Finite difference approach

Solving Newton s equation of motion requires a numerical procedure for integrating the differential equation. A standard method for solving ordinary differential equations, such as Newton s equation of motion, is the finite-difference approach. In this approach, the molecular coordinates and velocities at a time it + Ait are obtained (to a sufficient degree of accuracy) from the molecular coordinates and velocities at an earlier time t. The equations are solved on a step-by-step basis. The choice of time interval Ait depends on the properties of the molecular system simulated, and Ait must be significantly smaller than the characteristic time of the motion studied (Section V.B). 

The continuum treatment of electrostatics can also model salt effects by generalizing the Poisson equation (12) to the Poisson-Boltzmann equation. The finite difference approach to solving Eq. (12) extends naturally to treating the Poisson-Boltzmann equation [21], and the boundary element method can be extended as well [19]. 

In the examples to be discussed in Section 27.3, the toolbox of descriptors will be relatively compact, containing rj, S, fir), s(r), and r](r). The evaluation of these quantities will, in most cases, be done by the finite difference approach (for a review see Ref. [10]) leading to the working equations... 

Early design and simulation of large-diameter, melt-fed extruders were described by Fenner [17]. A numerical simulation of the axial pressure and temperature fora screw similar to that shown in Fig. 15.8 is shown in Fig. 15.10. This simulation was performed using a three-dimensional method using a finite difference approach. The process starts with an LDPE resin (2 dg/min, 2.16 kg, 190 °C) in the low-pressure separator at a pressure of 0.04 MPa (gauge) and a temperature of 230 °C. 

Meittinen et al. (1992) and Matsumiya et al. (1993) have also attempted to explicitly solve the diffusion equations using a finite difference approach for the diffusion of solute in the solid phase. Meittinen et al. (1992) used different approaches dependent on whether the solute was fast-moving or slow-moving and treated solidification involving 6-ferrite and austenite in different ways. This may give reasonable answers for steels but the programme then loses general applicability to other material types. 

Law et al. [15] determined the diffusion coefficient for benzyl penicillin in thin films of Palacos, Simplex and CMW cements assuming that antibiotic transport can be described by Fick s law using a finite difference approximation to quantify transient non-steady-state behaviour. These investigators found that the diffusion coefficient was increased in the presence of additives and proposed that the finite difference approach could be applied to determine release of antibiotic from preloaded PMMA beads. Dittgen and Stahlkopf [16] showed that incorporation of amino acids of varying solubilities also affected release of chloramphenicol from polymethacrylic... 

Although we will not belabor this point here, the folding technique can easily be adapted to the mathematical description. The reader should recognize that the presence of the barrier does not invalidate either of Fick s laws but does influence the boundary conditions chosen to obtain a specific solution, whether analytically or by the finite difference approach. 

In order to calculate the time derivatives of the state variables, we used a finite-difference approach, sampling the simulation data with a sample time r = 0.05 h ... 

Steady state equations fXr the adiabatic case corresponding to (1) through (4) were solved by the parameter mapping technique combined with the Newton-Fox shooting algorithm. The steady state nonadiabatic problems were solved by the finite-difference approach. 

Since we are considering the case of constant thermal conductivity, the heat flows may all be expressed in terms of temperature differentials. Equation (3-24) states very simply that the-net heat flow into any node is zero at steady-state conditions. In effect, the numerical finite-difference approach replaces the continuous temperature distribution by fictitious heat-conducting rods connected between small nodal points which do not generate heat. 

Considering a total of three, equally spaced nodes in the medium, two at the boundaries and one at the middle, estimate the exposed surface temperature of the plaje under steady conditions using the finite difference approach. 

The nodal temperatures in transient problems normally change during each time step, and you may be wondering whether to use temperatures at the previous time step i or the ir u time step i + 1 for the terms on the left side of Eq. 5-39. Well, both arc reasonable approaches and both are used in practice. The finite difference approach is called the explicit method in the first case and the iinplicit method in the second case, and they are expressed in the general form as (Hg. 5-39)... 

Oristaglio, M., and G. Hohmann, 1984, Diffusion of electromagnetic fields into a two-dimensional Earth A finite-difference approach Geophysics, 49, 870-894. 

Once the initial state of the system has been set up, the equations of motion (e.g., in the form of the Newton s equations) are solved numerically by means of the finite difference approach. Known atomic positions, velocities, and forces at time t are used to obtain the coordinates and velocities at time t + At, after which the procedure is repeated. The value of the time step that can be used depends on the specific model of the system and the accuracy of the integration algorithm. In general, the time step should be smaller than t/lO, where t is the minimum characteristic time in the system, e.g., the period of the highest frequency vibration. 

Two tests are available to ensure that the equations of motion are solved correctly in an MD simulation. The first test assesses the conservation of energy. In the absence of terms explicitly dependent on time, the Hamiltonian of the system must be a conserved quantity, and integrating Newton s equations of motion should conserve the total energy of the system E. In fact, small fluctuations of the energy will result from the finite difference approach and round-off errors. Numerical criteria often used for the energy fluctuations are... 

The last approximation corresponds to the finite differences approach to the first derivative, and I and A represent the ionization potential and electron affinity of the... 

In the finite difference approach, using the equation of the ideal model and appropriate numerical dispersion, we try to compensate exactly the axial dispersion term which is dropped from Eq. 11.7 by a numerical error term. We have shown in Chapter 10 (Section 10.3.5.2) that this is possible provided we choose values of the time, h, and the space, t, integration increments such that one of the following relationships is verified ... 

The finite difference approach is a widely used discretization technique because of its simplicity. Finite difference approximations of derivatives are obtained by using truncated Taylor series. The following Taylor expansions can be used ... 

As mentioned above, the goal of MD is to compute the phase-space trajectories of a set of molecules. We shall just say a few words about numerical technicalities in MD simulations. One of the standard forms to solve these ordinary differential equations i.s by means of a finite difference approach and one typically uses a predictor-corrector algorithm of fourth order. The time step for integration must be below the vibrational frequency of the atoms, and therefore it is typically of the order of femtoseconds (fs). Consequently the simulation times achieved with MD are of the order of nanoseconds (ns). Processes related to collisions in solids are only of the order of a few picoseconds, and therefore ideal to be studied using this technique. 

Equilibrium-stage processes are discrete steps. One approach to the analysis is an evaluation as a finite difference calculation where each stage is an equal and discrete interval in the process train. Obviously, a process simulator can be used. Finite difference approaches, including ones shown here, can be implemented on spreadsheets for rapid estimates. 

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