Appendix 8.2 Finite Difference Approximations

This section describes a number of finite difference approximations useful for solving second-order partial differential equations that is, equations containing terms such as d f jd d. The basic idea is to approximate f 2 z. polynomial in x and then to differentiate the polynomial to obtain estimates for derivatives such as df jdx and d f jdx -. The polynomial approximation is a local one that applies to some region of space centered about point x. When the point changes, the polynomial approximation will change as well. We begin by fitting a quadratic to the three points shown below. 

This is a second-order approximation and can be used to obtain derivatives up to the second. Differentiate to obtain 

The value of the hrst derivative depends on the position at which it is evaluated. Setting A = +Aa gives a second-order, forward difference.  

Setting A = 0 gives a second-order, central difference.  

Setting A = A gives a second-order, backward difference.  

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This approximation is called a forward difference since it involves the forward point, z + Az, as well as the central point, z. (See Appendix 8.2 for a discussion of finite difference approximations.) Equation (8.16) is the simplest finite difference approximation for a first derivative. 

The derivatives F r are called the first-order parametric sensitivities of the model. Their direct computation via Newton s method is implemented in Subroutines DDAPLUS (Appendix B) and PDAPLUS. Finite-difference approximations are also provided as options in GREGPLUS to produce the matrix A in either the Gauss-Newton or the full Newton form these approximations are treated in Problems 6.B and 6.C. 

Appendix 8.3 Finite-Difference Approximations 321 Evaluating them as a function of Ax gives the following ... 

In equation (1), Cp, T, t, K, v2, Qj., v< , v are density, heat capacity, temperature, time, thermal conductivity, Laplaclan operator, heat generation rates of cord and rubber, volume fractions of cord and rubber, respectively. The equation can be solved numerically by use of either the finite difference approximation or the finite element method. The solution of the equation with related boundary conditions provide the temperature profile In the tire wall cross section. Examples of such solutions are shown In Figure 2. The procedure of obtaining such solutions Is outlined In Appendix I. 

Given some initial conditions, these equations are solved using finite-difference approximations. The fluid is moved by a continuous mass transport method identical to the one described in Appendix C. 

Appendix 8.3 describes how finite differences can be used to approximate derivatives. Divide the radius of the tube into / increments, Ar = Rfl. For the radial direction we use second-order, central differences ... 

To summarize the finite difference method, all we have to do is to replace all derivatives in the equation to be solved by their appropriate approximations to yield a finite difference equation. Next, we deal with boundary conditions. If the boundary condition involves the specification of the variable y, we simply use its value in the finite difference equation. However, if the boundary condition involves a derivative, we need to use the fictitious point which is outside the domain to effect the approximation of the derivative as we did in the above example at x =. The final equations obtained will form a set of algebraic equations which are amenable to analysis by methods such as those in Appendix A. If the starting equation is linear, the finite difference equation will be in the form of tridiagonal matrix and can be solved by the Thomas algorithm presented in the next section. 

The discretization scheme, which leads to an error 0 h ) for second-order differential equations (without first derivative) with the lowest number of points in the difference equation, is the method frequently attributed to Nu-merov [494,499]. It can be efficiently employed for the transformed Poisson Eq. (9.232). In this approach, the second derivative at grid point Sjt is approximated by the second central finite difference at this point, corrected to order h, and requires values at three contiguous points (see appendix G for details). Finally, we obtain tri-diagonal band matrix representations for both the second derivative and the coefficient function of the differential equation. The resulting matrix A and the inhomogeneity vector g are then... 

Mathematically, the system consists of parabolic PDEs, which were solved numerically by discretization of the spatial derivatives with finite differences and by solving the ODEs thus created with respect to time (Appendix 2). Typically, 3-5-point difference formulae were used in the spatial discretization. The first derivatives of the concentrations originating from a plug flow (Equations 9.1 through 9.3) were approximated with BD formulae, whereas the first and second derivatives originating from axial dispersion in the bulk phases and diffusion inside the catalyst particles were approximated by central difference formulae. Some simple backward (Equation 9.14) and central difference (Equation 9.15) formulae are shown here as examples ... 

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