Solution techniques finite differences

Different techniques are commonly used to solve the diffusion equation (Carslaw and Jaeger, 1959). Analytic solutions can be found by variable separation, Fourier transforms or more conveniently Laplace transforms and other special techniques such as point sources or Green functions. Numerical solutions are calculated for the cases which have no simple analytic solution by finite differences (Mitchell, 1969 Fletcher, 1991), which is the simplest technique to implement, but also finite elements, particularly useful for complicated geometry (Zienkiewicz, 1977), and collocation methods (Finlayson, 1972). 

All numerical techniques require application of sampling theory. Briefly stated, one chooses a representative sample of points within the region of interest and at each point attempts to calculate iteratively the most accurate solution possible, guided by self-consistency of local solutions with each other and with the specified boundary conditions. We describe two seemingly contrasting techniques finite-difference and finite-element methods (1,2). 

Sun, W., T. Nousiainen, K. Muinonen, Q. Fu, N. G. Loeb and G. Videen, Light scatteri ng by Gaussian particles A solution with finite-difference time domain technique, J. Quant. Spectrosc. Radiative Transfer. 79-80, 1083-1090 (2003). 

The first result to be stressed is that, with the lEF approach, it is possible to reduce to an ASC form problems which are generally treated by resorting to 3D numerical techniques. Dielectrics with a tensorial permittivity c have been thus far treated by a combined Boundary Element - Finite Element Method (BEM/FEM) [36] and ionic solutions using Finite Difference (FD) approaches. 

Detailed Kinetic Modeling. Recent advances in computation techniques (11) have made it much easier to compute concentration-distance profiles for flame species. The one-dimensional isobaric flame equations are solved via a steady state solution using finite difference expressions. An added simplification is that the energy equation can be replaced with the measured temperature profile. In the adaptive mesh algorithm, the equations are first solved on a relatively coarse grid. Then additional grid points could be included if necessary, and the previous solution interpolated onto the new mesh where it served as the initial solution estimate. This process was continued until several termination criteria were satisfied. 

We shall discuss two techniques to find particular solutions for finite difference equations, both having analogs with continuous ODE solution methods ... 

The numerical error in the PDE initial value, boundary value solution by finite differences has been extensively explored for one PDE which is first order in the time derivative. In order to have an exactly known solution with which to compare the numerical techniques, a simple PDE boundary value problem must be considered. White it is sometimes dangerous to extrapolate too far from a known result, the results obtained here can provide some guidance with regard to the ex-... 

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. 

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... 

D.F. Hawken, J.J. Gottlieb, and J.S. Hansen, Review of Some Adaptive Node-Movement Techniques in Finite-Element and Finite-Difference Solutions of Partial Differential Equations, J. Comput. Phys. 95 (1991). 

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. 

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... 

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). 

The test technique shown in Fig. 7 is used to determine the variation of D with depth (i.e., length of the soil-solid sample) and with number of pore volumes (i. e.,PV) of passage of COM solution. By analyzing the various sections of a soil-solid sample, one can obtain the pollutant profile shown in Fig. 7. Casting Eq. (82) in the finite difference form yields the following ... 

Pollutants emitted by various sources entered an air parcel moving with the wind in the model proposed by Eschenroeder and Martinez. Finite-difference solutions to the species-mass-balance equations described the pollutant chemical kinetics and the upward spread through a series of vertical cells. The initial chemical mechanism consisted of 7 species participating in 13 reactions based on sm< -chamber observations. Atmospheric dispersion data from the literature were introduced to provide vertical-diffusion coefficients. Initial validity tests were conducted for a static air mass over central Los Angeles on October 23, 1968, and during an episode late in 1%8 while a special mobile laboratory was set up by Scott Research Laboratories. Curves were plotted to illustrate sensitivity to rate and emission values, and the feasibility of this prediction technique was demonstrated. Some problems of the future were ultimately identified by this work, and the method developed has been applied to several environmental impact studies (see, for example, Wayne et al. ). 

Gradient diffusion was assumed in the species-mass-conservation model of Shir and Shieh. Integration was carried out in the space between the ground and the mixing height with zero fluxes assumed at each boundary. A first-order decay of sulfur dioxide was the only chemical reaction, and it was suggested that this reaction is important only under low wind speed. Finite-difference numerical solutions for sulfur dioxide in the St. Louis, Missouri, area were obtained with a second-order central finite-difference scheme for horizontal terms and the Crank-Nicolson technique for the vertical-diffusion terms. The three-dimensional grid had 16,800 points on a 30 x 40 x 14 mesh. 

Since the stream function depends upon Reynolds number, the rate of transfer will depend upon both Re and Sc except in the limit Re 0 treated in Chapter 3. Solutions to Eq. (5-19) have been obtained using the techniques discussed earlier, i.e., finite-difference schemes (A6, D5, II, M6, W9), solution to the time-dependent problem (Hll), and series expansions (D5). 

Numerical solutions of the flow around and inside fluid spheres are again based on the finite difference forms of Eqs. (5-1) and (5-2) (BIO, H6, L5, L9). The necessity of solving for both internal and external flows introduces complications not present for rigid spheres. The boundary conditions are those described in Chapter 3 for the Hadamard-Rybczynski solution i.e., the internal and external tangential fluid velocities and shear stresses are matched at R = 1 (r = a), while Eq. (5-6) applies as R- co. Most reported results refer to the limits in which k is either very small (BIO, H5, H7, L7) or large (L9). For intermediate /c, solution is more difficult because of the coupling between internal and external flows required by the surface boundary conditions, and only limited results have been published (Al, R7). Details of the numerical techniques themselves are available (L5, R7). 

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