Partial elimination algorithm

For this aim, Lindborg et al [119] used a coupled solver in combination with the partial elimination algorithm (PEA) proposed by Spalding [177]. The combined PEA-coupled solver used in their work is outlined in the following. 

To deal with the strong coupling between the phasic momentum equations, the partial elimination algorithm (PEA)-method proposed by Spalding [18, 19] is frequently used. 

Karki KC, Patankar SV (2004) Application of the partial elimination algorithm for solving the coupled energy equations in porous media. Numer Heat Transf Part A 45 539-549... 

The most widely used algorithm is the systematic elimination or Gaussian elimination by partial pivoting. The success of this method is due to its stability, i.e. the algorithm produces small residuals r = Ax — b (x being the numerical solution of the system), despite round-off errors introduced by the computer during computations. The concept of stability of a numerical algorithm will be discussed in more detail in Sect. 4.5. 

Consider flow over a flat plate. The computation is started by assuming that uy = u,y, at the leading edge and v, = 0. The value of vy is needed in the explicit algorithm to move on to the i+1 level. It is not required to specify the initial values of vy in the formal mathematical formulation of the partial differential equation. A suitable initial distribution for vy can be obtained by using the continuity equation to eliminate du/5x from the x-momentum equation. For a laminar, incompressible flow, this means that... 

Initially, the algorithm considers each individual reaction step as a partial mechanism. Then, one intermediate after another are examined, and the set of partial mechanisms is modified so that the intermediate does not appear in the net stoichiometry the modification of mechanisms is carried out in a way that preserves the correct direction of irreversible reaction steps. By processing all intermediates in this way, a set of overall mechanisms is constructed. This final set of mechanisms might include duplicate mechanisms and even indirect ones these can be easily discarded. Similar action must be taken in the procedure of H S. Mavrovouniotis (1992) discusses procedures for eliminating such redundant mechanisms in the end, or even preventing their construction. 

The algorithm operates on a set of intermediate species N, the set of terminal species Nj-, and a set of partial mechanisms M, which is iteratively modified. Other information maintained by the algorithm during its operation is arranged in a convenient format in Fig. 1, which also explains how the entries are initialized. The algorithm proceeds from this setup by successively eliminating each intermediate species from the system. In order to eliminate an intermediate, we consider all the mechanisms whose reactions involve the intermediate species at hand. We can create combinations of two mechanisms at a time, with combination coefficients such that the intermediate vanishes from the reaction. The new row for a combination of mechanisms has elements (left portion of Fig. 1) and kj ( ght portion of Fig. 1) that are simply the linear combinations of the respective elements of the old mechanisms. To permit the best choice of intermediate to be made, the setup of the algorithm (Fig. 1) lists, below... 

Fig. 4. Application of the algorithm on the ammonia mechanism, after the elimination of At and 4, which yielded two new partial mechanisms, m,i and m, and eliminated mj, m, m, and mj. The number of combinations has been recomputed for each remaining intermediate species. The intermediate N,I (fl ) will be eliminated next. [Reprinted with permission from Mavrovouniotis, M. L., and Stephanopoulos, G. Synthesis of reaction mechanisms consisting of reversible and irreversible steps 1. A synthesis approach in the context of simple examples . Ind. Eng. Chem. Res. 31, 1625-1637, (1992). Copyright 1992 American Chemical Society.]...

The Rete algorithm stores not just exact matches but also partial matches this avoids reevaluating the complete set of facts when the rule base is changed. Additionally, it uses a decision-node-sharing technique, which eliminates certain redundancies. 

ALGORITHM 2. FORWARD ELIMINATION WITH UNSCALED PARTIAL PIVOTING. 

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