Collocation polynomial

This relationship can be extended to multivariate problems, even if, in this case, it is questionable how to subdivide the degrees of freedom Np, — Np among the different components. It should be also underlined that oo as No — Np 0 this clearly shows that using a too large number of parameters, or even resorting to a collocation polynomial, is not a proper scientific procedure. 

The orthogonal collocation polynomial approximation using a single parameter trial function was employed to solve equations (l)-(3), In addition to the solution for time concentration and activity profiles, effectiveness factors representing the combined effect of mass transfer resistance and poisoning in terms of pellet surface conditions were computed according to... 

To start the Newton iteration the collocation polynomial used in the last step is extrapolated to obtain guesses for the stage values ... 

Figure 4.7 The collocation polynomial in a typical step of RADAU5 acting on the unconstrained truck...

Note, that the order is defined via the global error in the discretization points tn-If we evaluate the collocation polynomial between these points and compare the result with the exact solution of the differential equation one usually gets a lower order (cf. Sec. 4.5). This phenomenon is called superconvergence of collocation methods. 

For implicit Runge-Kutta methods based on collocation polynomials these polynomials can serve as continuous representation of the solution. Unfortunately for many collocation methods the order of the continuous representation is q < p — 1, so that the requirement at the beginning of this section is not met. 

Theorem 4.5.2 The continuous representation based on a collocation polynomial u of an s-stage implicit Runge-Kutta method is of order s, i.e. 

First, we consider implicit methods and restrict ourselves to those related to collocation polynomials. For this end we have to extend Definition 4.3.1 to semi-explicit DAEs. 

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). 

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. ... 

The differential equation is evaluated at certain collocation points. The collocation points are the roots to an orthogonal polynomial, as first used by Lanczos [Lanczos, C.,/. Math. Phys. 17 123-199 (1938) and Lanczos, C., Applied Analysis, Prentice-Hall (1956)]. A major improvement was proposed by Villadsen and Stewart [Villadsen, J. V., and W. E. Stewart, Chem. Eng. Sci. 22 1483-1501 (1967)], who proposed that the entire solution process be done in terms of the solution at the collocation points rather than the coefficients in the expansion. This method is especially useful for reaction-diffusion problems that frequently arise when modeling chemical reactors. It is highly efficient when the solution is smooth, but the finite difference method is preferred when the solution changes steeply in some region of space. The error decreases very rapidly as N is increased since it is proportional to [1/(1 - N)]N 1. See Finlayson (2003) and Villadsen, J. V., and M. Michelsen, Solution of Differential Equation Models by Polynomial Approximations, Prentice-Hall (1978). 

Note that state variable profiles are one order higher than the controls because they have explicit interpolation coefficients defined at the beginning of each element. With this representation of Z(t) and U(t), we can extend this approach to piecewise polynomials and apply orthogonal collocation on NE finite elements (of length Aoc,). This leads to the following nonlinear algebraic equations ... 

A detailed analysis of polynomial approximation using collocation at Legendre roots by de Boor (1978) shows that the global error e(t) = Z(t) — z +i(t) satisfies the relation... 

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. 

Since the resulting system after radial collocation is still too complex for direct mathematical solution, the next step in the solution process is discretization of the two-dimensional system by orthogonal collocation in the axial direction. Although elimination of the spatial derivatives by axial collocation increases the number of equations,8 they become ordinary differential equations and are easily solved using traditional techniques. Since the position and number of points are the only factors affecting the solution obtained by collocation, any set of linearly independent polynomials may be used as trial functions. The Lagrangian polynomials of degree N based on the collocation points... 

The collocation points are calculated using programs given by Villadsen and Michelsen (1978) for calculating the zeros of an arbitrary Jacobi polynomial P% P x) that satisfies the orthogonality relationship... 

Two major forms of the OCFE procedure are common and differ only in the trial functions used. One uses the Lagrangian functions and adds conditions to make the first derivatives continuous across the element boundaries, and the other uses Hermite polynomials, which automatically have continuous first derivatives between elements. Difficulties in the numerical integration of the resulting system of equations occur with the use of both types of trial functions, and personal preference must then dictate which is to be used. The final equations that need to be integrated after application of the OCFE method in the axial dimension to the reactor equations (radial collocation is performed using simple orthogonal collocation) can be expressed in the form... 

Note that the collocation of a part of the switching region inside the sphere j in the Karplus scheme plays the same role as the polynomial normalization in TsLess. 

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