Solution of Partial Differential Equations Using Finite Differences

4 Solution of Partial Differential Equations Using Finite Differences 

In Chaps. 3 and 4, we developed the methods of finite differences and demonstrated that ordinary derivatives can be approximated, with any degree of desired accuracy, by replacing the differential operators with finite difference operators. 

In this section, we apply similar procedures in expressing partial derivatives in terms of finite differences. Since partial differential equations involve more than one independent variable, we first establish two-dimensional and three-dimensional grids, in two and three independent variables, respectively, as shown in Fig. 6.3. 

The notation (i,j) is used to designate the pivot point for the two-dimensional space and (i,j, k) for the three-dimensional space, where i, j, and k are the counters in the x, y, and z directions, respectively. For unsteady-state problems, in which time is one of the independent variables, the counter n is used to designate the time dimension. In order to keep the notation as simple as possible, we add subscripts only when needed. 

The distances between grid points are designated as Ax, Ay, and Az. When time is one of the independent variables, the time step is shown by At. 

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The previous chapter has discussed the solution of partial differential equations using the classical finite difference approach. This method of solution is most appropriate for physieal problems that match to a rectangular boundary area or that can be easily approximated by a rectangular boundary. One such class of PDEs is initial value problems in one spatial variable and one time variable. Other selected problems in two spatial dimensions are also amendable to this approach and selected examples are given in the previous chapter. 

The independent variable in ordinary differential equations is time t. The partial differential equations includes the local coordinate z (height coordinate of fluidized bed) and the diameter dp of the particle population. An idea for the solution of partial differential equations is the discretization of the continuous domain. This means discretization of the height coordinate z and the diameter coordinate dp. In addition, the frequently used finite difference methods are applied, where the derivatives are replaced by central difference quotient based on the Taylor series. The idea of the Taylor series is the value of a function f(z) at z + Az can be expressed in terms of the value at z. 

The material presented earlier was confined to steady-state flows over simply shaped bodies such as flat plates, with and without pressure gradients in the streamwise direction, or stagnation regions on blunt bodies. The simplicity of these flow configurations allows reduction of the problems to the solution of steady-state ordinary differential equations. The evaluation of convective heat transfer to more complex three-dimensional configurations, characteristic of real aerodynamic vehicles, involves the solution of partial differential equations. Even when the latter are confined to steady-state problems, they require extensive use of computers in the solution of finite difference or finite element formulations Nonsteady flows further complicate the problems by introducing another dimension, namely, time. 

The finite difference method is the oldest method for numerical solution of partial differential equations. It is also the easiest method used for simple geometries. The starting point is the conservation equation in differen-... 

In this chapter, we will present several alternatives, including polynomial approximations, singular perturbation methods, finite difference solutions and orthogonal collocation techniques. To successfully apply the polynomial approximation, it is useful to know something about the behavior of the exact solution. Next, we illustrate how perturbation methods, similar in scope to Chapter 6, can be applied to partial differential equations. Finally, finite difference and orthogonal collocation techniques are discussed since these are becoming standardized for many classic chemical engineering problems. 

Abstract Chapter 5 provides an examination of the numerical solutions of the dyeing models that can be applied to different conditions. Numerical simulation of the system involves the use of Matlab software to solve systems of highly non-linear simultaneous coupled partial differential equations. The finite difference and finite element methods are introduced The partition of the fibrous assembly geometry into small units of a simple shape, or mesh, is examined. Polygonal shapes used to define the element are briefly described. The defined geometries, boundary conditions, and mesh of the system enable solutions to the equations of flow or mass transfer models. 

This chapter has discussed the numerical solution of partial differential equations by the method of finite differences. This has made use of previously developed code for one dimensional equations. The FD method is most applicable to physi-... 

This chapter has discussed the numerical solution of partial differential equations by the method of finite elements. For some problems this is a complementary method to the finite difference method of the previous chapter. For PDFs involving one spatial dimension and one time dimension, either of these approaches can usually be used to obtain accurate solutions. The finite element approach really shines when one has a PDE and boundary problem involving a non-rectangular spatial region. The more general spatial element allowed by the FE approach makes it easy to describe general spatial boundaries and boundary conditions associated with the boundaries. 

E(u,v) is the inner product of u and v. In conventional OCFE calculation methods, the exponent of Az in Eq. 10.112 is 6, hence the degree of convergence between the calculated and the true profiles is of the sixth degree with respect to the space increment. One expects the value of C I4 to be rather small in the type of problems dealt with here. The fourth-order Runge-Kutta method used in the OCFE algorithm discussed here introduces an error of the fifth order. Accordingly, we may anticipate that the numerical solutions of the system of partial differential equations of chromatography calculated by an OCFE method will be more accurate than those obtained with a finite difference method [48] or even with the controlled diffusion method [49,50]. 

Finite difference methods Methods used for the calculation of ntunerical solutions of systems of partial differential equations. The differential elements in the differential equations are replaced by corresponding finite differences, giving difference equations. Stability and accuracy conditions must be satisfied (Chapter 10, Section 10.3). 

Several numerical methods, such as finite volume, finite difference, finite element, spectral methods, etc., are widely used for solving the complex set of partial differential equations. The latest computer technology allows us to obtain solutions with a mesh resolution on the order of millions of nodes. More-detailed discussion on numerical methodology is provided later. 

Numerical Solution. In the numerical formulation of THCC, Equations (2) and (3) are substituted into Equation (1). The resulting set of Nf, partial differential equations is transcribed into Nb finite-difference equations, using central differencing in space and the Crank-Nicolson method to obtain second-order accuracy in time. The set of unknowns consists of i = 1,..., ATft, and Pjt, k = 1,..., A/p, at each finite-difference node. Residue equations for the basis species are formed by algebraically summing all terms in the finite-difference forms of the transport equations. The finite-difference analogs of Equation (1) provide Ni, residue equations at each node the remaining Np residue equations are provided by the solubility products for the reactive solids. 

Similar global implicit formalisms can be developed to treat any form of partial differential equations in an atmospheric model. Accuracy of the solutions can be tested by increasing the temporal and spatial resolution (decreasing Ax, Ay, Az, and Ar) and repeating the calculation. However, because the full problem is solved as a whole, implicit formalisms require solution of very large systems, are slow, and require huge computational resources. Even if finite difference methods are easy to apply, they are rarely used for solution of the full (25.85) in atmospheric chemical transport models. 

There are many numerical approaches one can use to approximate the solution to the initial and boundary value problem presented by a parabolic partial differential equation. However, our discussion will focus on three approaches an explicit finite difference method, an implicit finite difference method, and the so-called numerical method of lines. These approaches, as well as other numerical methods for aU types of partial differential equations, can be found in the literature [5,9,18,22,25,28-33]. 

Errors are proportional to At for small At. When the trapezoid rule is used with the finite difference method for solving partial differential equations, it is called the Crank-Nicolson method. The implicit methods are stable for any step size but do require the solution of a set of nonlinear equations, which must be solved iteratively. The set of equations can be solved using the successive substitution method or Newton-Raphson method. See Ref. 36 for an application to dynamic distillation problems. 

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