Thomas algorithm

The Thomas algorithm begins by a for ward elimination, row by row starting down from the top row (j=l, the condenser stage), to give the following replacements shown in Fig. 13-50Z . For row T. 

The implicit Crank-Nicholson integration method was used to solve the equation. Radial temperature and concentrations were calculated using the Thomas algorithm (Lapidus 1962, Carnahan et al,1969). This program allowed the use of either ideal or non-ideal gas laws. For cases using real gas assumptions, heat capacity and heat of reactions were made temperature dependent. 

The books by Gelfand (1967), Samarskii and Nikolaev (1989) cover in full details the general theory of linear difference equations. Sometimes the elimination method available for solving various systems of algebraic equations is referred to, in the foreign literature, as Thomas algorithm and this... 

The Gaussian elimination can be used without pivoting because of diagonal dominance (ref. 1). Due to the many zeros the algorithm (sometimes called Thomas algorithm) is very easy to implement ... 

When all of the coefficients are known, this can be solved for the concentrations of component i in every stage. A straightforward method for solving a tridiagonal matrix is known as the Thomas algorithm to which references are made in Sec. 13.10, Basis for Computer Evaluation of Multicomponent Separations Specifications. ... 

Again, Thomas algorithm can be used, but in this case the convergence is proportional to ST2. 

For implicit schemes, we will obtain a system of linear algebraic equations that must be solved. As mentioned in Example 8.1, one-dimensional diffusion problems generate tri-diagonal matrices, that can be solved for using the Thomas algorithm or other fast matrix routines. Equation (8.83) can be written as... 

The solution methodology of the determinants is similar to that of the well-known Thomas algorithm used for the numerical solution of a differential equation with the finite-difference method [50]. An essential difference from the Thomas algorithm is that the first step ofthe algorithm here is a so-called backward process. This means that the calculation of T starts from the last sublayer, that is, from the Mth sublayer ofthe determinant and it is continued down to the 1st sublayer. Thus, the value of Ti is obtained directly, in the fist calculation step. Then, applying the known value of Ti, the value of Pi can be obtained by means of the fist boundary condition at X= 0, namely ... 

In general case Eqs. (4.60) and (4.61) present infinite sets of the five-term (pentadiagonal) recurrence relations with respect to the index l. In certain special cases (t - 0 or a - 0), they reduce to three-term (tridiagonal) recurrence relations. In this section the sweep procedure for solving such relations is described. This method, also known as the Thomas algorithm, is widely used for recurrence relations entailed by the finite-difference approximation in the solution of differential equations (e.g., see Ref. 61). In our case, however, the recurrence relation follows from the exact expansion (4.60) of the distribution function in the basis of orthogonal spherical functions and free of any seal of proximity, inherent to finite-difference method. Moreover, in our case, as explained below, the sweep method provides the numerical representation of the exact solution of the recurrence relations. 

The sets of the pentadiagonal recurrence relations (4.263) and (4.264) truncated at some large enough / - N, are solved with the aid of the generalized Thomas algorithm, as described in Section II.C.2. 

Taylor method, 189-192 Tellerette packing, 437, 485. 500 Thiele-Geddes method, 145, 146. 153 Thomas algorithm. 151, 152, 159 Thorogood tray maldistribution model, 386... 

Tridiagonal systems result from Eq. (10.24a). These can be solved efficiently using the Thomas algorithm, as discussed in the last section of this chapter. 

For tridiagonal matrices, the decomposition of the matrix into a product of a lower and an upper diagonal matrix leads to an efficient algorithm known as the Thomas algorithm. For a system of the form... 

The Thomas algorithm will always converge if the tridiagonal matrix is diagonally dominant. In other words, the matrix is such that... 

A subroutine in FORTRAN code is written below for the Thomas algorithm. 

Up to this point, the treatments have involved reactions for which the discrete form of the reaction-diffusion equations involve only terms in concentration of the species to which the discrete equation applies. That is, if there were two substances involved, O and R as above, then the discrete equation at a point i had terms only in C 0 for species O, and only C R for species R. This made it possible to use the Thomas algorithm to reduce a system like (6.27) to (6.28), treating the two species systems separately. They then get coupled through the boundary conditions. 

When homogeneous reactions take place, it often happens that some of the discrete equations contain terms in concentration for more than the one species, and it is then not generally possible to use the Thomas algorithm to reduce the systems. These systems are said to be coupled. An example will illustrate this situation. 

An interesting special case, mentioned in Chap. 3, is that of the second derivative on four points, u"(4). For arbitrarily (unequally) spaced points, this is a second-order accurate approximation and, as described in Chap. 9, it has some advantages. It allows the use of an efficient extended Thomas algorithm, rather than a pentadiagonal solver or a sparse solver required if... 

The above system, although leading to a quadradiagonal system of equations, can still be solved by a smallish extension of the Thomas algorithm [153]. Consider the last two equations of (8.33) and rewrite them, putting the bulk concentration terms on the right-hand side ... 

Thus far, this looks just like the Thomas algorithm for the tridiagonal system, as described above in Sect. 8.3. Prom here on, however, the processes diverge. We need to keep both substitutions for C N and C N 1 and use them in the third-last equation, which contains both. This process is continued backwards, reducing all equations with four unknowns to new ones with just two unknowns. The expressions resulting from this are the following ... 

Coupled equations are those in which some or all of the dynamic equations have terms in more than one of the variables (concentrations). This leads, upou discretisation, to systems of discrete equations that cannot usually be solved using the Thomas algorithm because, no matter how one orders the concentration vectors, the systems correspond to matrix equations that are... 

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