The Method of Weighted Residuals

The method of weighted residuals has been used in solving a variety of boundary value problems, ranging from fluid flow to heat and mass transfer problems. It is popular because of the interactive nature of the first step, that is, the user provides a first guess at the solution and this is then forced to satisfy the governing equations along with the conditions imposed at the boundaries. The left-over terms, called residuah, arise because the chosen form of solution does not exactly satisfy either the equation or the boundary conditions. How these residual terms are minimized provides the basis for parameter or function 

To illustrate the salient features of the method, we first consider the following boundary value problem in an abstract form, and then later attempt an elementary example of diffusion and reaction in a slab of catalyst material. We shall assume there exists an operator of the type discussed in Chapter 2 (Section 2.5) so that in compact form, we can write 

These boundary values could be initial values, but any boundary placement is allowed, for example dy(,0)/dx = 0, or y(l) = 0, where M(y) is simply a general representation of the operation on y as dy(0)/dx = 0 or y(l) = 0, respectively. 

The essential idea of the method of weighted residuals is to construct an approximate solution and denote it as y. Because of the approximate nature of the estimated solution, it may not, in general, satisfy the equation and the boundary conditions that is  

The method of weighted residuals will require two types of known functions. One is called the trial function, and the other is called the test function. The former is used to construct the trial solution, and the latter is used as a basis (criterion) to make the residual R small (a small residual leads to a small error in the approximate solution). To minimize the residual, which is usually a function of x, we need a means to convert this into a scalar quantity so that a 

In the general method of weighted residuals, it is desired to approximate f z) by a function/(z ai, t 2, fls. fln), which is a linear combination of the basis functions chosen from a linearly independent set. That is, the approximation of the solution is generally written as [76]  

For certain MWRs the coefficients are required to satisfy exactly the boundary conditions. Then, when substituted into the differential operator , the result of the operations is not, in general, g(z). Hence, a residual error will exist  

The notation in the MWR literature is to force the residual to zero in some average sense over the domain. That is, the MWR requires that the parameters a, are determined satisfying the integral  

There are several sub-groups of MWR methods, according to the particular choices for the weighting function Wi (z) employed. The performance of the resulting MWR is to a certain extent tied to the properties of the resulting coefficient matrix. To enable an efficient solution process it is desired that the coefficient matrix is symmetric, positive definite and characterized by a small condition number. At the same time the work needed to assemble the coefficient values should be minimized. 

In this method, the weighting functions Wi(z) are taken from the family of Dirac S functions in the domain. That is, Wi z) = S(z - Zi). The Dirac 5 functions are defined such that  

In some sense the method of weighted residuals (MWR) uses the integral form of the conservation or transport equations as its starting point. In particular. 

The key elements of the MWR are the expansion functions (also called the trail-, basis- or approximating functions) and the weight functions (also known as test functions). The trial functions are used as the basis functions for a truncated series expansion of the solution, which, when substituted into the differential equation, produces the residual. The test functions are used to ensure that the differential equation is satisfied as closely as possible by the truncated series expansion. This is achieved by minimizing the residual, i.e., the error in the differential equation produced by using the truncated expansion instead of the exact solution, with respect to a suitable norm. An equivalent requirement is that the residual satisfy a suitable orthogonality condition with respect to each of the test functions. 

The choice of test function distinguishes between the most commonly used spectral schemes, the Galerkin, tan, collocation, and least squares versions [22, 51, 84, 89] (see also [60, 132, 54, 17]). In the Galerkin approach, the test functions are the same as the trail functions, whereas in the collocation approach the test functions are translated Dirac delta functions centered at special, so-called collocation points. The collocation approach thus requires that the differential equation is satisfied exactly at the collocation points. Spectral tau methods are close to Galerkin methods, but they differ in the treatment of boundary conditions. 

Orthogonal collocation in the chemical engineering literature refers to the family of collocation methods with discretization grids associated to Gaussian quadrature methods [34, 204]. Spectral collocation methods for partial differential equations with an arbitrary distribution of collocation points are sometimes termed pseudo spectral methods [22]. 

In one view the choice of trial functions is one of the features which distinguishes the spectral methods (SMs) from the spectral element Methods (SEMs). The finite element methods (FEMs) can thus be regarded as SEMs with linear expansion- and weight functions. The trial functions for spectral methods are infinitely differentiable global functions. In the case of spectral element methods, the domain is divided into small elements, and the trail function is specified in each element. The trial and test functions are thus local in character, and well suited for handling complex geometries. 

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Finlayson, B. A. The Method of Weighted Residuals and Variational Frinciples, Academic, New York (1972). 

One of the most populax numerical methods for this class of problems is the method of weighted residuals (MWR) (7,8). For a complete discussion of these schemes several good numerical analysis texts are available (9,10,11). Orthogonal collocation on finite elements was used in this work to solve the model as detailed by Witkowski (12). 

A different approach in the use of orthogonal polynomials as a transformation method for the population balance is discussed in (8 2.) Here the error in Equation 11 is minimized by the Method of Weighted Residuals. This approach releases the restrictions on the growth rate and MSMPR operation, however, at the cost of the introduction of numerical integration of the integrals involved, which makes the method computationally unattractive. The applicability in determining state space models is presently investigated and results will be published elsewere. 

Although the finite difference technique is generally easily implemented and is quite robust, the procedure often becomes numerically prohibitive for packed bed reactor models since a large number of grid points may be required to accurately define the solution. Thus, since the early 1970s most packed bed studies have used one of the methods of weighted residuals rather than finite differences. 

The method of weighted residuals comprises several basic techniques, all of which have proved to be quite powerful and have been shown by Finlayson (1972, 1980) to be accurate numerical techniques frequently superior to finite difference schemes for the solution of complex differential equation systems. In the method of weighted residuals, the unknown exact solutions are expanded in a series of specified trial functions that are chosen to satisfy the boundary conditions, with unknown coefficients that are chosen to give the best solution to the differential equations ... 

The two most common of the methods of weighted residuals are the Galerkin method and collocation. In the Galerkin method, the weighting functions are chosen to be the trial functions, which must be selected as members of a complete set of functions. (A set of functions is complete if any function of a given class can be expanded in terms of the set.) Also according to Finlayson (1972),... 

Numerical methods such as the finite element method is based on the method of weighted residuals. To illustrate this method, we begin with boundary value partial differential equation (PDE) presented in the form... 

This approximation is commonly called the method of weighted residuals, it was first described by Crandall [7] and is fully explained in many references [11,12,13, 19, 23, 68], In principle, any set of independent functions, Wj, can be used for the purpose of weighting. According to the choice of the function, a different method is achieved. The most common choices are ... 

It is evident from these discussions that population balance equations are important in the description of dispersed-phase systems. However, they are still of limited use because of difficulties in obtaining solutions. In addition to the numerical approaches, solution of the scalar problem has been via the generation of moment equations directly from the population balance equation (H2, H17, R6, S23, S24). This approach has limitations. Ramkrishna and co-workers (H2, R2, R6) presented solutions of the population balance equation using the method of weighted residuals. Trial functions used were problem-specific polynomials generated by the Gram-Schmidt orthogonalization process. Their approach shows promise for future applications. 

Other methods may be more appropriate for equations with particular mathematical characteristics or when more accurate, robust, stable and efficient solutions are required. The alternative spectral methods can be classified as sub-groups of the general approximation technique for solving differential equations named the method of weighted residuals (MWR) [51]. The relevant spectral methods are called the collocation Galerkin, Tan- and Least squares methods. These methods can also be applied to subdomains. The subdomain... 

In the early 1970s, the standard finite element approximations were based upon the Galerkin formulation of the method of weighted residuals. This technique did emerge as a powerful numerical procedure for solving elliptic boundary value problems [102, 75, 53, 84, 50, 89, 17, 35]. The Galerkin finite element methods are preferable for solving Laplace-, Poisson- and and diffusion equations because they do not require that a variational principle exists for the problem to be analyzed. However, the power of the method is still best utilized in systems for which a variational principle exists, and it... 

The method of weighted residuals is adopted to derive the weak forms of the differential field equations for gas fluid pressure P, liquid saturation S and temperature T. 

Traditionally, all the methods that adopt the approximate functions v(a, ) are brought back to a single criterion the method of weighted residuals, according to which the P conditions needed to evaluate the P parameters are... 

In view of the solution of the potential problem - based on the method of weighted residual methods - a survey and discussion of the common solution techniques of the Laplace equation will be given. 

In Chapter Two, guided by the method of weighted residuals, a survey of the possibilities to solve the potential problem is given. The analytical and important numerical methods, namely the finite difference, the finite element and the boundary element method are classified and discussed. 

There are five widely used variations of the method of weighted residuals for engineering and science applications. They are distinguished by the choice of the test functions, used in the minimization of the residuals (Eq. 8.8). These five methods are... 

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