Orthogonal collocation Jacobi polynomial roots

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... 

The table below provides the roots of the Jacobi polynomials used as node points in orthogonal collocation, for some values of N. Values for X = 0 (i = 0) and X = 1 (i = N + 1) (the values are 0 and 1, resp.) are not included. The roots were computed using the subroutine JCOBI, modified from the original of Villadsen and Michelsen [562], discussed in Appendix C, using for a given N the call... 

If these N interior collocation points are chosen as roots of an orthogonal Jacobi polynomial of Vth degree, the method is called the orthogonal collocation method (Villadsen and Michelsen 1978). It is possible to use other orthogo-... 

Since the orthogonal collocation method will require roots of the Jacobi polynomial, we shall need to discuss methods for the computation of zeros of... 

Based on the above assumptions, the model equations are shown in Table 4. The mass balance equations at the pellet and crystal level are based in the double linear driving model equations or bidisperse model[30]. The solution of the set of parabolic partial differential equations showed in Table 4 was performed using the method of lines. The spatial coordinate was discretized using the method of orthogonal collocation in finite elements. For each element 2 internal collocation points were used and the basis polynomial were calculated using the shifted Jacobi polynomials with weighting function W x) = (a = Q,p=G) hat has equidistant roots inside each element [31]. The set of discretized ordinary differential equations are then solved with DASPK solver [32] which is based on backward differentiation formulas. 

The orthogonal collocation approach transforms the set of M nonlinear differential equations (3.6.3-7) into a set of MiVnonlinear algebraic equations in N internal collocation points which are the roots of the two parameter Jacobi polynomial of degree TV and with a and p chosen according to the... 

On each element, i, Np interior collocation points are chosen as the roots of an Npth degree orthogonal polynomial [65]. The first- and second-order differentials with respect to space are then approximated using two matrices, A and B, obtained by solving the Gaussian-Jacobi quadratures [63,65]. For boimdary problems, the endpoints can be included in the calculation of the spatial derivatives. Thus, at the fcth collocation point on the zth element, we have... 

In connection with the solution approximations, the term collocation points is normally associated with the act of assigning the error or residual at given points. If nodal basis functions are chosen for the implementation of a spectral method, the solution function is approximated at the collocation points. The location of the collocation points in the 12-domain is generally selected as the roots of one of the orthogonal polynomials in the Jacobi family [229]. 

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