Numerical methods collocation method

For the solntion of sophisticated mathematical models of adsorption cycles inclnding complex mnlticomponent eqnilibrinm and rate expressions, two nnmerical 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. 

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... 

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 ). 

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). 

In case an analytical solution of Eqs. (6) and (7) is not available, which is normally the case for non-linear isotherms, a solution for the equations with the proper boundary conditions can nevertheless be obtained numerically by the method of orthogonal collocation [38,39]. 

One of the most populax numerical methods for this class of problems is the method of weighted residuals (MWR) (7,8). For a complete discussion of these schemes several good numerical analysis texts are available (9,10,11). Orthogonal collocation on finite elements was used in this work to solve the model as detailed by Witkowski (12). 

Specifically, in Chapter 3 we create a surface for a transcendental function /(a, y) as an elevation matrix whose zero contour, expressed numerically as a two row matrix table of values, solves the nonlinear CSTR bifurcation problem. In Chapter 6 we investigate multi-tray processes via matrix realizations in Chapter 5 we benefit from the least squares matrix solution to find search directions for the collocation method that helps us solve BVPs and so on. Matrices and vectors are everywhere when we compute numerically. That is, after the laws of physics and chemistry and differential equations have helped us find valid models for the physico-chemical processes. 

The aim of this part of the book is to present the main and current numerical techniques that are used in polymer processesing. This chapter presents basic principles, such as error, interpolation and numerical integration, that serve as a foundation to numerical techniques, such as finite differences, finite elements, boundary elements, and radial basis functions collocation methods. 

J. Li and C.S. Chen. Some observations on unsymmetric RBF collocation methods for convection-diffusion problems. Inter. Journal for Numerical Methods in Eng., 57 1085-1094, 2003. 

Numerical Method. Both the isothermal FFB and the adiabatic MAT models are very stiff due to the coke deactivation terms >j. The spline orthogonal collocation technique was used to solve the above models (19). Typically, the distance x was divided into two regions (0collocation points in each region. At the interface between the two regions, both the concentration and the mass flux were taken as continuous. The value of y varied with the degree of stiffness. 

The model was solved by a numerical method based on BAND and orthogonal collocation.21 This method was suitable for solving this model, but in the case of the highest applied potential exhibited oscillatory, fluctuating behavior. Figure 21 shows the typical potential and concentration distributions under medium potential conditions for 2-D model. 

The usual numerical methods for solving integral equations can be classified in two groups the collocation methods and the Galerkin methods. 

Wajge et al. (1997) attempted to develop rigorous PDAE model for packed batch distillation with and without chemical reaction and used finite difference and orthogonal collocation techniques to solve such model. The main purpose of the study was to investigate the efficiencies of the numerical methods employed. The authors observed that the collocation techniques are computationally more efficient compared to the finite difference method, however the order of approximating polynomial needs to be carefully chosen to achieve a right balance between accuracy and efficiency. See the original reference for further details. 

When the gap width between two particles becomes very small, numerical calculations involved in both the bispherical coordinate method and the boundary collocation technique are computationally intensive because the number of terms in the series required to be retained to achieve a desired accuracy increases tremendously. To solve this near-contact motion more effectively and accurately, Loewenberg and Davis [43] developed a lubrication solution for the electrophoretic motion of two spherical particles in near contact along their line of centers with the assumption of infinitely thin ion cloud. The axisymmetric motion of the two particles in near contact can be approximated as the pairwise motion of the spheres in point contact plus a deviation stemming from their relative motion caused by the contact force. The lubrication results agree very well with those obtained from the collocation method. It is shown that near contact electrophoretic interparticle... 

In addition to several empirical correlation s, various numerical approximations have also been presented [12-14]. Even generalized numerical estimation procedures are given, such as the collocation method of Fmlayson [15]. Ibanez [16] and Namjoshi et al. [17] have defined transformations for several forms of kinetics, which can be used to make numerical approximation easier. 

When radial dispersion is included, even the steady state equations are partial differential equations — in the axial and radial space variables. The dispersion model equations can be numerically solved by finite-difference schemes, or more efficiently, by orthogonal collocation methods (14, 15). 

An orthogonal collocation method for elliptic partial differential equations is presented and used to solve the equations resulting from a two-phase two-dimensional description of a packed bed. Comparisons are made between the computational results and experimental results obtained from earlier work. Some qualitative discrimination between rival correlations for the two-phase model parameters is possible on the basis of these comparisons. The validity of the numerical method is shown by applying it to a one-phase packed-bed model for which an analytical solution is available problems arising from a discontinuity in the wall boundary condition and from the semi-infinite domain of the differential operator are discussed. 

A new mathematical model was developed to predict TPA behaviors of hydrocarbons in an adsorber system of honeycomb shape. It was incorporated with additional adsorption model of extended Langmuir-Freundlich equation (ELF). LDFA approximation and external mass transfer coefficient proposed by Ullah, et. al. were used. In addition, rate expression of power law model was employed. The parameters used in the power model were obtained directly from the conversion data of hydrocarbons in adsorber systems. To get numerical solutions for the proposed model, orthogonal collocation method and DVODE package were employed. 

In ref. 167 the preservation of some structure properties of the flow of differential systems by numerical exponentially fitted Runge-Kutta (EFRK) methods is investigated. The sufficient conditions on symplecticity of EFRK methods are presented. A family of symplectic EFRK two-stage methods with order four has been produced. This new method includes the symplectic EFRK method proposed by Van de Vyver and a collocation method at variable nodes that can be considered as the natural collocation extension of the classical RK Gauss method. 

In one of the first articles on this subject [8], the general analytical solution of Eq. (3) was derived. This general solution is easy to find, but it contains infinite series and (integration) constants that depend on the boundary conditions. Those were determined for the central cells of square and triangular arrays, using the boundary collocation method [8]. More recent publications on this subject are based mostly on complete numerical solution using finite-element methods. 

Peric, M., Kessler, R. and Scheuerer, G. (1988), Comparison of finite volume numerical methods with staggered and collocated grids, Comput. Fluids, 16(4), 389-403. 

The problem of decreased catalyst activity due to irreversible thiophene adsorption was solved numerically using an orthogonal collocation method with three internal points. The numerical results were compared with experimental data obtained by... 

A variety of methods can be used to derive numerical solutions of Eq. 10.61. These methods include mainly finite-difference methods and methods of orthogonal collocation on finite elements. We discuss briefly these methods, the properties of the solutions obtained, and some of the problems of numerical analysis encotmtered in the development and use of algorithms for the computation of solutions of Eq. 10.61 [49,50]. 

In the case of a breakthrough curve for a binary mixture (step injection), Liapis and Rippin [32,33] used an orthogonal collocation method to calculate numerical solutions of a kinetic model including axial dispersion, intraparticle diffusion, and surface film diffusion, and assuming constant coefficients of diffusion and... 

---