Collocation

In a first discretization step, we apply a suitable spatial discretization to Schrodinger s equation, e.g., based on pseudospectral collocation [15] or finite element schemes. Prom now on, we consider tjj, T, V and H as denoting the corresponding vector and matrix representations, respectively. The total... 

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). 

Packages exist that use various discretizations in the spatial direction and an integration routine in the time variable. PDECOL uses B-sphnes for the spatial direction and various GEAR methods in time (Ref. 247). PDEPACK and DSS (Ref. 247) use finite differences in the spatial direction and GEARB in time (Ref. 66). REACOL (Ref. 106) uses orthogonal collocation in the radial direction and LSODE in the axial direction, while REACFD uses finite difference in the radial direction both codes are restricted to modeling chemical reactors. 

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. 

Madsen N. K., Sincovec R. E. (1979) PDECOL General Collocation Software for Partial Differential Equations, ACAf Trawx. Math. 5 326-351. 

Arthur M. Squires, Origins of the Fast Fluid Bed Yu Zhiqing, Application Collocation Youchu Li, Hydrodynamics Li Jinghai, Modeling... 

Ascher U., Christiansen J., Russell R. D. (1979) A Collocation Solver for Mixed Order Systems of Boundary Value Problems, Math Comput 33 659-679. 

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. 

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). 

Useful reviews of these basic elements of CFD can be found with Patankar (1980), Abbott and Basco (1989), Shaw (1992), and Ranade (2002). In the meantime, substantial progress has been realized in developing more effective and powerful numerical techniques. Several of them have made it into the common commercial CFD packages. Just as an example, several of the commercial vendors have incorporated the option of collocated grids. A few more important issues should be highlighted here. 

The partial differential equations representing material and energy balances of a reaction in a packed bed are rarely solvable by analytical means, except perhaps when the reaction is of zero or first order. Two examples of derivation of the equations and their analytical solutions are P8.0.1.01 and P8.01.02. In more complex cases analytical, approximations can be made (by "Collocation" or "Perturbation", for instance), but these usually are quite sophisticated to apply. Numerical solutions, on the other hand, are simple in concept and are readily implemented on a computer. Two such methods that are suited to nonlinear kinetics problems will be described. 

This problem can be solved using a combined optimization and constraint model solution strategy (Muske and Edgar, 1998) by converting the differential equations to algebraic constraints using orthogonal collocation or some other model discretization approach. 

The superposition principle allows us to assume just a single scatterer in the view of the radar. The transmitted signal hits this scatterer whose distance (we measure distance and time in the same units) from the (collocated) transmitter and receiver is r. Assume that the scatterer is stationary. The return signal will be a delayed version of the original, delayed by the total round trip time from the radar to the scatterer. Specifically the signal voltage at the antenna of the receiver is... 

The simultaneous solution strategy offers several advantages over the sequential approach. A wide range of constraints may be easily incorporated and the solution of the optimization problem provides useful sensitivity information at little additional cost. On the other hand, the sequential approach is straightforward to implement and also has the advantage of well-developed error control. Error control for numerical integrators (used in the sequential approach) is relatively mature when compared, for example, to that of orthogonal collocation on finite elements (a possible technique for a simultaneous approach). 

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