Method collocation

These methods are based on the concept of interpolation of unequally spaced points that is, choosing a function, usually a polynomial, that approximates the solution of a differential equation in the range of integration, Xq x and determining the coefficients of that function from a set of base points. 

Suppose that the. solutions y (jr) and y ix) of Eq. (5.106) can be approximated by the following polynomials, which we call trial functions.  

Numerical Solution of Ordinary Differential Equations Chapter 5 

The collocation method chooses the weighting functions to be the Dirac delta (unit impulse) function  

This implies that at a given number of collocation points, I A = 0. 1,. . . , n, the coefficients of the polynomials (5.126) are chosen so that Eq. (5.133) is satisfied that is, the polynomials are exact solutions of the differential equations at those collocation points (note that x = Xj). The larger the number of collocation points, the closer the trial function would resemble the true solution y (jf) of the differential equations. 

A method defined by (4.3.3) is called an implicit Runge-Kutta method. Unlike (4.2.2) the stages Xi are defined here implicitly. 

Collocation methods can be related to quadrature formulas by applying them to the special case x t) = f t),x tn) = Xn or, equivalently, to 

This requirement defines the collocation points Ci and by (4.3.2) and (4.3.4) also the other Runge-Kutta coefficients a j and bi. The maximal degree k can be achieved 

---

Parabolic Equations m One Dimension By combining the techniques apphed to initial value problems and boundary value problems it is possible to easily solve parabolic equations in one dimension. The method is often called the method of lines. It is illustrated here using the finite difference method, but the Galerldn finite element method and the orthogonal collocation method can also be combined with initial value methods in similar ways. The analysis is done by example. 

Other methods can be used in space, such as the finite element method, the orthogonal collocation method, or the method of orthogonal collocation on finite elements (see Ref. 106). Spectral methods employ Chebyshev polynomials and the Fast Fourier Transform and are quite useful for nyperbohc or parabohc problems on rec tangular domains (Ref. 125). 

Full rate modeling Accurate description of transitions Appropriate for shallow beds, with incomplete wave development General numerical solutions by finite difference or collocation methods Various to few... 

For both the finite difference and collocation methods a set of coupled ordinaiy differential eqiiafions results which are integrated forward in time using the method of hnes. Various software packages implementing Gear s method are popular. 

Orthogonal Collocation The orthogonal collocation method has found widespread application in chemical engineering, particularly for chemical reaction engineering. In the collocation method, the dependent variable is expanded in a series of orthogonal polynomials. See "Interpolation and Finite Differences Lagrange Interpolation Formulas. ... 

Different techniques are commonly used to solve the diffusion equation (Carslaw and Jaeger, 1959). Analytic solutions can be found by variable separation, Fourier transforms or more conveniently Laplace transforms and other special techniques such as point sources or Green functions. Numerical solutions are calculated for the cases which have no simple analytic solution by finite differences (Mitchell, 1969 Fletcher, 1991), which is the simplest technique to implement, but also finite elements, particularly useful for complicated geometry (Zienkiewicz, 1977), and collocation methods (Finlayson, 1972). 

In addition to stability considerations, the order of the approximation error is also a function of the system index. From the results of Brenan and Petzold (1987), systems of equations of higher index can be considered simply by choosing the appropriate IRK method with the appropriate integration error constraints. Based on error and stability considerations, Logsdon and Biegler (1989) concluded that minimum order requirements for collocation methods are the following ... 

In the collocation method, the weighting functions are chosen to be the Dirac delta function... 

Of the various methods of weighted residuals, the collocation method and, in particular, the orthogonal collocation technique have proved to be quite effective in the solution of complex, nonlinear problems of the type typically encountered in chemical reactors. The basic procedure was used by Stewart and Villadsen (1969) for the prediction of multiple steady states in catalyst particles, by Ferguson and Finlayson (1970) for the study of the transient heat and mass transfer in a catalyst pellet, and by McGowin and Perlmutter (1971) for local stability analysis of a nonadiabatic tubular reactor with axial mixing. Finlayson (1971, 1972, 1974) showed the importance of the orthogonal collocation technique for packed bed reactors. 

The orthogonal collocation method has several important differences from other reduction procedures. Jn collocation, it is only necessary to evaluate the residual at the collocation points. The orthogonal collocation scheme developed by Villadsen and Stewart (1967) for boundary value problems has the further advantage that the collocation points are picked optimally and automatically so that the error decreases quickly as the number of terms increases. The trial functions are taken as a series of orthogonal polynomials which satisfy the boundary conditions and the roots of the polynomials are taken as the collocation points. A major simplification that arises with this method is that the solution can be derived in terms of its value at the collocation points instead of in terms of the coefficients in the trial functions and that at these points the solution is exact. 

The differential equations (7.138) and (7.139) for the top-fired furnace, however, describe a boundary value problem with the boundary conditions (7.140) and (7.141) that can be solved via MATLAB s built-in BVP solver bvp4c that uses the collocation method, or via our modified BVP solver bvp4cf singhouseqr. m which can deal with singular Jacobian search matrices referred to in Chapter 5. On the other hand, the differential equation (7.142) is a simple first-order IVP. 

Specifically, in Chapter 3 we create a surface for a transcendental function /(a, y) as an elevation matrix whose zero contour, expressed numerically as a two row matrix table of values, solves the nonlinear CSTR bifurcation problem. In Chapter 6 we investigate multi-tray processes via matrix realizations in Chapter 5 we benefit from the least squares matrix solution to find search directions for the collocation method that helps us solve BVPs and so on. Matrices and vectors are everywhere when we compute numerically. That is, after the laws of physics and chemistry and differential equations have helped us find valid models for the physico-chemical processes. 

The catalyst intraparticle reaction-diffusion process of parallel, equilibrium-restrained reactions for the methanation system was studied. The non-isothermal one-dimensional and two-dimensional reaction-diffusion models for the key components have been established, and solved using an orthogonal collocation method. The simulation values of the effectiveness factors for methanation reaction Ch4 and shift reaction Co2 are fairly in agreement with the experimental values. Ch4 is large, while Co2 is very small. The shift reaction takes place as direct and reverse reaction inside the catalyst pellet because of the interaction of methanation and shift reaction. For parallel, equilibrium-restrained reactions, effectiveness factors are not able to predict the catalyst internal-surface utilization accurately. Therefore, the intraparticle distributions of the temperature, the concentrations of species and so on should be taken into account. 

Applying orthogonal collocation method, the above equations become... 

Simulation results of one-dimensional and two-dimensional models by orthogonal collocation method... 

The aim of this part of the book is to present the main and current numerical techniques that are used in polymer processesing. This chapter presents basic principles, such as error, interpolation and numerical integration, that serve as a foundation to numerical techniques, such as finite differences, finite elements, boundary elements, and radial basis functions collocation methods. 

In Chapter 11 of this book we will use the thin spline radial function to develop the radial basis functions collocation method (RBFCM). A well known property of radial interpolation is that it renders a convenient way to calculate derivatives of the interpolated function. This is an advantage over other interpolation functions and it is used in other methods such us the dual reciprocity boundary elements [43], collocation techniques [24], RBFCM, etc. For an interpolated function u,... 

In collocation methods, different sets of basis functions can also be used, for example radial basis functions, which are functions that depend only on the euclidean distance between collocation points i and j, ri j. 

The Chebyshev-collocation method. This example problem uses the Chebyshev -collocation method to approximate u(x,t) in a domain x e [-1,1] as a solution to the PDE... 

This allows us to represent partial differential equations as found in the balance equations using the collocation method. Equation (11.47) is a solution to a partial differential equation represented by a system of linear algebraic equations, formed by the interpolation coefficients, oij, and the operated radial functions. The interpolation coefficients are solved for using matrix inversion techniques to approximately satisfy the partial differential equation... 

O.A. Estrada, I.D. Lopez-Gomez, and T.A. Osswald. Modeling the non-newtonian calendering process using a coupled flow and heat transfer radial basis functions collocation method. Journal of Polymer Technology, 2005. 

J. Li and C.S. Chen. Some observations on unsymmetric RBF collocation methods for convection-diffusion problems. Inter. Journal for Numerical Methods in Eng., 57 1085-1094, 2003. 

The usual numerical methods for solving integral equations can be classified in two groups the collocation methods and the Galerkin methods. 

---