, in that no attempt is made to describe the microscopic details of the geometry. Volume-averaged properties are used to describe the electrode kinetics, species concentrations, etc. One-dimensional expressions are written for the fluxes of electroactive species in terms of concentration gradients, preferably using the concentrated solution theory of Newman (24). Expressions are also written for the species continuity conditions, which relate the time dependence of concentrations to interfacial current density and the spatial variation of the flux. These equations are combined with expressions for the interfacial current density (heterogeneous rate equation), electroneutrality condition, potential drop in the electrode, and potential drop in the electrolyte (which includes spatial variation of the electrolyte concentration). These coupled equations are linearized using finite-difference techniques and then solved numerically. 

The three modes of numerical solution techniques are finite difference, finite element, and spectral methods. These methods perform the following steps ... 

Problems of inclusions in solids are also treated by exact elasticity approaches such as Muskhelishvili s complex-variable-mapping techniques [3-9]. In addition, numerical solution techniques such as finite elements and finite differences have been used extensively. 

These coupled second-order partial differential equations do not have a closed-form solution. Accordingly, the approximate numerical technique of finite differences is employed. First, however, the boundary conditions must be prescribed in order to complete the formulation of the problem. Symmetry of the laminate about several planes permits reduction of the region of consideration to a quarter of the laminate cross section in the y-z plane at any value of x as shown in Figure 4-52. There, along the stress-free upper surface. 

Without a solution, formulated mathematical systems (models) are of little value. Four solution procedures are mainly followed the analytical, the numerical (e.g., finite different, finite element), the statistical, and the iterative. Numerical techniques have been standard practice in soil quality modeling. Analytical techniques are usually employed for simplified and idealized situations. Statistical techniques have academic respect, and iterative solutions are developed for specialized cases. Both the simulation and the analytic models can employ numerical solution procedures for their equations. Although the above terminology is not standard in the literature, it has been used here as a means of outlining some of the concepts of modeling. 

Models of the above have been presented by various researchers of the U.S. Geological Survey (USGS) and the academia. The above equation has been solved principally (a) numerically over a temporal and spatial discretized domain, via finite difference or finite element mathematical techniques (e.g., 11) (b) analytically, by seeking exact solutions for simplified environmental conditions (e.g., 12) or (c) probabilistically (e.g., 13). 

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