Collocation Method trial function

The orthogonal collocation technique is a simple numerical method which is easy to program for a computer and which converges rapidly. Therefore it is useful for the solution of many types of second order boundary value problems. This method in its simplest form as presented in this section was developed by Villadsen and Stewart (1967) as a modification of the collocation methods. In collocation methods, trial function expansion coefficients are typically determined by variational principles or by weighted residual methods (Finlayson, 1972). The orthogonal collocation method has the advantage of ease of computation. This method is based on the choice of suitable trial series to represent the solution. The coefficients of trial series are determined by making the residual equation vanish at a set of points called collocation points , in the solution domain. 

The two most common of the methods of weighted residuals are the Galerkin method and collocation. In the Galerkin method, the weighting functions are chosen to be the trial functions, which must be selected as members of a complete set of functions. (A set of functions is complete if any function of a given class can be expanded in terms of the set.) Also according to Finlayson (1972),... 

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. 

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... 

For the solution of sophisticated mathematical models of adsorption cycles including complex multicomponent equilibrium and rate expressions, two numerical methods are popular. These are finite difference methods and orthogonal collocation. The former vary in the manner in which distance variables are discretized, ranging from simple backward difference stage models (akin to the plate theory of chromatography) to more involved schemes exhibiting little numerical dispersion. Collocation methods are often thought to be faster computationally, but oscillations in the polynomial trial function can be a problem. The choice of best method is often the preference of the user. 

This expression is used as a trial-function expansion for T in much the same way as the Lagrange interpolation polynomial is in the polynomial collocation method of Villadsen and Stewart (10,1 1 There are four unknown constants associated with each node, giving a total of 4(n+l)(nH-1) unknowns in the expansion. 

In the historical survey of the spectral methods given by Canute et al [22], it was assumed that Lanczos [101] was the first to reveal that a proper choice of trial functions and distribution of collocation points is crucial to the accuracy of the solution of ordinary differential equations. Villadsen and Stewart [203] developed this method for boundary value problems. The earliest applications of the spectral collocation method to partial differential equations were made for spatially periodic problems by Kreiss and Oliger [94] and Orszag [139]. However, at that time Kreiss and Oliger [94] termed the novel spectral method for the Fourier method while Orszag [139] termed it a pseudospectral method. 

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. 

When using the method of weighted residuals, the most important factor is the choice of the trial function. Important guidelines come from the boundary conditions and the symmetrj of the problem. A further extension of the method is choosing collocation points as roots of orthogonal polynomials. This is known as the method of orthogonal collocation. The basic property of any orthogonal polynomial is... 

There are two common values for the weighting w x ). These are u = 1 and w = —x. Roots for these Jacobi polynomials are presented in Table 8.8 for all three geometries, both values of w and various orders of N. The roots presented in Table 8.8 specify specific collocation points to be used in the method of weighted residuals. The collocation method satisfies the describing differential equation at the collocation points. In order to implement this, we need to evaluate the function and its derivatives at the collocation points. The trial function of equation (8.12.21) can also be written as... 

In other words, the function approximation methods find a solution by assuming a particular type of function, a trial (basis) function, over an element or over the whole domain, which can be polynomial, trigonometric functions, splines, etc. These functions contain unknown parameters that are determined by substituting the trial function into the differential equation and its boundary conditions. In the collocation method, the trial function is forced to satisfy the boundary conditions and to satisfy the differential equation exactly at some discrete points distributed over the range of the independent variable, i.e. the residual is zero at these collocation points. In contrast, in the finite element method, the trial functions are defined over an element, and the elements, are joined together to cover an entire domain. 

The collocation method is one of a general class of approximate methods known as the method of weighted residuals (Ames 1965). The method involves expanding the temperature and concentration in a series 2af(z)Fj(r) of known functions of radius, Pi(r), multiplied by unknown functions of z. The trial functions are substituted into the partial differential equations that are satisfied at discrete radial collocation points, r. This gives a set of ordinary differential equations governing Ui(z). The trial functions are orthogonal polynomials, e.g.r... 

The orthogonal collocation method, which is an extension of the method just described, provides a mechanism for automatically picking the collocation points by making use of orthogonal polynomials. This method chooses the trial functions yj(j ) and y x) to be the linear combination... 

There are numerous variations of the method of weighted residuals relating to the choice of the trial basis functions as well as the weight functions for orthogonalizing the residual. In particular, the method of orthogonal collocation depends upon equating the residual to zero at selected interpolation points in the interval in which the function is defined. 

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