Finite-difference representations

How do we represent the above equations with FD methods on a grid The answer is that a function is Taylor expanded about a central point Xi in the forward and backward directions  

This shows that the error in om 3-point formula for the second derivative is of order h. Higher order forms can be easily derived/ with a gain in accuracy of two orders with each pair of extra terms. Such high-order forms have been used extensively in electronic structure calculations. It is clear from the above approximation that the sign and magnitude of f xi) over the domain will be important for determining the value of the total energy. 

Using the above formula for the second derivative, we can represent the Poisson equation as 

In an iterative procedure that applies the appropriate update matrix to the function many times, the boundary values of the function (at points 1 and 6 above) are typically fixed (or periodic boundaries are enforced). We can write Eq. [13] in the simple matrix form 

During the solution process, the progress can be monitored by computing the residual 

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The truncation error in the first two expressions is proportional to Ax, and the methods are said to be first-order. The truncation error in the third expression is proportional to Ax, and the method is said to be second-order. Usually the last equation is used to insure the best accuracy. The finite difference representation of the second derivative is ... 

Figure 4-52 Finite Difference Representation and Boundary Conditions (After Pipes and Pagano [4-12])...

The task of the problem-independent chemistry software is to make evaluating the terms in Equations (6-10) as straightforward as possible. In this case subroutine calls to the Chemkin software are made to return values of p, Cp, and the and hk vectors. Also, subroutine calls are made to a Transport package to return the ordinary multicomponent diffusion matrices Dkj, the mixture viscosities p, the thermal conductivities A, and the thermal diffusion coefficients D. Once this is done, finite difference representations of the equations are evaluated, and the residuals returned to the boundary value solver. 

In the text, however, the numerical problem is formulated using momentum and total-energy balances on a finite control volume. The intent of this problem is to write a numerical simulation that is based on a finite-difference representation of the differential equations. 

Fig. 7.3 Illustration of the finite-difference representations of the heat equation in both standard form and differential-algebraic form.

Discuss the relationship between the continuity equation (Eq. 7.44) and Eq. 7.60 that represents the relationship between the physical radial coordinate and the stream function. Note that one is a partial differential equation and that the other is an ordinary differential equation. Formulate a finite-difference representation of the continuity equation in the primative form. Be sure to respect the order of the equation in the discrete representation. 

If the diffusion process is coupled with other influences (chemical reactions, adsorption at an interface, convection in solution, etc.), additional concentration dependences will be added to the right side of Equation 2.11, often making it analytically insoluble. In such cases it is profitable to retreat to the finite difference representation and model the experiment on a digital computer. Modeling of this type, when done properly, is not unlike carrying out the experiment itself (provided that the discretization error is equal to or smaller than the accessible experimental error). The method is known as digital simulation, and the result obtained is the finite difference solution. This approach is described in more detail in Chapter 20. 

A finite difference representation for a derivative can be introduced by recalling the definition of the derivative of a function 4> x)... 

Typically, these methods arrive at the same finite difference representation for a given problem. However, we feel that Taylor-series expansions are easy to illustrate and we will therefore use them here in the derivation of finite difference equations. We encourage the student of polymer processing to look up the other techniques in the literature, for instance, integral methods and polynomial fitting from Tannehill, Anderson and Pletcher [26] or from Milne [16] and finite volume approach from Patankar [18], Versteeg and Malalasekera [27] or from Roache [20]. 

As we can imagine, most of these issues are directly related to the order of the approximation used in the finite difference representation. In fact, the truncation error (as shown in Chapter 7) is the difference between the PDE and the FD representation, which is represented by the terms collapsed in 0(Axn). For problems represented by PDEs with more than one independent variable the truncation error will be the sum of the truncation error for each FD representation. For example, for a transient one dimensional PDE, where we use a first order approximation for the time derivative and a second order for the spatial derivative, we will have a truncation error that is O(At) + 0 Ax2), which can also be written as 0(At, Ax2). 

The consistency of a finite difference approximation is the behavior of this representation when the mesh is refined. In a one dimensional case, for example, the mesh will indicate the value for Ax, which, as we discussed above, dictates the value of the truncation error. Thus, a finite difference representation of a PDE is said to be consistent if the truncation error goes to zero as the grid size (or Ax) goes to zero. 

Models for the reacting polydispersed particles contain stiff ordinary differential equations. The stiffness is due partly to the wide range of thermal time constants of the particles and partly to the high temperature dependence of reactions like combustion and devolatilization. As an alternative to the established solution techniques based on Gear s method an iterative approach is developed which uses the finite difference representations of the differential equations. The finite differences are obtained by... 

The simplest finite-difference representation of the mixed boundary condition (8-44) may be readily obtained by considering for the spatial derivative the forward-... 

An improved 0(h2) finite-difference representation of the boundary condition (8-44) results by approximating the solution in the vicinity of the boundary by the second order Lagrange interpolating polynomial passing through the points (xi,cj), (x2,cg), and (x3,cg) (equally spaced gridpoints are assumed) ... 

FORTRAN computer program that predicts the species, temperature, and velocity profiles in two-dimensional (planar or axisymmetric) channels. The model uses the boundary layer approximations for the fluid flow equations, coupled to gas-phase and surface species continuity equations. The program runs in conjunction with CHEMKIN preprocessors (CHEMKIN, SURFACE CHEMKIN, and TRAN-FIT) for the gas-phase and surface chemical reaction mechanisms and transport properties. The finite difference representation of the defining equations forms a set of differential algebraic equations which are solved using the computer program DASSL (dassal.f, L. R. Petzold, Sandia National Laboratories Report, SAND 82-8637, 1982). 

If we divide the airshed into L cells and consider N species, LN ordinary differential equations of the form (15) constitute the airshed model. As might be expected, this model bears a direct relation to the partial differential equations of conservation (7). If we allow the cell size to become small, it can be shown that (15) is the same as the first-order spatial finite difference representation of (7) in which turbulent diffusive transport is neglected—i.e,. 

There are a number of numerical algorithms to solve the difference equation representation of the partial differential equations. Implicit algorithms such as Crank-Nicolson scheme where the finite difference representations for the spatial derivatives are averaged over two successive times, t = nAt and t = n + l)At, are frequently used because they are usually unconditionally stable algorithms. Most conservation laws lead to equations of the form... 

Such an equation is written for each interior grid point shown in Figure F. 1. At the boundaries, the hnite difference method uses the boundary conditions, again with the finite difference representation of derivatives. 

Fig. 15 Example calculation of temperature dependence of temperature sensitivity for double base propellant-N showing the difference between continuous functional representation of ( p(To) (solid line) and finite difference representation (data point at 300 K) for increasing Op with Tg, the latter over-estimates the former as shown by the single data point value at 300 K, 0.54 %/K, relative to the continuous function value at the same temperature, 0.35 %/K.

The first term on the right is Pick s second law, and we have seen that its finite difference representation is given by (B.1.6). Thus, we can immediately write the finite difference analog to (B.3.3) as... 

Table 2 Convergence of the Finite Difference Representation for the Harmonic Oscillator...

Second-Order-Correct Finite-Difference Representations of First and Second Derivatives... 

Solution of equations (23-22) yields second-order-correct finite difference representations for first- and second-derivatives of /(x) at x=xi. These generalized results for nonequispaced data points are ... 

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