Method of collocation

Apply the methods of collocation and Galerkin to obtain the approximate solutions. 

Equation (1), with the associated boundary conditions, is a nonlinear second-order boundary-value ODE. This was solved by the method of collocation with piecewise cubic Hermite polynomial basis functions for spatial discretization, while simple successive substitution was adequate for the solution of the resulting nonlinear algebraic equations. The method has been extensively described before [9], and no problems were found in this application. 

In this section we introduce the concept of implicit Runge-Kutta methods of collocation type without aiming for completeness. The use of implicit Runge-Kutta methods is quite recent in multibody dynamics. There are higher order A-stable methods in this class for simulating highly oscillatory problems as well as methods specially designed for constrained multibody systems, see Sec. 5.4.2. 

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). 

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. 

Boundary value methods provide a description of the solution either by providing values at specific locations or by an expansion in a series of functions. Thus, the key issues are the method of representing the solution, the number of points or terms in the series, and how the approximation converges to the exact answer, i.e., how the error changes with the number of points or number of terms in the series. These issues are discussed for each of the methods finite difference, orthogonal collocation, and Galerkin finite element methods. 

In case an analytical solution of Eqs. (6) and (7) is not available, which is normally the case for non-linear isotherms, a solution for the equations with the proper boundary conditions can nevertheless be obtained numerically by the method of orthogonal collocation [38,39]. 

The second step in this equation involves a property called Green s identity. Using either method brings one to the point where the solutions of both require the same basic approaches solving a matrix problem. As in the case of collocation, the L sample points are used to generate the rows of the A matrix and b vector whose elements are written m,k = y) (x, y) dx dy and = b x, y)[Pg.257]

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). 

The resulting set of model partial differential equations (PDEs) were solved numerically according to the method of lines, applying orthogonal collocation techniques to the discretization of the unknown variables along both the z and x coordinates and integrating the resulting ordinary differential equation (ODE) system in time. 

A significant step in the numerical solution of packed bed reactor models was taken with the introduction of the method of orthogonal collocation to this class of problems (Finlayson, 1971). Although Finlayson showed the method to be much faster and more accurate than that based on finite differences and to be easily applicable to two-dimensional models with both radial temperature and concentration gradients, the finite difference technique remained the generally accepted procedure for packed bed reactor model solution until about 1977, when the analysis by Jutan et al. (1977) of a complex butane hydrogenolysis reactor demonstrated the real potential of the collocation procedure. 

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),... 

Another potential solution technique appropriate for the packed bed reactor model is the method of characteristics. This procedure is suitable for hyperbolic partial differential equations of the form obtained from the energy balance for the gas and catalyst and the mass balances if axial dispersion is neglected and if the radial dimension is first discretized by a technique such as orthogonal collocation. The thermal well energy balance would still require a numerical technique that is not limited to hyperbolic systems since axial conduction in the well is expected to be significant. 

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. 

There are three parameters in exponential collocation p N and a characteristic time, At. Different values of p have been used, and no difference has been observed p - 0 was used in the simulations that follow. The number of collocation points (N + 1) is equivalent to the reciprocal of the step size in finite difference methods the more points used, the greater is the accuracy, but the more time consuming the solution. The sensitivity of the solution to N and At is shown in Figure 2, which shows the movement of the combustion zone in an adiabatic dry ash gasifier when the throughput is reduced to 80% of full load at constant feed ratios. 

In the last section we considered explicit expressions which predict the concentrations in elements at (t + At) from information at time t. An error is introduced due to asymmetry in relation to the simulation time. For this reason implicit methods, which predict what will be the next value and use this in the calculation, were developed. The version most used is the Crank-Nicholson method. Orthogonal collocation, which involves the resolution of a set of simultaneous differential equations, has also been employed. Accuracy is better, but computation time is greater, and the necessity of specifying the conditions can be difficult for a complex electrode mechanism. In this case the finite difference method is preferable7. 

The exact numerical results for s at various ajlat were obtained by Keh and Yang [16] using the collocation results and are presented in Table 2. The corresponding approximate results from (93) are also shown in the same Table for comparison. It can be found that the results of s predicted by the method of reflections have significant errors when the ratio ajlat is small. For equilibrium double layer of finite thickness, an approximate analytic expression for atj was obtained by Ennis and White [56] using the reflection results, but no exact solution is available as yet. 

This is one of the variants of the finite element methods. The essence of orthogonal collocation (OC) is that a set of orthogonal polynomials is fitted to the unknown function, such that at every node point, there is an exact fit. The points are called collocation points, and the set of polynomials is chosen suitably, usually as Jacobi polynomials. The optimal choice of collocation points is to make them the roots of the polynomials. There are tables of such roots, and thus point placements, in Appendix A. The notable things here are the small number of points used (normally, about 10 or so will do), their... 

A30/Ai = 0, k /k2 = 1. This problem was first analyzed numerically by Brian and Beaverstock (12). Approximate analytical solution to this problem is also available (Ramachandran (13)). Figure 3 has been mainly presented to test the method for multi-step reaction scheme and hence detailed comparison of the accuracy of the solution and the influence of the number of collocation points was not done for this case. 

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