Collocation points

Index 2 problems three-point collocation (L-stable if many high index elements) ... 

With four-point collocation using gaussian roots (the A-stable case), we fail to find an optimal solution. Because of this, we include an additional constraint on the control profile error (third time derivative of the control profile less than a tolerance, see Russell and Christiansen, 1978). The resulting... 

Use rough and ready methods but do not carry them beyond their point of usefulness. A one-point collocation solution, for example, may not be very accurate, but may give an insight. A sketched phase plane may be qualitatively correct even though its numerical value is way off (see The Phase Plane in Chapter 4). 

Point collocation. The weighting functions are equal to delta functions located at n points, Wj(x) = Sj(x), which is equivalent to forcing the residual to zero at the n points [24]. 

The problem of decrease in catalyst activity due an irreversible adsorption of poison was solved numerically using a single point collocation approximation. The numerical results are compared with experimental data obtained by measuring concentration changes due to thiophene poisoning of Ni/AljO in benzene hydrogenation. 

The finite difference method can be obtained as follows. Define in the domain a number of grid points and replace (approximate) the Laplace equation by a finite difference expression. When a Dirac delta weight function is used at each grid point i, one forces the finite difference trial to be zero at each point. Therefore, the finite difference method (FDM) can be considered as point collocation method. 

Boundary value problems are encountered so frequently in modelling of engineering problems that they deserve special treatment because of their importance. To handle such problems, we have devoted this chapter exclusively to the methods of weighted residual, with special emphasis on orthogonal collocation. The one-point collocation method is often used as the first step to quickly assess the behavior of the system. Other methods can also be used to treat boundary value problems, such as the finite difference method. This technique is considered in Chapter 12, where we use this method to solve boundary value problems and partial differential equations. 

PARSIM applies a 4-point collocation method within every element. Between the elements continuity of the fluxes (mass, energy) or optional continuity of the derivatives is obtained by continuity conditions [7]. 

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. 

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). 

Different techniques are commonly used to solve the diffusion equation (Carslaw and Jaeger, 1959). Analytic solutions can be found by variable separation, Fourier transforms or more conveniently Laplace transforms and other special techniques such as point sources or Green functions. Numerical solutions are calculated for the cases which have no simple analytic solution by finite differences (Mitchell, 1969 Fletcher, 1991), which is the simplest technique to implement, but also finite elements, particularly useful for complicated geometry (Zienkiewicz, 1977), and collocation methods (Finlayson, 1972). 

To overcome these limitations, Vasantharajan and Biegler (1990) propose a new formulation based on the residual function (see Russell and Christiansen, 1978), Since the ODE residual equations are satisfied only at the collocation points, a straightforward way to compute an error estimate... 

Index 1 problems two-point orthogonal collocation (4-stable) ... 

Ascher, U and Bader, G., Stability of collocation at Gaussian points," SIAM J. Numer. Anal. 23(2), 412-422 (1986). 

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]

This technique then forces the residual to be zero at N specified collocation points. As N increases, the residual is zero at more and more points and presumably approaches zero everywhere. 

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. 

The first step in the solution procedure is discretization in the radial dimension, which involves writing the three-dimensional differential equations as an enlarged set of two-dimensional equations at the radial collocation points with the assumed profile identically satisfying the radial boundary conditions. An examination of experimental measurements (Valstar et al., 1975) and typical radial profiles in packed beds (Finlayson, 1971) indicates that radial temperature profiles can be represented adequately by a quadratic function of radial position. The quadratic representation is preferable to one of higher order since only one interior collocation point is then necessary,6 thus not increasing the dimensionality of the system. The assumed radial temperature profile for either the gas or solid is of the form... 

Application of the procedure, presented subsequently, of satisfying the boundary conditions with the assumed profile leads to the coefficients dt(9, Q for the profiles in terms of the temperatures at the collocation point, within the... 

Along with two boundary points, there are three collocation points. 

Although additional radial collocation points increase the dimensionality of the resulting model, they may be necessary to accurately express the radial concentration profiles. Preliminary analysis in this section considers only one interior radial concentration collocation point, although a detailed analysis of this assumption is presented in Section VI,E. 

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... 

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