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Additional comments on the article "Solution of boundary-value problems by orthogonai collocation"

(Received 15 January 1996)

1 understand that the article "Solution of Boundary-Value Problems by Orthogonal Collocation" (Villadsen and Stewart, 1967) is being reprinted by your Journal as a Cita- tion Classic*. This recognition is appreciated.

The Commentary by Villadsen (1981) which accompanied the article expresses the views of the junior author only. My perspective as major author is offered here.

The Commentary implies that the method of orthogonal collocation was a 'chance discovery'. Actually, it did not happen that way. The development of this method was a natural extension of the investigations on variational and weighted residual methods begun by Louis Snyder (Snyder, 1964; Snyder and Stewart, 1966) under my direction. Toward the end of his research, I remarked to Snyder that Galerkin's method would reduce to a simple collocation scheme if the needed integrals were represented as minimum-point opti- mal quadratures, rather than the higher-order Gaussian quadratures he had used in his packed-bed transport compu- tations. Snyder's basis functions were too complicated to be treated this way, but the idea appeared promising for simpler geometries where polynomial basis functions might be used. This idea was saved for another time, and was presented to John Villadsen when he joined me in September 1965. After some initial debate, John accepted the idea when I demon- strated it with the one-point approximations for effectiveness factors that are given in Table 5 of the paper.

In our 1967 paper, we avoided the conventional view of collocation as a weighted residual method with weights of the form 6(x - xO. That view, still prevalent in the literature and in textbooks, gives no idea how to choose good collo- cation points. A more useful approach is to select the collo- cation points by optimizing the remainder term of an inter- polation formula for the residual. This approach was used in the 1967 paper to obtain a collocation analog of Galerkin's method, and continues to be useful in designing collocation grids for various objectives as outlined by Stewart (1984).

Since the 1967 paper, many advances have been made in orthogonal collocation theory, procedures and applications. A few of these will be mentioned here. Finite-element ver- sions of the method have been given by various authors, including Stewart and Sorensen (1972), de Boor and Swartz (1973), and Carey and Finlayson (1975). Improved algo- rithms for computing grid points and weights for interior orthogonal collocation with polynomials have been pro- vided by Villadsen and Michelsen (1978). These develop- ments for continuous media have been followed by corres- ponding methods for finite-dimensional systems, such as staged separation columns (Stewart et al., 1985; Swartz and Stewart, 1987) and particulate catalyst beds (Marr et al., 1993).

*Reprinted in Chem. Engn,q Sci. 50 (24).

Our initial paper of 1967 was not meant to be exhaustive, but rather to introduce our collocation strategy clearly by means of simple examples, It turned out that we could do this very well without going beyond n = 3, but we also gave formulas for computing higher-order tables and solutions. The construction of software for the points and weights was deferred to another time, and was carried out by Villadsen and Michelsen (1978). The citation record suggests that our readers were largely satisfied with these decisions, and that the 1967 paper gave a useful foundation for many applica- tions.

WARREN E. STEWART Department of Chemical Engineering University of Wisconsin Madison W I 53706, U.S.A.

REFERENCES

de Boor, C. and Swartz, B., 1973, Collocation at Gaussian points. SIAM d. Numer. Anal. 10, 582-606.

Carey, C. F. and Finlayson, B. A, 1975, Orthogonal collo- cation on finite elements. Chem. Engng Sci. 30, 587.

Marr, D. F., Gola-Galimidi, A. M. and Stewart, W. E., 1993, Transport modelling of packed-tube reactors--III. Initial data-based functions and application. Chem. Engng Sei. 48, 3945 3950.

Snyder, L. J., 1964, Ph.D. Thesis, University of Wisconsin. Snyder, L. J. and Stewart, W. E., 1966, Velocity and pressure

profiles for Newtonian creeping flow in regular packed beds of spheres, A.I.Ch.E.J. 12, 167-173, 620.

Stewart, W. E. and Sorensen, J. P., 1972, Transient reactor analysis by orthogonal collocation, Fifth European Sympo- sium on Chemical Reaction Engineering, pp. B8-75, C2 8, C2 9. Elsevier, Amsterdam.

Stewart, W. E., 1984, Simulation and estimation by ortho- gonal collocation, 3M Award Lecture. Chem. Engng Ed., 204 212.

Stewart, W. E., Levien, K. L. and Morari, M., 1985, Simula- tion of fractionation by orthogonal collocation. Chem. En(Ing Sci. 40, 409 421.

Swartz, C. L. E. and Stewart, W. E., 1987, A collocation approach to distillation column design. A.I.Ch.E.J. 32, 1832 1838.

Villadsen, J. V. and Stewart, W. E., 1967, Solution of bound- ary-value problems by orthogonal collocation. Chem. Engng Sci. 22, 1483-1502; 23, 1515.

Villadsen, J. V., 1981, Commentary on the foregoing article. Current Contents, 21 Sept.

Villadsen, J. V. and Michelsen, M. L., 1978, Solution of Differential Equation Models by Polynomial Approxima- tion. Prentice-Hall, Englewood Cliffs, NJ.

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