Orthogonal collocation boundary values

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. 

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. 

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. One simply combines the methods for ordinary differential equations (see Ordinary Differential Equations—Boundary Value Problems ) with the methods for initial-value problems (see Numerical Solution of Ordinary Differential Equations as Initial Value Problems ). Fast Fourier transforms can also be used on regular grids (see Fast Fourier Transform ). 

Villadsen, J., and Stewart, W. E. (1967). Solution of boundary-value problems by orthogonal collocation. Chem. Eng. Sci. 22, 1483-1501. 

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. 

For the nonlinear case, the nonlinear two-point boundary value differential equation(s) for the catalyst pellet can be solved using the same method as used for the axial dispersion model in Section 5.1, i.e., by the orthogonal collocation technique of MATLAB s bvp4c. m boundary value solver. 

The modelling equations (8-12) form two nonlinear boundary value problems which we solve by orthogonal collocation. This... 

Versteeg HK, Malalasekera W (1996) An Introduction to Computational Eluid Dynamics The Finite Volume Method. Longman, Harlow Versteeg HK, Malalasekera W (2007) An Introduction to Computational Fluid Dynamics The Finite Volume Method. Pearson Prentice Hall, Harlow Villadsen JV, Stewart WE (1967) Solution of boundary-value problems by orthogonal collocation. Chem Eng Sci 22 1483-1501... 

The appendices present the parameters and empirical correlations necessary for the models discussed in the book. They also give basic information on the use of the orthogonal collocation technique for the solution of non-linear two-point boundary value differential equations which arise in the modelling of porous catalyst pellets and the estimation of clFectiveness factors. The application of orthogonal collocation techniques to equations resulting from the Fickian type model as well as models based on the more rigorous Stefan-Maxwell equations are presented. 

Non-isothermal ejfectiveness factor The non-isothermal effectiveness factor can be obtained numerically only by integrating the two points boundary value differential equations using different numerical techniques, the most efficient of these techniques is the orthogonal collocation method. 

The pellet mass and heat balances are described by second order differential equations of the two point boundary value type. For this case the reaction is neither too fast nor highly exothermic and therefore the concentration and temperature gradients inside the pellet are not very steep. Therefore the orthogonal collocation method with one internal collocation point was found sufficient to transform the differential equation into a set of algebraic equations which were solved numerically using the bisectional method (Rice,... 

The non-linear two-point boundary value differential equation of the catalyst pellet is best solved using the orthogonal collocation method as described by Elnashaie et al. (1988a). The bulk phase initial value differential equations should be solved using standard integration routines with automatic step size to ensure accuracy (e.g. DGEAR-IMSL library). 

The effectiveness factors at each point along the length of the reactors are calculated for the key components methane and carbon dioxide, using the dusty gas model and simplified models I and II. The catalyst equations resulting from the use of the dusty gas model are complicated two-point boundary value differential equations and are solved by global orthogonal collocation technique (Villadsen and Michelsen, 1978 Kaza and Jackson, 1979). The solution of the catalyst pellet equations of the simplified models 1 and 2 at each point... 

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. 

Table 6.9 presents detailed simulation results for an industrial ammonia converter formed of three beds with interstage cooling between the beds. The simulation results are presented for both the empirical and the diffusion-reaction approaches for computing t]. For the diffusion-reaction approach two techniques are used for the solution of the two point boundary value differential equations, namely the shooting technique and the more efficient orthogonal collocation technique. 

The behaviour of the system is described by equations 7.26-7.30. The non-linear coupled two-point boundary value differential equations are difficult to solve as a part of the maximization procedure due to the excessive computational effort involved. The solid phase equations will therefore be recast into an equivalent set of non-linear algebraic equations using the orthogonal collocation method (Villad-sen and Michelsen, 1978). The application of this method to this problem is explained in Appendix C. 

When the Fickian diffusion model is used many reaction-diffusion problems in porous catalyst pellet can be reduced to a two-point boundary value differential equation of the form of equation B.l. This is not a necessary condition for the application of this simple orthogonal collocation technique. The technique in principle, can be applied to any number of simultaneous two-point boundary value differential equations. 

The two-point boundary value differential equations of the catalyst pellet, are reduced to a set of algebraic equations at a number of points inside and at the surface of the pellet using the efficient orthogonal collocation technique (Villadsen and Stewart, 1967). 

Chapter 8 Approximate Methods for Boundary Value Problems Table Computations Using Orthogonal Collocation Effect of Boundary Resistance... 

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. 

In Chapters 7 and 8, we presented numerical methods for solving ODEs of initial and boundary value type. The method of orthogonal collocation discussed in Chapter 8 can be also used to solve PDEs. For elliptic PDEs with two spatial domains, the orthogonal collocation is applied on both domains to yield a set of algebraic equations, and for parabolic PDEs the collocation method is applied on the spatial domain (domains if there are more than one) resulting in a set of coupled ODEs of initial value type. This set can then be handled by the methods provided in Chapter 7. 

The domain u e(0,l) is now represented discretely by N interior collocation points. Taking the boundary point (u=l) as the (N+l)-th point, we have a total of N+1 interpolation points. According to the orthogonal collocation method, the first and second derivatives at these interpolation points are related to the functional values at all points as given below ... 

Due to the variable gas velocity and the nonlinear rate law the model equations represent a set of coupled nonlinear algebraic and differential equations of boundary value type which must be solved numerically. For this purpose the nonlinear equations are entirely linearized using the cjuasilinearization technique (12) and the linearized differential equations are solved using the orthogonal collocation method based on shifted Legendre polynomials (13). 

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