Partial Differential Equations
In this section, the application of the finite difference method for solution of partial differential type models is illustrated by several examples.
Contents
The Method of Finite Differences
The method of finite differences is one of many methods available for numerical solution of partial differential equations. Because of its simplicity and convenience, the method of finite differences is the most frequently used numerical method for solution of differential equations. This method provides algebraic approximations to derivatives so that differential equations can be transformed into a set of algebraic equations, which can be solved by appropriate numerical procedures. Although the finite difference approximations can be derived by various methods, a simple method based on the power series approach is presented here to avoid complicated mathematical derivation. Interested readers may resort to many excellent textbooks and literature available on the finite difference method. The information provided in this chapter is sufficient for many applications and for the purpose of this book. Most transport phenomenological models involve first and second order derivatives. Therefore, the following derivation is limited to the development of the first and second order derivative formulae. However, the higher order derivative formulae can be readily derived by the same approach presented in this chapter.
First Order Derivatives
In general, a function can be approximated by a power series as:
in which a0,a1,a2,... are some fitting coefficients. To determine the fitting coefficients, consider any set of three discrete function values fi_1, fi, and fi+l located at the sample points xi_1, xi, and xi+l, respectively,
More points could be considered for better accuracy. Higher order finite difference formulae can be derived easily using the quadrature method
as described by Civan (1994). With three points, we can write the following three quadratic approximations at i -1, i, i +1:
If the middle point is considered as a reference point, then the locations of the three points are given by:
Thus, substituting Eq. 16-23 into Eqs. 16-20 through 22, and then solving the resultant three algebraic equations simultaneously yields the following
expressions for the fitting coefficients of the quadratic expression:
On the other hand, the derivative of Eq. 16-19 for quadratic approximation is given by:
Thus, the following forward difference formula is obtained by substituting Eqs. 16-25 and 26 into Eq. 16-27 for a1,a2 at x = xi_1 =-Δx:
The central difference formula is obtained as, by substituting Eqs. 16- 25 and 26 for a1,a2 into Eq. 16-27 at x = x1; =0:
The backward difference formula is obtained as, by substituting Eqs. 16-25 and 26 for a1,a2 into Eq. 16-27 at x = xi+1 = Δx:
Second Order Derivatives
A similar procedure can be applied to derive the second (and higher) order derivative approximations. Thus, consider a power series expansion as:
Expressions similar to Eqs. 16-24 through 26 are obtained for the fitting coefficients, given by:
The derivative of the quadratic equation is obtained from Eq. 16-31 as:
Thus, the forward difference formula is obtained as, by substituting Eqs. 16-33 and 34 for b1,b2 at x = xi_1 =-Δx into Eq. 16-35:
The central difference formula is obtained as, by substituting Eqs. 16-33 and 34 for b1,b2 at x = xi=0 into Eq. 16-35:
The backward difference formula is obtained as, by substituting Eqs. 16-33 and 34 for b1,b2 into Eq. 16-35 for x = xi+1 =Δx:
However, only the central second order derivative formula is used in our models. Thus, substituting the first order forward and backward difference
formulae given by Eqs. 16-28 and 30 into Eq. 16-37, the central second order difference formula is obtained as:
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